1. Introduction
Stateoftheart technological information processing happens mainly within the digital realm. Numerical values are quantized, and calculations are performed in discretetime computational cycles. In contrast, the information carried by the physical world surrounding any technological device is analog and continuous. Twelve years after Alan Turing formalized the notion of digital computing [
1], Claude Shannon established the theoretical foundation of sampling and interpolation theory [
2]. Since then, scientists and engineers have extended Turing’s and Shannon’s theories and refined the relevant hardware, making the analog world increasingly accessible to digital information processing.
As follows from Shannon’s sampling theorem, suitable (infinite) interpolation series uniquely restore any bandlimited continuoustime signal, provided that the signal’s energy is finite and the samples are taken at least at the Nyquist rate. In its purest form, this result is present in signal processing and communications technology in the context of analog–digital/digital–analog conversion. In principle, however, any computational discretization method, such as finiteelement algorithms, employs a similar paradigm: A set of discrete points, sufficiently dense for the inbetween to become negligible, represents some continuous, physical object (c.f. [
3] for a recent example in which the physical “object” is an electromagnetic field).
Another recent concept in the domain of digital information processing is known as digital twinning. According to the formalization established in [
4], digital twinning commonly involves a physical entity to be twinned, a machinereadable description (in some machinereadable language) that represents the entity virtually on an appropriate hardware platform, and an interaction between the entity and its description through measurement and control. In this context, the machinereadable description is the entity’s digital twin. More generally, if the particular type of computing hardware is not specified, we refer to the entity’s virtual representation as a virtual twin. Furthermore, the term “entity” indicates that the virtual twin’s physical counterpart does not necessarily have to be an actual object. In theory, any abstract formation—such as, for example, an entire communication network [
5], or, as above, an electromagnetic field—qualifies for digital twinning, provided we can characterize it by a suitable mathematical model. Despite the different contexts, digital twinning resembles traditional Shannon Sampling and Interpolation (SSI) in some aspects. Both approaches represent a physical entity (in the context of SSI, a bandlimited signal) using digital data, digitalizing the relevant analog information through a sequential measurement process. However, classical SSI is geared towards completely restoring the physical entity, while digital twinning primarily aims to recover the entity’s relevant properties. Within the employed mathematical model, relevant properties usually take the form of a (mathematical) function or relation with a particular interpretation on the practical level, such as the position of a material object at a given time or the total energy contained in an electromagnetic field.
Originally associated primarily with Industry 4.0 [
6], digital twinning is attracting significant interest in many areas of modern technology. As part of the internet’s anticipated evolution towards a unified metaverse, even more facets of the physical environment will connect to virtual space. In particular, information processing will increasingly incorporate the interaction between human multimodalities (human senses) and the digital domain, c.f. [
7]. In order to make human senses experienceable, the computational infrastructure will have to coordinate, process, and distribute the relevant information in real time. This requirement imposes engineers with unprecedented technological challenges regarding optimization, control, and decisionmaking. In this regard, research and development advocates digital twinning as one of several critical enablers. The novel technological applications that researchers envision in the context of digital twinning are just as ambitious. Medical research, for example, considers applications such as diseasetrajectory estimation, optimization of medicalcare timing, identification of biomarkers or elucidation of drug mechanisms, and patienttailored prediction of treatment effects, employing digital twins of, e.g., a patient’s immune system [
8].
Given the potential for hazardous impacts of future digitaltwinning applications on sensitive aspects of human wellbeing, the need to follow strict specifications on privacy, integrity, reliability, and safety is manifest. The upcoming 6G industry standard for communication technologies, which incorporates large parts of the technological infrastructure for the metaverse and other digitaltwinning applications, summarizes such requirements by the term trustworthiness [
9]. Depending on the potential hazards of an application, the physical entity’s relevant properties must be reliably recoverable from the entity’s digital twin. In practice, technological systems for critical applications must undergo technology assessment, which evaluates the implementation with regard to criteria of provable performance. When expressed in mathematical terms, e.g., by a margin of error, the recovered property must almost surely meet, such criteria entail “sufficient” and “insufficient” ways of representing a physical entity in virtual space. That is, the employed machinereadable language must satisfy specific structural characteristics, such that the relevant properties can be reliably computed from any of the entity’s machinereadable descriptions. We summarize this observation in terms of the following fundamental problem statement, which we aim to elaborate on throughout this article.
Given an application that requires the processing of analog information, find a sufficient way to represent the information on the chosen hardware platform.

The problem statement refers to general hardware platforms. As previously indicated, future virtualtwinning applications will not necessarily be limited to digital technology, c.f. [
4]. However, in the scope of this article, we will only discuss traditional digital computing. Hence, in the problem statement, we may replace “the chosen hardware platform” with “digital hardware” in the context of the subsequent sections.
So far, our discussion on the fundamental problem statement and the associated concepts has been abstract, without a clear picture of how they translate to actual engineering problems. Throughout the subsequent sections, we aim to draw a precise picture of machinereadable languages, relevant properties, and proper representations for signalprocessing and communicationsengineering applications that involve traditional SSI. Aside from the conceptual similarities between digital twinning and SSI we discussed above, SSI has relevant direct applications in digital twinning. In the context of general virtual twinning, ref. [
10] discussed an actual implementation of such an application, c.f.
Figure 1.
Recall that traditional SSI aims at restoring the physical entity (i.e., a bandlimited signal) entirely. The relevant analytic result is known as (generalized) Plancherel–Pólya Theorem, c.f.
Section 2 and
Figure 2: The bandlimited signal uniquely determines the corresponding sequence of samples, and vice versa.
Accordingly, we expect that any property of the bandlimited signal should be recoverable from the sequence of sampling values. At this point, Turing’s theory of digital computing enters the stage: A priori, the (generalized) Plancherel–Pólya Theorem is a purely analytic result. For it to hold on the algorithmic level, effectiveness in the sense of computable analysis is required. In this context, we will analyze two machinereadable languages emerging from the (generalized) Plancherel–Pólya Theorem for their structural properties. We employ the theory of Turing machines and effective analysis, classifying our results in terms of digital twinning and the article’s fundamental problem statement. Particularly, we provide formal definitions of the terms machinereadable languages and machinereadable descriptions, and discuss formal examples of relevant properties. After the mathematical part of the article, we provide a brief subsumption and interpretation of our results, together with some prospects of how they affect nearfuture digital informationprocessing technology.
The remainder of the article is structured as follows. In
Section 2, we provide some mathematical background on SSI, introducing the signal spaces
${\mathcal{l}}_{0}^{\infty},{\mathcal{l}}_{0}^{1},{\mathcal{B}}_{0,\pi}^{\infty},$ and
${\mathcal{B}}_{\pi}^{1}$, and formally establishing the (generalized) Plancherel–Pólya Theorem in terms of the Banachspace operators
${\mathit{S}}_{\u2605}:\mathcal{B}(\u2605)\to \mathcal{l}(\u2605)$ and
${\mathit{T}}_{\u2605}:={\mathit{S}}_{\u2605}^{1}$. Applying the theory of Turing computability and effective analysis, we continue to develop a framework of machinereadable languages for
${\mathcal{B}}_{0,\pi}^{\infty}$ and
${\mathcal{B}}_{\pi}^{1}$. This framework formalizes the traditional theory of digital signal processing for communications engineering based on a mathematically rigorous notion of computability. Particularly, we define the machinereadable languages
${\mathfrak{X}}_{1}$ and
${\mathfrak{X}}_{\infty}$, which mirror the implicit quasistandard in digital signal processing, and the machinereadable languages
${\mathfrak{F}}_{1}$ and
${\mathfrak{F}}_{\infty}$, which take the relevant signal’s continuoustime behavior into account. In
Section 3, we provide (for didactic purposes) a mathematical model of digitaltwinning systems such as the one shown in
Figure 1, marking the Banachspace norms
${\parallel \xb7\parallel}_{\infty}$ and
${\parallel \xb7\parallel}_{1}$ as a relevant property of signals
$\mathit{f}\in {\mathcal{B}}_{0,\pi}^{\infty}$,
$\mathit{f}\in {\mathcal{B}}_{\pi}^{1}$, respectively. Guided by the exemplary application case, we establish our main results: The (generalized) Plancherel–Pólya Theorem does
not hold true on the algorithmic level. Depending on which of the established machinereadable languages we choose, we either can or cannot compute
${\parallel \mathit{f}\parallel}_{\infty}$,
${\parallel \mathit{f}\parallel}_{1}$, respectively, despite all languages determining the relevant signals uniquely in the (analytic) sense of the (generalized) Plancherel–Pólya Theorem. Finally,
Section 4 discusses several other signal properties our theory can analyze and closes the article by interpreting our results as indicated above.
2. Materials and Methods
In the following, we provide a concise introduction to the mathematics of sampling and interpolation, which are primarily based on the theory of Banach spaces and linear operators. To this end, we introduce the Banach spaces
${\mathcal{l}}_{0}^{\infty},\phantom{\rule{3.33333pt}{0ex}}{\mathcal{l}}^{p},\phantom{\rule{3.33333pt}{0ex}}{\mathcal{B}}_{0,\sigma}^{\infty},$ and
${\mathcal{B}}_{\sigma}^{p}$,
$1\le p<\infty $,
$0<\sigma <\infty $. Commonly,
${\mathcal{B}}_{0,\sigma}^{\infty}$ and
${\mathcal{B}}_{\sigma}^{p}$ are referred to as Bernstein spaces. For a comprehensive introduction, we refer the reader to [
11,
12].
By
${\mathcal{l}}_{0}^{\infty}$, we denote the set of all complexvalued sequences indexed by
$\mathbb{Z}$ that vanish at infinity. That is, we have
for all
$\mathit{x}={\left(\mathit{x}\left[k\right]\right)}_{k\in \mathbb{Z}}\in {\mathcal{l}}_{0}^{\infty}$. Equipped with the uniform norm
${\parallel \mathit{x}\parallel}_{\infty}:={sup}_{k\in \mathbb{Z}}\left\mathit{x}\left[k\right]\right$ the set
${\mathcal{l}}_{0}^{\infty}$ becomes a Banach space. Further, by
${\mathcal{l}}^{p}$,
$1\le p<\infty $, we denote the Banach space of
pthpowersummable sequences with the
pnorm
A function $\mathit{f}:\mathbb{C}\to \mathbb{C},z\mapsto \mathit{f}\left(z\right)$ is called entire if it is welldefined and holomorphic on all of $\mathbb{C}$. For entire functions that are (essentially) bounded on the real line, we define the essentialsupremum norm ${\parallel \mathit{f}\parallel}_{\infty}:={ess\; sup}_{t\in \mathbb{R}}\left\mathit{f}\left(t\right)\right$. The space ${\mathcal{B}}_{0,\sigma}^{\infty},0\sigma \infty ,\phantom{\rule{3.33333pt}{0ex}}$ consists of all entire functions $\mathit{f}$ that satisfy the following conditions:
Equipped with the essential supremum norm, the space
${\mathcal{B}}_{0,\sigma}^{\infty}$ becomes a Banach space. Further, the Banach spaces
${\mathcal{B}}_{\sigma}^{p}$,
$1\le p<\infty $, consists of all functions in
${\mathcal{B}}_{0,\sigma}^{\infty}$ that are
pthpower integrable on the real line, equipped with the
pnorm
Pure mathematics studies all of the spaces
${\mathcal{l}}_{0}^{\infty},\phantom{\rule{3.33333pt}{0ex}}{\mathcal{l}}^{p},\phantom{\rule{3.33333pt}{0ex}}{\mathcal{B}}_{0,\sigma}^{\infty},$ and
${\mathcal{B}}_{\sigma}^{p}$,
$1\le p<\infty $,
$0<\sigma <\infty $. In contrast, only the spaces
${\mathcal{l}}^{1},\phantom{\rule{3.33333pt}{0ex}}{\mathcal{l}}^{2},\phantom{\rule{3.33333pt}{0ex}}{\mathcal{l}}_{0}^{\infty}$ and
${\mathcal{B}}_{\sigma}^{1},\phantom{\rule{3.33333pt}{0ex}}{\mathcal{B}}_{\sigma}^{2},\phantom{\rule{3.33333pt}{0ex}}{\mathcal{B}}_{0,\sigma}^{\infty}$ occur frequently throughout signal processing and communications engineering, the arguably most “wellknown” ones being
${\mathcal{l}}^{2}$ and
${\mathcal{B}}_{\sigma}^{2}$. They consist of discrete and, respectively, bandlimited continuoustime signals with finite energy and form the mathematical basis for the seminal results in SSI, established before the relevant theory was extended to general Bernstein spaces. Fourier analysis provides a bijective isometry between
${\mathcal{l}}^{2}$ and
${\mathcal{B}}_{\sigma}^{2}$: defining
${\mathit{x}}_{\mathit{f}}\left[k\right]:=\mathit{f}(k\pi /\sigma ),\phantom{\rule{3.33333pt}{0ex}}k\in \mathbb{Z}$, we have
for all
$\mathit{f}\in {\mathcal{B}}_{\sigma}^{2}$, where
$\mathcal{F}\mathit{f}$ denotes the Fourier Transform of
$\mathit{f}$ on the real line. Through the definition of the sincfunction,
and the linearity of the Fourier Transform, the isometry provides Shannon’s original sampling theorem for the spaces
${\mathcal{l}}^{2}$ and
${\mathcal{B}}_{\sigma}^{2}$, we have
Since
$\mathcal{F}\mathit{f}$ is zero outside the interval
$[\sigma ,\sigma ]$,
$\mathit{f}$ is called bandlimited with bandwidth
$\sigma $. The spaces
${\mathcal{B}}_{\sigma}^{2}$,
$0<\sigma <\infty $, thus correspond to the traditional notion of bandlimited signals. Through the definition of exponential types (Point 2 of the requirements above), this notion is generalized to a significantly larger class of functions. For
$1<p<\infty $ arbitrary, the Plancherel–Pólya theorem (Theorem 3, p. 152 in [
11]) provides a nontrivial extension to the Shannon’s sampling theorem.
Theorem 1 (Plancherel–Pólya).
Let $1<p<\infty $. For all sequences $\mathit{x}\in {\mathcal{l}}^{p}$, $1<p<\infty $, there exists a unique function $\mathit{f}\in {\mathcal{B}}_{\sigma}^{p}$, such thatis satisfied. In particular, $\mathit{f}$ is the unique solution to the interpolation problem $\mathit{f}(k\pi /\sigma )=\mathit{x}\left[k\right]$, $k\in \mathbb{Z}$. Conversely, for all signals $\mathit{f}\in {\mathcal{B}}_{\sigma}^{p}$, $1<p<\infty $, the sequence ${\left(\mathit{f}(k\pi /\sigma )\right)}_{k\in \mathbb{Z}}$ belongs to ${\mathcal{l}}^{p}$ and there exist constants ${C}_{L}\left(p\right)>0$ and ${C}_{R}\left(p\right)>0$, independent of $\mathit{f}$, such thatholds true. For the spaces
${\mathcal{l}}^{1},\phantom{\rule{3.33333pt}{0ex}}{\mathcal{B}}_{\sigma}^{1}$ and
${\mathcal{l}}_{0}^{\infty},\phantom{\rule{3.33333pt}{0ex}}{\mathcal{B}}_{0,\sigma}^{\infty}$,
$0<\sigma <\infty $, Theorem 1 does
not hold to its full extent. The (generalized) Plancherel–Pólya Theorem, which we will subsequently refer to as generalized Shannon equivalence, provides the following: For all
$\mathit{f}\in {\mathcal{B}}_{\pi}^{1}$,
$\mathit{f}\in {\mathcal{B}}_{0,\pi}^{\infty}$, respectively, and
${\left(\mathit{f}(k\pi /\sigma )\right)}_{k\in \mathbb{Z}}={\mathit{x}}_{\mathit{f}}$, we have
$\mathit{f}\equiv 0$ if and only if we also have
${\mathit{x}}_{\mathit{f}}\equiv 0$. Furthermore, interpolation on the basis of sincfunctions provides uniform convergence on all compact subsets of
$\mathbb{C}$, i.e., for all
$C>0$, we have
However, there exist sequences
$\mathit{x}\in {\mathcal{l}}^{1}$,
$\mathit{x}\in {\mathcal{l}}_{0}^{\infty}$, respectively, such that
no function
$\mathit{f}\in {\mathcal{B}}_{\sigma}^{1}$,
$\mathit{f}\in {\mathcal{B}}_{0,\sigma}^{\infty}$, respectively, satisfies the interpolation condition
$\mathit{x}\left[k\right]=\mathit{f}\left(k\right)$ for all
$k\in \mathbb{Z}$. For
${\mathcal{l}}^{1}$, the Kroneckerdelta family
${\mathit{\delta}}_{i}\in {\mathcal{l}}^{1}$,
$i\in \mathbb{Z}$, defined by
forms a simple example of such sequences. For an example in
${\mathcal{l}}_{0}^{\infty}$, see (
A2). In other words, the inclusions
$\{{\mathit{x}}_{\mathit{f}}={\left(\mathit{f}\left(k\right)\right)}_{k\in \mathbb{Z}}:\mathit{f}\in {\mathcal{B}}_{\pi}^{1}\}\subset {\mathcal{l}}^{1}$ and
$\{{\mathit{x}}_{\mathit{f}}={\left(\mathit{f}\left(k\right)\right)}_{k\in \mathbb{Z}}:\mathit{f}\in {\mathcal{B}}_{0,\sigma}^{\infty}\}\subset {\mathcal{l}}_{0}^{\infty}$ are proper. For further details regarding the generalized Shannon equivalence, we refer to [
11,
12] (Lecture 21, pp. 155–162; Chapter 6, pp. 48–66).
The results established in the present article hold true for all spaces
${\mathcal{B}}_{\sigma}^{1}$,
${\mathcal{B}}_{0,\sigma}^{\infty}$,
$0<\sigma <\infty $. In particular, the specific choice of
$\sigma $ is irrelevant. Therefore, without loss of generality, we will restrict ourselves to the case of
$\sigma =\pi $ in the following, and denote
Further, most definitions and results hold analogously for both
${\mathcal{B}}_{\pi}^{1}$ and
${\mathcal{B}}_{0,\pi}^{\infty}$. For the sake of brevity, we will thus employ the symbol ‘★’ as a placeholder that may be (uniformly) replaced by ‘1’ or ‘
∞’ within every appropriate scope (such as a definition, a lemma, a theorem, or a proof), and write
with some abuse of notation.
For notational convenience, we introduce the sampling operator
${\mathit{S}}_{\u2605}:\mathcal{B}(\u2605)\to \mathcal{l}(\u2605),$$\mathit{f}\mapsto {\mathit{S}}_{\u2605}\mathit{f}:={\left(\mathit{f}\left(k\right)\right)}_{k\in \mathbb{Z}}$ and its inverse
${\mathit{S}}_{\u2605}^{1}=:{\mathit{T}}_{\u2605}$, which we refer to as interpolation operator. Subsequently, we will formalize the notion of computability for the spaces
$\mathcal{B}(\u2605)$ and
$\mathcal{l}(\u2605)$. Then, the (informal) question of whether the generalized Shannon equivalence holds true on the algorithmic level corresponds to the (formal) question of whether the operators
${\mathit{S}}_{\u2605}$ and
${\mathit{T}}_{\u2605}$ are computable in the chosen machinereadable language, which we will address in
Section 3. Observe that the sampling operator
${\mathit{S}}_{\u2605}$ is bounded and injective. Thus, the interpolation operator
${\mathit{T}}_{\u2605}$ is welldefined on a linear subspace
of
$\mathcal{B}(\u2605)$. However,
${\mathit{T}}_{\u2605}$ is unbounded, and the subspace
$\mathrm{dom}\left({\mathit{T}}_{\u2605}\right)$ is
not closed, c.f. (
A1) and (
A4). Therefore, the set
$\mathrm{dom}\left({\mathit{T}}_{\u2605}\right)$ is
not a Banach space itself, which is essential in deriving the main results of our work.
Having established the analytic theory of sampling and interpolation, we will now turn to the formalization of its computable variant. To this end, we provide a concise introduction to the theory of Turing machines [
1],
$\mu $recursive functions [
13], and computable analysis [
14,
15,
16,
17]. Although mature topics in the field of computer science, they have not yet received much attention within the signalprocessing community.
Turing machines form an abstract mathematical model for digital computing. In fact, the widely accepted Church–Turing Thesis implies that they form a definitive and universal model of digital computing, i.e., any mathematical problem can (in principle) be solved through a realworld digital computer if and only if it can theoretically be solved by a Turing machine. Hence, if a certain algorithmic problem cannot be solved on a Turing machine, it can definitely not be solved on an actual digital hardware. The algorithms a Turing machine can compute is equivalent to the class of
$\mu $recursive functions, c.f. [
18] for the proof of equivalence.
By $\mathbb{N}$, we denote the set of natural numbers including zero. Throughout this article, it will occasionally be necessary to exclude zero from $\mathbb{N}$ to obtain a meaningful mathematical expression. As the reader may easily detect such a necessity from the relevant context, we avoid indicating them explicitly by a distinguished notation.
We call a mapping naturalnumber function if it is of the form
$g:{\mathbb{N}}^{n}\supseteq \to \mathbb{N}$,
$n\in \mathbb{N}$, where the symbol “
$\supseteq \to $” denotes partiality. That is, we have
$\mathrm{dom}\left(g\right)\subseteq {\mathbb{N}}^{n}$. A partial naturalnumber function is called total if the inclusion is improper, i.e., if we have
$\mathrm{dom}\left(g\right)\subseteq {\mathbb{N}}^{n}$. Then, the set of
$\mu $recursive functions
$\mathcal{U}$ consists of all those naturalnumber functions that we can construct from the sucessor function, constant functions, and projection functions through application of composition, primitive recursion, and unbounded search (Definition 2.1, p. 8, Definition 2.2, p. 10 in [
14]). By
$\mathcal{U}\left(n\right)$,
$n\in \mathbb{N}$, we denote the set of
$\mu $recursive functions in
n arguments, where
$\mathcal{U}\left(0\right)$ can be understood as the set
$\mathbb{N}$ itself, i.e., constant functions in zero arguments. Accordingly, we have
$\mathcal{U}\left(0\right)\cup \mathcal{U}\left(1\right)\cup \mathcal{U}\left(2\right)\cup \dots =\mathcal{U}$.
Observe that, generally speaking, $\mu $recursive functions are partial. When Turing machines are modeled as actual statebased machines that perform computations in sequential processing steps, the domain of the corresponding $\mu $recursive function equals the set of inputs for which the Turing machine halts its computation in finite time. A set $\mathsf{\Omega}\subseteq \mathbb{N}$ is called recursively enumerable if it is either empty or the domain of a $\mu $recursive function. Consequently, a set $\mathsf{\Omega}\subseteq \mathbb{N}$ is recursively enumerable if and only if it is either empty or the range of a total $\mu $recursive function. Furthermore, if $\mathsf{\Omega}$ is recursively enumerable, there exists a (total) $\mu $recursive function ${g}_{\mathsf{\Omega}}:{\mathbb{N}}^{2}\to \{0,1\}$ that satisfies the following for all $n\in \mathbb{N}$:
There exists a number $m\in \mathbb{N}$ such that ${g}_{\mathsf{\Omega}}(n,m)=1$ is satisfied if and only if $n\in \mathsf{\Omega}$ holds true;
If ${g}_{\mathsf{\Omega}}(n,m)=1$ holds true for a number $m\in \mathbb{N}$, then ${g}_{\mathsf{\Omega}}(n,k)=1$ holds true for all $k\in \mathbb{N}$ that satisfy $k>m$.
We call such a function a runtime function for
$\mathsf{\Omega}$. A set
$\mathsf{\Omega}\subset \mathbb{N}$ is called recursive if both
$\mathsf{\Omega}$ and
$\mathbb{N}\setminus \mathsf{\Omega}$ are recursively enumerable which, in turn, holds true if and only if the indicator function
of
$\mathsf{\Omega}$ is a (total)
$\mu $recursive function.
Alan Turing introduced the concept of computable real numbers in [
1]. Our definition of computable real numbers, and, subsequently, computable complex numbers, is based on computable sequences of rational numbers (p. 14 in [
15]).
Definition 1. A sequence of rational numbers, ${\left({r}_{m}\right)}_{m\in \mathbb{N}}$, is called a computable sequence of rational numbers if there exist (total) μrecursive functions $g,{h}_{1},{h}_{2}:\mathbb{N}\to \mathbb{N}$ such thatis satisfied for all $m\in \mathbb{N}$. Analogously, for $n\in \mathbb{N}$, an nfold computable sequence of rational numbers is defined through (total) μrecursive functions $g,{h}_{1},{h}_{2}:{\mathbb{N}}^{n}\to \mathbb{N}$ in n arguments. Definition 2. A sequence of complex numbers, ${\left({s}_{m}\right)}_{m\in \mathbb{N}}$, is called a computable sequence of rationalcomplex numbers if there exist a pair $({\left({r}_{1,m}\right)}_{m\in \mathbb{N}},{\left({r}_{2,m}\right)}_{m\in \mathbb{N}})$ of computable sequences of rational numbers such that ${s}_{m}={r}_{1,m}+j{r}_{2,m}$ is satisfied for all $m\in \mathbb{N}$. Analogously, for $n\in \mathbb{N}$, an nfold computable sequence of rationalcomplex numbers is defined through a pair of nfold computable sequences of rational numbers.
Definition 3. A real number, x, is called computable if there exist a computable sequence of rational numbers ${\left({r}_{n}\right)}_{n\in \mathbb{N}}$ and a (total) μrecursive function $\xi :\mathbb{N}\to \mathbb{N}$ such that $x{r}_{n}<{2}^{M}$ holds true for all $n,M\in \mathbb{N}$ that satisfy $n\ge \xi \left(M\right)$. For a triple $(x,{\left({r}_{n}\right)}_{n\in \mathbb{N}},\xi )$ of this kind, we write ${[{\left({r}_{n}\right)}_{n\in \mathbb{N}},\xi ]}_{\mathfrak{R}}=x$. Further, we denote the set of computable real numbers by ${\mathbb{R}}_{\mu}$.
Definition 4. A complex number, z, is called computable if there exist a computable sequence of rationalcomplex numbers ${\left({s}_{n}\right)}_{n\in \mathbb{N}}$ and a (total) μrecursive function $\xi :\mathbb{N}\to \mathbb{N}$ such that $z{s}_{n}<{2}^{M}$ holds true for all $n,M\in \mathbb{N}$ that satisfy $n\ge \xi \left(M\right)$. For a triple $(z,{\left({s}_{n}\right)}_{n\in \mathbb{N}},\xi )$ of this kind, we write ${[{\left({s}_{n}\right)}_{n\in \mathbb{N}},\xi ]}_{\mathfrak{C}}=z$. Further, we denote the set of computable complex numbers by ${\mathbb{C}}_{\mu}$.
In Definitions 3 and 4, the $\mu $recursive function $\xi $ provides a computable way to control the approximation error $x{r}_{n}$, $n\in \mathbb{N}$, $z{s}_{n}$, $n\in \mathbb{N}$, respectively. In this case, the convergence of ${\left({r}_{n}\right)}_{n}$ and ${\left({s}_{n}\right)}_{n\in \mathbb{N}}$ to x and z, respectively, is referred to as effective, and the function $\xi $ is called a corresponding effective modulus of convergence.
In the following, we will extend the established concepts of computability to the spaces
$\mathcal{l}(\u2605)$ and
$\mathcal{B}(\u2605)$. To this end, we analogously write
with the relevant formal definitions following below. Further, we employ an enumeration
$\nu :\mathbb{N}\to \mathbb{Z},\phantom{\rule{3.33333pt}{0ex}}n\mapsto \nu \left(n\right)$ of the integers
$\mathbb{Z}$, defined through
Definition 5. A sequence, $\mathit{x}\in \mathcal{l}(\u2605)$, is called computable in $\mathcal{l}(\u2605)$ if there exist a computable double sequence of rationalcomplex numbers ${\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}}$ and a (total) μrecursive function $\xi :\mathbb{N}\to \mathbb{N}$, such thatis satisfied for all $M\in \mathbb{N}$. We denote the set of all such sequences by $\mathcal{C}\mathcal{l}(\u2605)$. Further, if we have $\mathit{f}={\mathit{T}}_{\u2605}\mathit{x}$ for some $\mathit{f}\in \mathcal{CB}(\u2605)$ (c.f. Definition 6), we write ${[{\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}},\xi ]}_{\mathfrak{X}}^{\u2605}=\mathit{f}$ for the triple $(\mathit{f},{\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}},\xi )$. Definition 6. A function, $\mathit{f}\in \mathcal{B}(\u2605)$, is called computable in $\mathcal{B}(\u2605)$ if there exist a computable double sequence of rationalcomplex numbers ${\left({s}_{n,m}^{\prime}\right)}_{n,m\in \mathbb{N}}$ and a (total) μrecursive function ${\xi}^{\prime}:\mathbb{N}\to \mathbb{N}$, such thatis satisfied for all $M\in \mathbb{N}$. We denote the set of all such functions by $\mathcal{CB}(\u2605)$. Further, we write ${[{\left({s}_{n,m}^{\prime}\right)}_{n,m\in \mathbb{N}},{\xi}^{\prime}]}_{\mathfrak{F}}^{\u2605}=\mathit{f}$ for the triple $(\mathit{f},{\left({s}_{n,m}^{\prime}\right)}_{n,m\in \mathbb{N}},{\xi}^{\prime})$. Observe that, generally speaking, linear combinations of sincfunctions are
not elements of
${\mathcal{B}}_{\pi}^{1}$. However, specific linear combinations of sincfunctions with rationalcomplex coefficients are, in fact, elements of
${\mathcal{B}}_{\pi}^{1}$, and the set of these linear combinations is dense in
${\mathcal{B}}_{\pi}^{1}$. For details, we refer the reader to Appendix B, p. 6363f in [
19] (upon minor adjustments, the proof presented in the reference holds true for the restricted case of rationalcomplex coefficients).
A sequence
$\mathit{x}\in \mathcal{C}\mathcal{l}(\u2605)$ is called an elementary computable if there exists a rationalcomplex
$(2L+1)$tuple
${\left({z}_{k}\right)}_{k\in \mathcal{I}},\mathcal{I}:=\{0,\dots ,2L\}$, such that we have
Analogously, a function $\mathit{f}\in \mathcal{CB}(\u2605)$ is called an elementary computable if there exists an elementary computable sequence $\mathit{x}\in \mathcal{C}\mathcal{l}(\u2605)$ such that we have $\mathit{f}={\mathit{T}}_{\u2605}\mathit{x}$. Hence, elementary computable functions are exactly those functions that we can represented by a finite interpolation series with rationalcomplex coefficients ${\left({z}_{k}\right)}_{k\in \mathcal{I}}$ in the sense of traditional SSI.
For
$\mathit{x}\in \mathcal{C}\mathcal{l}(\u2605)$, a computable double sequence of rationalcomplex numbers
${\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}}$, and a (total)
$\mu $recursive function
$\xi :\mathbb{N}\to \mathbb{N}$, let (
1) be satisfied. Then, for all
$M\in \mathbb{N}$, the sequence
is an elementary computable with coefficients
${z}_{n},\phantom{\rule{3.33333pt}{0ex}}n\in \mathbb{N}$. The sequence of sequences
${\left({\mathit{x}}_{M}\right)}_{M\in \mathbb{N}}$ is called a computable sequence of elementary computable sequences. Further, for all
$M\in \mathbb{N}$, we have
$\parallel \mathit{x}{\mathit{x}}_{M}{\parallel}_{\u2605}<{2}^{M}$. In general, a sequence
$\mathit{x}\in \mathcal{l}(\u2605)$ is computable in
$\mathcal{l}(\u2605)$ if and only if there exists a computable sequence of elementary computable sequences
${\left({\mathit{x}}_{M}\right)}_{M\in \mathbb{N}}$ that converges effectively towards
$\mathit{x}$, with respect to
${\parallel \xb7\parallel}_{\u2605}$ and a suitable effective modulus of convergence.
For
$\mathit{f}\in \mathcal{CB}(\u2605)$, a computable double sequence of rationalcomplex numbers
${\left({s}_{n,m}^{\prime}\right)}_{n,m\in \mathbb{N}}$, and a (total)
$\mu $recursive function
${\xi}^{\prime}:\mathbb{N}\to \mathbb{N}$, let (
2) be satisfied. Analogously, for all
$M\in \mathbb{N}$, the function
is an elementary computable with coefficients
${z}_{n}^{\prime},\phantom{\rule{3.33333pt}{0ex}}n\in \mathbb{N}$. The sequence of functions
${\left({\mathit{f}}_{M}\right)}_{M\in \mathbb{N}}$ is called a computable sequence of elementary computable functions. Further, for all
$M\in \mathbb{N}$, we have
$\parallel \mathit{f}{\mathit{f}}_{M}{\parallel}_{\u2605}<{2}^{M}$. In general, a function
$\mathit{f}\in \mathcal{B}(\u2605)$ is computable in
$\mathcal{B}(\u2605)$ if and only if there exists a computable sequence of elementary computable functions
${\left({\mathit{f}}_{M}\right)}_{M\in \mathbb{N}}$ that converges effectively towards
$\mathit{f}$, with respect to
${\parallel \xb7\parallel}_{\u2605}$ and a suitable effective modulus of convergence.
Throughout the remainder of this article, we will prove the following: There exist
$\mathit{f}={\mathit{T}}_{\u2605}\mathit{x}\in \mathcal{CB}(\u2605)$ and
${\left({\mathit{x}}_{M}\right)}_{M\in \mathbb{N}}$ as above, such that we have
${lim}_{M\to \infty}\parallel \mathit{f}{\mathit{T}}_{\u2605}{\mathit{x}}_{M}\parallel \ne 0$, despite
${\left({\mathit{x}}_{M}\right)}_{M\in \mathbb{N}}$ converging effectively towards
$\mathit{x}$. In other words, even if
${\left({\mathit{x}}_{M}\right)}_{M\in \mathbb{N}}$ converges effectively towards
$\mathit{x}={\mathit{S}}_{\u2605}\mathit{f}$, the computable sequence of elementary computable functions
${\left({\mathit{T}}_{\u2605}{\mathit{x}}_{M}\right)}_{M\in \mathbb{N}}$ does
not necessarily converge towards
${\mathit{T}}_{\u2605}\mathit{x}$.
Section 3 will discuss the consequences of this observation extensively. To this end, we will now establish two preliminary lemmas, the proofs of which we provide in
Appendix A.
Lemma 1. Let $\mathsf{\Omega}\subset \mathbb{N}$ be a recursively enumerable set. There exists a (not
necessarily computable) sequence ${\left({\mathit{f}}_{m}\right)}_{m\in \mathbb{N}}$ of elementary computable functions in ${\mathcal{B}}_{0,\pi}^{\infty}$ that satisfies the following: the sequence ${\left({\mathit{x}}_{m}\right)}_{m\in \mathbb{N}}={\left({\mathit{S}}_{\infty}{\mathit{f}}_{m}\right)}_{m\in \mathbb{N}}$ is a computable sequence of sequences in $\mathcal{C}{\mathcal{l}}_{0}^{\infty}$, and, for all $m\in \mathbb{N}$, we have Lemma 2. Let $\mathsf{\Omega}\subset \mathbb{N}$ be a recursively enumerable set. There exists a (not
necessarily computable) sequence ${\left({\mathit{f}}_{m}\right)}_{m\in \mathbb{N}}$ of elementary computable functions in ${\mathcal{B}}_{\pi}^{1}$ that satisfies the following: the sequence ${\left({\mathit{x}}_{m}\right)}_{m\in \mathbb{N}}={\left({\mathit{S}}_{1}{\mathit{f}}_{m}\right)}_{m\in \mathbb{N}}$ is a computable sequence of sequences in $\mathcal{C}{\mathcal{l}}^{1}$, and for all $m\in \mathbb{N}$, we have For the general definition of computable sequences of abstract objects (such as those used in Lemmas 1 and 2), see below.
Recall this article’s fundamental problem statement from
Section 1: Given an application that requires the processing of analog information, find a sufficient way to represent the information on digital hardware. In order to provide a solution, we require a general formalization of how to represent information on Turing machines, employing the natural numbers as their “atomic” numerical object. The authors advise readers that this formalization is somewhat abstract, but necessary for a mathematically rigorous treatment. After establishing the formalization in its abstract form, we will put it in the context of SSI, allowing for a less cumbersome and more intuitive treatment.
For two $\mu $recursive functions ${g}_{1},{g}_{2}:{\mathbb{N}}^{n}\supseteq \to \mathbb{N}$, we write ${g}_{1}={g}_{2}$ if $\mathrm{dom}\left({g}_{1}\right)=\mathrm{dom}\left({g}_{2}\right)$ is satisfied, and for all $({m}_{1},\dots ,{m}_{n})\in \mathrm{dom}\left({g}_{1}\right)$, we have ${g}_{1}({m}_{1},\dots ,{m}_{n})={g}_{2}({m}_{1},\dots ,{m}_{n})$. Furthermore, for ease of notation, we will make use of anonymous mappings. In general, an explicit definition of a mapping is of the form $G:\mathcal{A}\supseteq \to \mathcal{B}$, $a\mapsto G\left(a\right):=$ “$\mathrm{EXPR}\left(a\right)$”, where $\mathcal{A}$ and $\mathcal{B}$ are arbitrary sets, and “$\mathrm{EXPR}\left(a\right)$” is the term defining G, such as, for example, “$a+a$”, “${a}^{2}$”, “$ln\left(a\right)$”, and so on. If the context determines $\mathcal{A}$ and $\mathcal{B}$ without ambiguity, but does not require providing an explicit definition, we simply write $(a\mapsto $ “$\mathrm{EXPR}\left(a\right)$”) to denote the respective mapping.
The formalization of how to represent information on Turing machines builds upon the existence of universal
$\mu $recursive functions
$U:\mathbb{N}\times \mathbb{N}\supseteq \to \mathbb{N}$, an arbitrary one of which we fix for the remainder of this article. Then, for every
$\mu $recursive function
$g:{\mathbb{N}}^{n}\supseteq \to \mathbb{N},\phantom{\rule{3.33333pt}{0ex}}n\in \mathbb{N}$, the following holds true: there exists a “program”
$M\in \mathbb{N}$ such that the function
${U}_{M}^{n}:{\mathbb{N}}^{n}\supseteq \to \mathbb{N},\phantom{\rule{3.33333pt}{0ex}}({m}_{1},\dots ,{m}_{n})\mapsto {U}_{M}^{n}({m}_{1},\dots ,{m}_{n})$, defined through
satisfies
$g={U}_{M}^{n}$. Accordingly, for all
$n\in \mathbb{N}$, the universal
$\mu $recursive function
U provides an equivalence relation on
$\mathbb{N}$: for
$M,K\in \mathbb{N}$, we have
$M\equiv K$ if
${U}_{M}^{n}={U}_{K}^{n}$. Evidently, the equivalencerelation’s quotient set
$\left\{\right\{K\in \mathbb{N}:K\equiv M\}:M\in \mathbb{N}\}$ is in onetoone correspondence with the the set
$\mathcal{U}\left(n\right)$. We denote
which hints towards the usual notation for quotient sets in the context of equivalence relations: we have
${\left[M\right]}_{\mathfrak{U}}^{n}={\left[K\right]}_{\mathfrak{U}}^{n}$ if and only if
$M\equiv K$. We call the set
a machinereadable language for the set
$\mathcal{U}\left(n\right)$, and
$M\in \mathbb{N}$ is a machinereadable description of
${U}_{M}^{n}\in \mathcal{U}\left(n\right)$. Furthermore, for
$n,m\in \mathbb{N}$, a mapping
$G:\mathfrak{U}\left(m\right)\supseteq \to \mathfrak{U}\left(n\right)$ is called computable if there exists a
$\mu $recursive function
$g:\mathbb{N}\supseteq \to \mathbb{N}$ such that, for all
$M\in \mathbb{N}$ with
${U}_{M}^{m}\in \mathrm{dom}\left(G\right)$, we have
For
${m}_{1},\dots ,{m}_{k}\in \mathbb{N}$, we can extend the principle to analogously mappings of the form
$G:\mathfrak{U}\left({m}_{1}\right)\times \dots \times \mathfrak{U}\left({m}_{k}\right)\supseteq \to \mathfrak{U}\left(n\right)$. Observe that arithmetic operations on
$\mu $recursive functions, such as
and so on, as well as composition, primitive recursion, and unbounded search (see above), when seen as operations on
$\mu $recursive functions, provide computable mappings in the sense of the definition above. Throughout the remainder of the article, we will make implicit use of the computability of mappings of the form
$G:\mathfrak{U}\left({m}_{1}\right)\times \dots \times \mathfrak{U}\left({m}_{k}\right)\supseteq \to \mathfrak{U}\left(n\right)$ on many occasions. For details, we refer to the SMNTheorem, c.f. Theorem 3.5, p. 16 in [
14].
Following the principle of
$\mathfrak{U}\left(n\right)$,
$n\in \mathbb{N}$, we can now define general machinereadable languages and general computable mappings through an inductive scheme. For all
$n\in \mathbb{N}$, the set of
ntuples of natural numbers,
${\mathbb{N}}^{n}$, is an atomic machinereadable language, and for all
$n,k\in \mathbb{N}$, a mapping
$G:{\mathbb{N}}^{n}\supseteq \to {\mathbb{N}}^{k},\phantom{\rule{3.33333pt}{0ex}}({m}_{1},\dots ,{m}_{n})\mapsto G({m}_{1},\dots ,{m}_{n})$ is called atomically computable if there exist functions
${g}_{1},\dots ,{g}_{k}\in \mathcal{U}\left(n\right)$ such that
$\mathrm{dom}\left(G\right)$ is a (possibly improper) subset of
$\mathrm{dom}\left({g}_{1}\right)\cap ...\cap \mathrm{dom}\left({g}_{k}\right)$, and we have
for all
$({m}_{1},\dots ,{m}_{n})\in \mathrm{dom}\left(G\right)$. A (
nonatomic) machinereadable language for the (abstract) set
$\mathcal{A}$ is of the form
where
${\mathsf{\Lambda}}_{\mathfrak{A}}$ is a machinereadable language and
${[\xb7]}_{\mathfrak{A}}:{\mathsf{\Lambda}}_{\mathfrak{A}}\supseteq \to \mathcal{A}$ is a partial surjective mapping. Further,
$\lambda \in \mathrm{dom}\left({[\xb7]}_{\mathfrak{A}}\right)$ is called a machinereadable description of
${\left[\lambda \right]}_{\mathfrak{A}}\in \mathcal{A}$. Again, “
${[\xb7]}_{\mathfrak{A}}$” hints towards the usual notation for quotient sets: for
${\lambda}_{1},{\lambda}_{2}\in \mathrm{dom}\left({[\xb7]}_{\mathfrak{A}}\right)$, we have
${\lambda}_{1}\equiv {\lambda}_{2}$ if and only if
${\left[{\lambda}_{1}\right]}_{\mathfrak{A}}={\left[{\lambda}_{2}\right]}_{\mathfrak{A}}$, i.e.,
${\lambda}_{1}$ and
${\lambda}_{2}$ are two machinereadable descriptions of the same abstract object. However, since
${[\xb7]}_{\mathfrak{A}}$ is generally partial, so is the induced equivalence relation. Finally, mapping
${G}_{1}:\mathfrak{A}\supseteq \to \mathfrak{B}$, where
$\mathfrak{A}$ and
$\mathfrak{B}$ are machinereadable languages, is called (
nonatomically) computable if there exists a computable mapping
${G}_{2}:{\mathsf{\Lambda}}_{\mathfrak{A}}\supseteq \to {\mathsf{\Lambda}}_{\mathfrak{B}}$ such that, for all
$\lambda \in \mathrm{dom}\left({[\xb7]}_{\mathfrak{A}}\right)$ with
$({\left[\lambda \right]}_{\mathfrak{A}},{\mathsf{\Lambda}}_{\mathfrak{A}},{[\xb7]}_{\mathfrak{A}})\in \mathrm{dom}\left({G}_{1}\right)$, we have
Unless defined otherwise, a sequence
${\left({\mathfrak{a}}_{n}\right)}_{n\in \mathbb{N}}$ of elements of
$\mathfrak{A}$ is called computable if there exists a (total) computable mapping
$(n\mapsto {\mathfrak{a}}_{n})$. Observe that if
$\mathfrak{A}$ and
$\mathfrak{B}$ are machinereadable languages for arbitrary abstract sets
$\mathcal{A}$ and
$\mathcal{B}$, respectively,
${G}_{1}$ naturally induces a mapping
${G}_{1}:\mathcal{A}\supseteq \to \mathcal{B}$ according to
and vice versa. If, according to the specific context, there is no danger of ambiguity, we will not distinguish between
${G}_{1}:\mathfrak{A}\supseteq \to \mathfrak{B}$ and
${G}_{1}:\mathcal{A}\supseteq \to \mathcal{B}$.
In essence, a machinereadable language is a formal specification of how to represent abstract information on digital hardware, such that we can (in principle) trace this specification down to the level of tuples of natural numbers and fundamental operations thereon. Upon fixing a suitable
$\mu $recursive pairing function (Chapter 1.4, p. 12 in [
16]), i.e., a bijective mapping
with
${\langle \xb7\rangle}_{1},{\langle \xb7\rangle}_{2}\in \mathcal{U}\left(1\right)$, every machinereadable language
$\mathfrak{A}$ exhibits a canonical numbering, i.e., computable surjective mapping
${\phi}_{\mathfrak{A}}:\mathbb{N}\supseteq \to \mathfrak{A}$, defined in an inductive manner:
If
$\mathfrak{A}$ is an atomic machinereadable language, i.e., we have
$\mathfrak{A}={\mathbb{N}}^{n}$ for some number
$n\in \mathbb{N}$, we define
where
${\langle m\rangle}_{2}^{n}$ denotes the
nfold successive application of
${\langle \xb7\rangle}_{2}$ to
m;
For a
nonatomic machinereadable language
$\mathfrak{A}=\{(a,\mathfrak{B},{[\xb7]}_{\mathfrak{A}}):a\in \mathcal{A}\}$ and a general machinereadable language
$\mathfrak{B}$ with canonical numbering
${\phi}_{\mathfrak{B}}:\mathbb{N}\supseteq \to \mathfrak{B}$, we define
with
$\mathrm{dom}\left({\phi}_{\mathfrak{A}}\right):=\{m\in \mathrm{dom}\left({\phi}_{\mathfrak{B}}\right):{\phi}_{\mathfrak{B}}\left(m\right)\in \mathrm{dom}\left({[\xb7]}_{\mathfrak{A}}\right)\}$ accordingly.
Among other things, and together with the relevant pairing function $({\langle \xb7\rangle}_{1},{\langle \xb7\rangle}_{2})$, the canonical numbering facilitates the definition of machinereadable languages for tuples of the form $({a}_{1},{a}_{2})\in {\mathcal{A}}_{1}\times {\mathcal{A}}_{2}$, provided we have already defined machinereadable languages for the abstract sets ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$.
Referring to this article’s fundamental problem statement, if we want to represent abstract information on a digital machine in a sufficient way, we need to specify a machinereadable language for the relevant abstract set, and then investigate the language’s structural properties. Albeit rarely explicit, this principle is used throughout the literature of computable analysis. In the context of Banach spaces, it is strongly related to the definitions of computability structures (Chapter 2.1, p. 80ff in [
15]). Further, any canonical numbering
${\phi}_{\mathfrak{A}}$ as defined above is essentially a numbering in the sense of a concept that is fundamental in computability theory (Chapter 1.4, p. 12 in [
16]). As indicated before, formal approaches of this form are necessary for a mathematically rigorous theory of computable analysis. Yet, they are somewhat cumbersome in use. In Definition 3, for example, we have implicitly introduced a machinereadable language for the set of computable real numbers by defining the relation
${[{\left({r}_{n}\right)}_{n\in \mathbb{N}},\xi ]}_{\mathfrak{R}}=x$. This convention is an abuse of notation regarding the justestablished formalization of machinereadable languages. Strictly speaking, we first have to define machinereadable languages for the set of triples
$(g,{h}_{1},{h}_{2}):g,{h}_{1},{h}_{2}\in \mathcal{U}\left(1\right)$. Then, we have to define a machinereadable language for the set of computable sequences of rational numbers. Finally, we have to define a machinereadable language for the set of pairs
$({\left({r}_{n}\right)}_{n\in \mathbb{N}},\xi )$ as above, based on which we can define the machinereadable language for the set of computable real numbers in the sense of Definition 3. Intuitively, on the other hand, it is evident from Definition 3 that we describe a computable real number by a suitable pair
$({\left({r}_{n}\right)}_{n\in \mathbb{N}},\xi )$, and we can implement computable mappings on computable real numbers by applying
$\mu $recursive functions to the “programs” (with respect to the universal
$\mu $recursive function
U) of the underlying quadruple
$(g,{h}_{1},{h}_{2},\xi )$. Keeping the formal definition in mind, we will, with some abuse of nomenclature and notation, employ the following conventions for mathematical ease:
A standard description of computable real number x consists of a pair $({\left({r}_{n}\right)}_{n\in \mathbb{N}},\xi )$ that characterizes x in the sense of Definition 3, and we write $x={[{\left({r}_{n}\right)}_{n\in \mathbb{N}},\xi ]}_{\mathfrak{R}}$. We denote the associated standard machinereadable language by $\mathfrak{R}$;
A standard description of computable complex number z consists of a pair $({\left({s}_{n}\right)}_{n\in \mathbb{N}},\xi )$ that characterizes z in the sense of Definition 4, and we write $z={[{\left({s}_{n}\right)}_{n\in \mathbb{N}},\xi ]}_{\mathfrak{C}}$. We denote the associated standard machinereadable language by $\mathfrak{C}$.
For the set $\mathcal{CB}(\u2605)$, the same convention applies. However, based on the generalized form of Theorem 1, we have two different machinereadable languages available:
A discretetime description of $\mathit{f}\in \mathcal{CB}(\u2605)$ consists of a pair $({\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}},\xi )$ that characterizes $\mathit{f}$ in the sense of Definition 5, and we write $\mathit{f}={[{\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}},\xi ]}_{\mathfrak{X}}^{\u2605}$. We denote the associated discretetime machinereadable language by ${\mathfrak{X}}_{\u2605}$;
A continuoustime description of $\mathit{f}\in \mathcal{CB}(\u2605)$ consists of a pair $({\left({s}_{n,m}^{\prime}\right)}_{n,m\in \mathbb{N}},{\xi}^{\prime})$ that characterizes $\mathit{f}$ in the sense of Definition 6, and we write $\mathit{f}={[{\left({s}_{n,m}^{\prime}\right)}_{n,m\in \mathbb{N}},{\xi}^{\prime}]}_{\mathfrak{F}}^{\u2605}$. We denote the associated continuoustime machinereadable language by ${\mathfrak{F}}_{\u2605}$.
Returning to the abstract theory of machinereadable language once more, we can consider the general case of an abstract set
$\mathcal{A}$ with more than one associated machinereadable language: in fact, even though any machinereadable language has necessarily only countably many elements, there exists an uncountable number of machinereadable languages for any nontrivial abstract set. Consider machinereadable languages
${\mathfrak{A}}_{1}$ and
${\mathfrak{A}}_{2}$ for the set
$\mathcal{A}$, and define the corresponding identity mapping
We can now define a partial quasiorder on the class of machinereadable languages for the set
$\mathcal{A}$ as follows:
Intuitively, if ${\mathfrak{A}}_{1}\u2ab0{\mathfrak{A}}_{2}$, we can find an algorithm that transforms any description ${\lambda}_{1}$ of any object $a\in \mathfrak{A}$ in the language ${\mathfrak{A}}_{1}$ into a description ${\lambda}_{2}$ of the same object in the language ${\mathfrak{A}}_{2}$. For any computable mapping $G:{\mathfrak{A}}_{2}\supseteq \to \mathfrak{B}$, where $\mathfrak{B}$ is an arbitrary machinereadable language, the composition $G\circ {\mathrm{Id}}_{1,2}$ is computable as well. Thus, any computational problem we can solve by means of the language ${\mathfrak{A}}_{2}$, we can also solve by means of the language ${\mathfrak{A}}_{1}$. In view of this article’s fundamental problem statement, we can distinguish four cases:
If ${\mathfrak{A}}_{1}\succ {\mathfrak{A}}_{2}$, descriptions in the language ${\mathfrak{A}}_{1}$ contain more information than descriptions in the language ${\mathfrak{A}}_{2}$;
If ${\mathfrak{A}}_{1}\prec {\mathfrak{A}}_{2}$, descriptions in the language ${\mathfrak{A}}_{1}$ contain less information than descriptions in the language ${\mathfrak{A}}_{2}$;
If ${\mathfrak{A}}_{1}\simeq {\mathfrak{A}}_{2}$, descriptions in the language ${\mathfrak{A}}_{1}$ contain the same information as descriptions in the language ${\mathfrak{A}}_{2}$;
If neither of the previous cases holds, descriptions in the language ${\mathfrak{A}}_{1}$ contain different information than descriptions in the language ${\mathfrak{A}}_{2}$.
The remainder of this article will address the relationship between the languages
${[\xb7]}_{\mathfrak{X}}^{\u2605}$ and
${[\xb7]}_{\mathfrak{F}}^{\u2605}$. The generalized Shannon equivalence motivates the engineering paradigm that processing any (bandlimited) analog signal can be entirely moved to the discretetime domain, provided that we have a sequence of sampling values with sufficient quantization accuracy available. However, as stated before, the generalized Shannon equivalence is an abstract analytical concept, formalized in terms of the Banachspace operators
${\mathit{S}}_{\u2605}$ and
${\mathit{T}}_{\u2605}$. Previously in this section, we have stated that the (informal) question of whether the generalized Shannon equivalence also holds true on the algorithmic level corresponds to the (formal) question of whether the operators
${\mathit{S}}_{\u2605}$ and
${\mathit{T}}_{\u2605}$ are computable (in the machinereadable language under consideration). In
Section 3, we will establish that the computability of
${\mathit{S}}_{\u2605}$ and
${\mathit{T}}_{\u2605}$ is essentially a rephrasing of the relationship between
${[\xb7]}_{\mathfrak{X}}^{\u2605}$ and
${[\xb7]}_{\mathfrak{F}}^{\u2605}$.
Before concluding the present section, observe that we have
${\mathcal{CB}}_{\pi}^{1}\subset {\mathcal{CB}}_{0,\pi}^{\infty}$. Further, for all
$\mathit{x}\in {\mathcal{l}}^{1}$, we have
${\parallel \mathit{x}\parallel}_{1}\ge {\parallel \mathit{x}\parallel}_{\infty}$, and for all
$\mathit{f}\in {\mathcal{B}}_{\pi}^{1}$, we have
${\parallel \mathit{f}\parallel}_{1}\ge {\parallel \mathit{f}\parallel}_{\infty}$, implying
for the restrictions
${\mathfrak{X}}_{\infty}{\mathcal{CB}}_{\pi}^{1}$ and
${\mathfrak{F}}_{\infty}{\mathcal{CB}}_{\pi}^{1}$ of
${\mathfrak{X}}_{\infty}$ and
${\mathfrak{F}}_{\infty}$ to elements of
${\mathcal{CB}}_{\pi}^{1}$. We will briefly return to these inequalities in
Section 3.
3. Results
In the scope of our theory, digital twinning involves an abstract set
$\mathcal{A}$ and a corresponding machinereadable language
$\mathfrak{A}$, both of which are results of how the relevant technological system is modeled from an engineering perspective. When the system is operated, it gives rise to a (
not necessarily computable) sequence
${\left({\mathfrak{a}}_{t}\right)}_{t\in \mathbb{N}}$ of elements of
$\mathfrak{A}$, which ultimately emerges from a successive measurement process. In each instance
$t\in \mathbb{N}$,
${\mathfrak{a}}_{t}$ should, in one way or another, correspond to the instantaneous state of the physical technological system. The details of this correspondence are, again, a result of modeling. For example, referring to
Figure 1, denote the robot’s instantaneous position at time
$t\in \mathbb{N}$ by
${\overrightarrow{\varkappa}}_{t}\in {\mathbb{R}}^{2}$. Further, denote by
a suitable machinereadable language corresponding to a countable set
${\tilde{\mathbb{R}}}^{2}\subset {\mathbb{R}}^{2}$ of our choice, and define
Then, we might require that for a suitable computable mapping
$G:\mathfrak{A}\supseteq \to \mathfrak{K},\mathfrak{a}\mapsto G\left(\mathfrak{a}\right)$ and some
$\u03f5>0$, we have
for all
$t\in \mathbb{N}$. Intuitively, we consider the robot’s instantaneous position a relevant property, and thus want to be able to recover it from the robot’s virtual twin up to a certain error at any instance in time.
Recall that
Figure 1 (Right) illustrates the instantaneous discretetime impulse response of the transmission channel between the robot and the receiving end. Commonly, wireless transmission channels are assumed linear, i.e., their behavior is determined entirely by the relevant impulse response. Further, wireless communication systems are commonly restricted to a specific frequency range, i.e., the transmission is bandlimited. Accordingly, for all
$t\in \mathbb{N}$, the instantaneous physical transmission channel uniquely corresponds to a bandlimited signal
${\mathit{f}}_{\mathrm{ph},t}\in {\mathcal{B}}_{\sigma}^{1}$,
$0<\sigma <\infty $. Without loss of generality, we again consider
$\sigma =\pi $ in the following. For
$(\mathit{f},{\mathsf{\Lambda}}_{\mathfrak{X}},{[\xb7]}_{\mathfrak{X}}^{1})\in {\mathfrak{X}}_{1}$ and
$(\mathit{f},{\mathsf{\Lambda}}_{\mathfrak{F}},{[\xb7]}_{\mathfrak{F}}^{1})\in {\mathfrak{F}}_{1}$, define
Then, for some
${\u03f5}_{\mathrm{ph}}>0$ and suitable computable mappings
${G}_{\mathrm{dt}}:\mathfrak{A}\supseteq \to {\mathfrak{X}}_{1},\mathfrak{a}\mapsto {G}_{\mathrm{dt}}\left(\mathfrak{a}\right)$ and
${G}_{\mathrm{ct}}:\mathfrak{A}\supseteq \to {\mathfrak{F}}_{1},\mathfrak{a}\mapsto {G}_{\mathrm{ct}}\left(\mathfrak{a}\right)$, we might again require
respectively, to hold for all
$t\in \mathbb{N}$. Observe that (
4) is a purely analytical relation, describing the requirement that
${G}_{\mathrm{dt}}\left({\mathfrak{a}}_{t}\right)$ and
${G}_{\mathrm{ct}}\left({\mathfrak{a}}_{t}\right)$ at each time
$t\in \mathbb{N}$ provide a sufficiently accurate approximation to the instantaneous properties of the physical channel. The “true” sequence
${\left({\mathit{f}}_{\mathrm{ph},t}\right)}_{t\in \mathbb{N}}$ of channel characteristics does
not need to consist of computable components. In the design process of a digitaltwin system, such as that shown in
Figure 1, the responsible engineer has to prove—based on mathematical modeling—that, during the system’s operation, the sequence
${\left({G}_{\mathrm{dt}}\left({\mathfrak{a}}_{t}\right)\right)}_{t\in \mathbb{N}}$,
${\left({G}_{\mathrm{ct}}\left({\mathfrak{a}}_{t}\right)\right)}_{t\in \mathbb{N}}$, respectively, will satisfy (
4). Recall that, by definition, both
${\mathfrak{X}}_{1}$ and
${\mathfrak{F}}_{1}$ are machinereadable languages for the set
${\mathcal{CB}}_{\pi}^{1}$. Hence, according to
Section 2,
${G}_{\mathrm{dt}}$ and
${G}_{\mathrm{ct}}$ each induce a mapping
${G}_{\mathrm{dt}}:\mathcal{A}\supseteq \to {\mathcal{CB}}_{\pi}^{1}$,
${G}_{\mathrm{ct}}:\mathcal{A}\supseteq \to {\mathcal{CB}}_{\pi}^{1}$, respectively. In the following, we assume
${G}_{\mathrm{dt}}$ and
${G}_{\mathrm{ct}}$ to be the same, and we denote
${G}_{\mathrm{dt}}\left({a}_{t}\right)={G}_{\mathrm{ct}}\left({a}_{t}\right)=:{\mathit{f}}_{t}$. The system’s design process will then include the choice between implementing
${\left({\mathit{f}}_{t}\right)}_{t\in \mathbb{N}}$ using
${\left({G}_{\mathrm{dt}}\left({\mathfrak{a}}_{t}\right)\right)}_{t\in \mathbb{N}}$—i.e., approximating
${\left({\mathit{f}}_{\mathrm{ph}}\right)}_{t\in \mathbb{N}}$ through discretetime descriptions—or using
${\left({G}_{\mathrm{ct}}\left({\mathfrak{a}}_{t}\right)\right)}_{t\in \mathbb{N}}$—i.e., approximating
${\left({\mathit{f}}_{\mathrm{ph}}\right)}_{t\in \mathbb{N}}$ through continuoustime descriptions—the implications of which we will analyze subsequently.
Motivated by the generalized Shannon equivalence, the textbook approach considers discretetime descriptions of bandlimited signals. As indicated before, the evident advantage of this paradigm consists of computational “convenience”. In a simplified manner, the standard (abstract) engineering model—that is, without considering questions of computability, yet—for the wireless communication (sub)system from
Figure 1 may look as follows. At time
$t\in \mathbb{N}$, the robot aims to transmit one of
$M\in \mathbb{N}$ messages to the receiving end, for which he employs an encoding scheme
${\mathcal{E}}_{t}:\{1,\dots ,M\}\to \mathrm{dom}\left({\mathit{T}}_{\infty}\right),m\mapsto {\mathcal{E}}_{t}\left(m\right):={\mathit{y}}_{t,m}$. For reasons of simplicity, we summarize processes such as encoding (in the sense of information theory), channel precoding, modulation, and pulse shaping in this step. Setting
${\mathit{x}}_{\mathrm{ph},t}:={\mathit{S}}_{1}{\mathit{f}}_{\mathrm{ph},t}$, the signal at the receiving end is of the form
where
$\mathit{w}\in {\mathcal{l}}_{0}^{\infty}$ is a sequence of additive noiselike disturbances, and
${\mathit{x}}_{\mathrm{ph},t}\ast {\mathit{y}}_{t,m}$ denotes the convolution of
${\mathit{x}}_{\mathrm{ph},t}$ and
${\mathit{y}}_{t,m}$. The receiving end then employs a decoding scheme
${\mathcal{D}}_{t}:{\mathcal{l}}_{0}^{\infty}\to \{1,\dots ,M\},\phantom{\rule{3.33333pt}{0ex}}{\mathit{y}}_{\mathrm{re},t}\mapsto {\mathcal{D}}_{t}\left({\mathit{y}}_{\mathrm{re},t}\right)$. Again, for reasons of simplicity, we summarize processes such as demodulation, filtering, and decoding (in the sense of information theory) in this step. Regionbased decoding is a common way of implementing
${\mathcal{D}}_{t}$, in which case
${\mathcal{D}}_{t}$ is of the form
where
${\mathit{y}}_{\mathrm{de},t,1},\dots ,{\mathit{y}}_{\mathrm{de},t,N}\in {\mathcal{l}}_{0}^{\infty},\phantom{\rule{3.33333pt}{0ex}}N\in \mathbb{N},$ is a list of reference signals and
$D:\{1,\dots ,N\}\to \{1,\dots ,M\}$ is a mapping that assigns each reference signal an inferred message.
In the setup depicted in
Figure 1, the choice of the pair
$({\mathcal{E}}_{t},{\mathcal{D}}_{t})$ will generally be based on the robot’s digital twin, i.e., (assuming both the receiving end and the robot itself have access to the sequence
${\left({\mathfrak{a}}_{t}\right)}_{t\in \mathbb{N}}$), it will be implemented through computable mappings
${\mathfrak{a}}_{t}\mapsto {\mathcal{E}}_{t}$,
${\mathfrak{a}}_{t}\mapsto {\mathcal{D}}_{t}$, involving an optimization of some kind. For example, we may aim to choose
${\mathit{y}}_{t,1},\dots ,{\mathit{y}}_{t,M}$ and
${\mathit{y}}_{\mathrm{de},t,1},\dots ,{\mathit{y}}_{\mathrm{de},t,N}$ such that
for all
$m\in \{1,\dots ,M\}$ and all
$\mathit{w}\in {\mathcal{l}}_{0}^{\infty}$ that satisfy
${\parallel \mathit{w}\parallel}_{\infty}<{\u03f5}_{\mathit{w}}$ for some
${\u03f5}_{\mathit{w}}>0$. Accordingly, upon implementing the communication system, we require
${\mathit{y}}_{\mathrm{de},t,1},\dots ,{\mathit{y}}_{\mathrm{de},t,N}\in \mathcal{C}{\mathcal{l}}_{0}^{\infty}$ and
${\mathit{y}}_{t,1},\dots ,{\mathit{y}}_{t,M}\in \mathrm{dom}\left({\mathit{T}}_{\infty}\right)\cap \mathcal{C}{\mathcal{l}}_{0}^{\infty}$. In theory, we can then entirely neglect the analog part of the real system, i.e., the transmission of the signals through the physical (analog) medium, in the design of our signal processing algorithms.
Again, recall that both
${\mathfrak{X}}_{\u2605}$ and
${\mathfrak{F}}_{\u2605}$ are machinereadable languages for the set
$\mathcal{CB}(\u2605)$. In particular, per its definition,
${\mathfrak{X}}_{\u2605}$ does
not include the entirety of
$\mathcal{C}\mathcal{l}(\u2605)$, which appears unnecessary following the discussion above. In fact, there does not seem to be an obvious reason to consider the space
$\mathcal{CB}(\u2605)$ at all. Assuming we are able to prove that the implementation of our systems satisfies (
4), we even can perform the optimization of
$({\mathcal{E}}_{t},{\mathcal{D}}_{t})$ within the discretetime domain. Upon closer inspection, we find that despite the discussion above, there exists a variety of reasons why we cannot neglect the analog part of the communication system. We will discuss several of them in the following:
As mentioned above, any real implementation of our system will involve a steps of digitaltoanalog conversion of the transmission signal ${\mathit{y}}_{t,m}$. However, any realworld digitaltoanalog converter will not be able to synthesize the signal ${\mathit{T}}_{\infty}{\mathit{y}}_{t,m}$ perfectly. More realistically, we will be able to synthesize some signal ${\tilde{\mathit{f}}}_{t,m}$, that exhibits distortion from effects such as quantization and imperfect filtering. Depending on the application, it may be necessary to compute the signal ${\tilde{\mathit{f}}}_{t,m}$ in the first place, or at least compute the error $\parallel {\tilde{\mathit{f}}}_{t,m}{\mathit{T}}_{\infty}{\mathit{y}}_{t,m}{\parallel}_{\infty}$, in order to ensure the proper transmission of messages.
The generalized Shannon equivalence, which motivates the processing of signals in the digital domain, is applicable as longs as the entirety of the considered system is linear. In practical wireless communications systems,
nonlinear distortions are a common issue. In particular, the analog subsystems of both the transmitter and the receiving end have to operate within a certain dynamic range, which sets an upper limit to the
${\parallel \xb7\parallel}_{\infty}$norm of the analog signals they can process properly. Accordingly, in order to avoid
nonlinear distortions, we need to be able to compute or at least upperbound the values
$\parallel {\tilde{\mathit{f}}}_{t,m}{\parallel}_{\infty}$ and
The details of this issue are investigated in the context of boundedinputboundedoutput (BIBO) stability analysis and the peaktoaverage power ratio (PAPR) problem.
Analogous to the situation at the transmitter, we will not be able to measure the signal ${\mathit{y}}_{\mathrm{re},t}$ perfectly at the receiving end. Due to finite quantization accuracy and imperfect filtering, we obtain an approximate signal ${\tilde{\mathit{y}}}_{\mathrm{re},t}$. Since the overall duration of sampling is finite, we can assume ${\tilde{\mathit{y}}}_{\mathrm{re},t}$ and ${\mathit{T}}_{\infty}{\tilde{\mathit{y}}}_{\mathrm{re},t}=:{\tilde{\mathit{f}}}_{\mathrm{re},t}$ are elementary computable. Choosing ${\mathit{y}}_{\mathrm{de},t,n}\in \mathrm{dom}\left({\mathit{T}}_{\infty}\right)\cap \mathcal{C}{\mathcal{l}}_{0}^{\infty},n\in \{1,\dots ,N\}$, and denoting ${\mathit{f}}_{\mathrm{de},t,n}:={\mathit{T}}_{\infty}{\mathit{y}}_{\mathrm{de},t,n},n\in \{1,\dots ,N\}$, we generally have $\parallel {\tilde{\mathit{y}}}_{\mathrm{re},t}{\mathit{y}}_{\mathrm{de},t,n}{\parallel}_{\infty}\ne {\parallel {\tilde{\mathit{f}}}_{\mathrm{re},t}{\mathit{f}}_{\mathrm{de},t,n}\parallel}_{\infty}$. Aside from computational convenience, there is no a priori reason to perform decoding based on ${\tilde{\mathit{y}}}_{\mathrm{re},t}$ and ${\mathit{y}}_{\mathrm{de},t,1},\dots ,{\mathit{y}}_{\mathrm{de},t,N}$ rather than ${\tilde{\mathit{f}}}_{\mathrm{re},t}$ and ${\mathit{f}}_{\mathrm{de},t,1},\dots ,{\mathit{f}}_{\mathrm{de},t,N}$. In view of the mentioned limitations of realworld systems, this observation becomes even more relevant: since effects such as quantization are generally nonlinear, information may actually be lost if the decoding is performed in the space $\mathcal{C}{\mathcal{l}}_{0}^{\infty}$ rather than ${\mathcal{CB}}_{0,\pi}^{\infty}$. Unless proven otherwise for a specific case, the same argument holds true when we consider decoding schemes other than regionbased ones.
From a modelbased perspective, taking imperfect sampling at the receiving end into account raises another issue when considering the entire system within
$\mathcal{C}{\mathcal{l}}^{1}$. As indicated above, we may want to design the system with a specified marginoferror, i.e., we want to guarantee that proper message transmission is possible as long as we have
${\parallel \mathit{w}\parallel}_{\infty}<{\u03f5}_{\mathit{w}}$. If we instead require
$\parallel {\mathit{T}}_{\infty}{\mathit{w}\parallel}_{\infty}<{\u03f5}_{\mathit{w}}$, we can provide the continuoustimedomain upper bound
for the reconstruction error, which can then be transferred to the discretetime domain. Requiring
${\parallel \mathit{w}\parallel}_{\infty}<{\u03f5}_{\mathit{w}}$ alone is insufficient, since
${\mathit{T}}_{\infty}\mathit{w}$ may be arbitrarily large in this case nevertheless. Thus, we can provide an estimate for the reconstruction error only if we consider the continuoustime domain of the system.
Consequently, we consider signals
${\mathit{f}}_{\mathrm{ph},t}\in {\mathcal{B}}_{\pi}^{1}$,
${\mathit{f}}_{t}\in {\mathcal{CB}}_{\pi}^{1}$,
${\tilde{\mathit{f}}}_{t,m},{\mathit{f}}_{\mathrm{de},t,n}\in {\mathcal{CB}}_{0,\pi}^{\infty}$,
$t\in \mathbb{N}$,
$m\in \{1,\dots ,M\},n\in \{1,\dots ,N\}$, and a communications system of the form
in the following. Observe that we do not aim at actually implementing signal processing in the analog domain. We merely aim at finding find proper digital representatives for the continuoustime signals
${\mathit{f}}_{t}$,
${\tilde{\mathit{f}}}_{t,1},\dots ,{\tilde{\mathit{f}}}_{t,M}$,
${\mathit{f}}_{\mathrm{de},t,1},\dots ,{\mathit{f}}_{\mathrm{de},t,N}$,
${\mathit{x}}_{t},{\tilde{\mathit{f}}}_{t,1},\dots ,{\tilde{\mathit{f}}}_{t,M}$,
${\mathit{f}}_{\mathrm{de},t,1},\dots ,{\mathit{f}}_{\mathrm{de},t,N}$, since only considering their discretetime counterparts is insufficient for the reasons we mentioned above. Nevertheless,
${\mathfrak{X}}_{\u2605}$ is, in principle, a valid way to represent
${\mathit{f}}_{t}$,
${\tilde{\mathit{f}}}_{t,1},\dots ,{\tilde{\mathit{f}}}_{t,M}$,
${\mathit{f}}_{\mathrm{de},t,1},\dots ,{\mathit{f}}_{\mathrm{de},t,N}$,
${\mathit{x}}_{t},{\tilde{\mathit{f}}}_{t,1},\dots ,{\tilde{\mathit{f}}}_{t,M}$,
${\mathit{f}}_{\mathrm{de},t,1},\dots ,{\mathit{f}}_{\mathrm{de},t,N}$, as follows from the generalized Shannon equivalence: for each
$({\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}},\xi )\in \mathrm{dom}\left({[\xb7]}_{\mathfrak{X}}^{\u2605}\right)$, there exists exactly one
$\mathit{f}\in \mathcal{CB}(\u2605)$ such that
${[{\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}},\xi ]}_{\mathfrak{X}}^{\u2605}=\mathit{f}$ holds true, leaving us with the decision of whether to implement the signal processing for
${\mathit{f}}_{t}$,
${\tilde{\mathit{f}}}_{t,1},\dots ,{\tilde{\mathit{f}}}_{t,M}$,
${\mathit{f}}_{\mathrm{de},t,1},\dots ,{\mathit{f}}_{\mathrm{de},t,N}$ based on
${\mathfrak{X}}_{1}$ and
${\mathfrak{X}}_{\infty}$ or
${\mathfrak{F}}_{1}$ and
${\mathfrak{F}}_{\infty}$.
From
Section 2, recall the inequalities
${\mathfrak{X}}_{1}\u2ab0{\mathfrak{X}}_{\infty}{\mathcal{CB}}_{\pi}^{1}$ and
${\mathfrak{F}}_{1}\u2ab0{\mathfrak{F}}_{\infty}{\mathcal{CB}}_{\pi}^{1}$. Hence, if possible, it may be beneficial to implement the signal processing for all of the signals
${\tilde{\mathit{f}}}_{t,1},\dots ,{\tilde{\mathit{f}}}_{t,M}$,
${\mathit{f}}_{\mathrm{de},t,1},\dots ,{\mathit{f}}_{\mathrm{de},t,N}$ based on
${\mathfrak{X}}_{1}$ or
${\mathfrak{F}}_{1}$ as well, since we can always recover corresponding signal descriptions in
${\mathfrak{X}}_{\infty}{\mathcal{CB}}_{\pi}^{1},\phantom{\rule{3.33333pt}{0ex}}{\mathfrak{F}}_{\infty}{\mathcal{CB}}_{\pi}^{1}$, respectively, if needed. On the other hand, depending on the specific application, we may have to choose
${\tilde{\mathit{f}}}_{t,1},\dots ,{\tilde{\mathit{f}}}_{t,M}$,
${\mathit{f}}_{\mathrm{de},t,1},\dots ,{\mathit{f}}_{\mathrm{de}t,N}\in {\mathcal{CB}}_{0,\pi}^{\infty}\setminus {\mathcal{CB}}_{\pi}^{1}$, in which case we necessarily have to resort to using either
${\mathfrak{X}}_{\infty}$ or
${\mathfrak{F}}_{\infty}$. However, in any of the above cases, we must first and foremost be able to compute the relevant norms of the involved signals to implement the communication system. Before mathematical analysis, we summarize the relevant requirements as follows:
In a communication system of the form (5) and (6), we consider $\parallel {\mathit{f}}_{t}{\parallel}_{1}$, $\parallel {\tilde{\mathit{f}}}_{t,m}{\parallel}_{\infty}$, $\parallel {\mathit{f}}_{\mathrm{de},t,n}{\parallel}_{\infty}$, $\parallel {\mathit{f}}_{t}\ast {\tilde{\mathit{f}}}_{t,m}{\parallel}_{\infty}$, $\parallel {\tilde{\mathit{f}}}_{\mathrm{re},t}{\mathit{f}}_{\mathrm{de},t,n}{\parallel}_{\infty}$, and $\parallel {\tilde{\mathit{f}}}_{\mathrm{re},t}({\mathit{f}}_{t}\ast {\tilde{\mathit{f}}}_{t,m}){\parallel}_{\infty}$ relevant properties for all $t\in \mathbb{N}$, $m\in \{1,\dots ,M\}$, $n\in \{1,\dots ,N\}$. Thus, regarding any sufficient representation of signals $\mathit{f}\in \mathcal{CB}(\u2605)$ on digital hardware, we require to be able to recover ${\parallel \mathit{f}\parallel}_{\u2605}$. In other words, the mapping ${\parallel \xb7\parallel}_{\u2605}:\mathcal{CB}(\u2605)\to {\mathbb{R}}_{\mu},\phantom{\rule{3.33333pt}{0ex}}\mathit{f}\mapsto {\parallel \mathit{f}\parallel}_{\u2605}$ has to be computable in the employed machinereadable language.

Theorem 2. The mapping ${\parallel \xb7\parallel}_{\u2605}:{\mathfrak{F}}_{\u2605}\to \mathfrak{R},\phantom{\rule{3.33333pt}{0ex}}(\mathit{f},{\mathsf{\Lambda}}_{\mathfrak{F}},{[\xb7]}_{\mathfrak{F}}^{\u2605})\mapsto {(\parallel \mathit{f}\parallel}_{\u2605},{\mathsf{\Lambda}}_{\mathfrak{R}},{[\xb7]}_{\mathfrak{R}})$ is computable.
Proof. Observe that for
$\mathit{f}\in \mathcal{CB}(\u2605)$ elementary computable, the mapping
$\mathit{f}\mapsto {\parallel \mathit{f}\parallel}_{\u2605}$ is computable. That is, for
$N,M\in \mathbb{N}$ and a rationalcomplex
$(M+1)$tuple
$\mathit{z}:={\left({z}_{m}\right)}_{m\in \mathcal{I}}$,
$\mathcal{I}=\{0,\dots M\}$, there exists a computable mapping
$(N,\mathit{z})\mapsto {G}_{\u2605}(N,\mathit{z})\in \mathbb{Q}$, such that
holds true (provided the relevant norm exists). For
$\mathit{f}={[{\left({s}_{n,m}^{\prime}\right)}_{n,m\in \mathbb{N}},{\xi}^{\prime}]}_{\mathfrak{F}}^{\u2605}$ arbitrary, let the computable sequence
${\left({\mathit{f}}_{M}\right)}_{M\in \mathbb{N}}$ of elementary computable sequences satisfy (
3). Further, for
$M\in \mathbb{N}$, define
Then,
${\left({r}_{M}\right)}_{M\in \mathbb{N}}$ is a computable sequence of rational numbers. Employing the triangle inequality, we obtain
Thus, defining
$\xi :\mathbb{N}\to \mathbb{N},M\mapsto M+1$, we have
${[{\left({r}_{M}\right)}_{M\in \mathbb{N}},\xi ]}_{\mathfrak{R}}={\parallel \mathit{f}\parallel}_{\u2605}$. Further, it follows from the SMNTheorem (c.f.
Section 2) that the mapping
$({\left({s}_{n,m}^{\prime}\right)}_{n,m\in \mathbb{N}},{\xi}^{\prime})\mapsto ({\left({r}_{M}\right)}_{M\in \mathbb{N}},\xi )$ (with
$({\left({s}_{n,m}^{\prime}\right)}_{n,m\in \mathbb{N}},{\xi}^{\prime})$ and
$({\left({r}_{M}\right)}_{M\in \mathbb{N}},\xi )$ as above) is computable, which concludes the proof. □
Theorem 3. The mapping ${\parallel \xb7\parallel}_{\u2605}:{\mathfrak{X}}_{\u2605}\to \mathfrak{R},\phantom{\rule{3.33333pt}{0ex}}(\mathit{f},{\mathsf{\Lambda}}_{\mathfrak{X}},{[\xb7]}_{\mathfrak{X}}^{\u2605})\mapsto {(\parallel \mathit{f}\parallel}_{\u2605},{\mathsf{\Lambda}}_{\mathfrak{R}},{[\xb7]}_{\mathfrak{R}})$ is not computable.
Proof. The statement follows by contradiction from Lemmas 1 and 2, respectively. To this end, assume the mapping
${\parallel \xb7\parallel}_{\u2605}:{\mathfrak{X}}_{\u2605}\to \mathfrak{R},\phantom{\rule{3.33333pt}{0ex}}(\mathit{f},{\mathsf{\Lambda}}_{\mathfrak{X}},{[\xb7]}_{\mathfrak{X}}^{\u2605})\mapsto {(\parallel \mathit{f}\parallel}_{\u2605},{\mathsf{\Lambda}}_{\mathfrak{R}},{[\xb7]}_{\mathfrak{R}})$ is computable and let
${\left({\mathit{x}}_{k}\right)}_{k\in \mathbb{N}}$ be a sequence that satisfies Lemma 1, Lemma 2, respectively, for some recursively enumerable
nonrecursive set
$\mathsf{\Omega}\subset \mathbb{N}$. Then, there exists a computable mapping
such that for all
$k\in \mathbb{N}$, the pair
$({\left({s}_{n,m}\left(k\right)\right)}_{n,m\in \mathbb{N}},{\xi}_{k})$ determines
${\mathit{x}}_{k}$ in the sense of Definition 5, and we have
If the mapping
${\parallel \xb7\parallel}_{\u2605}:{\mathfrak{X}}_{\u2605}\to \mathfrak{R},\phantom{\rule{3.33333pt}{0ex}}(\mathit{f},{\mathsf{\Lambda}}_{\mathfrak{X}},{[\xb7]}_{\mathfrak{X}}^{\u2605})\mapsto {(\parallel \mathit{f}\parallel}_{\u2605},{\mathsf{\Lambda}}_{\mathfrak{R}},{[\xb7]}_{\mathfrak{R}})$ is indeed computable, there must also exist a computable mapping
$({\left({s}_{n,m}\left(k\right)\right)}_{n,m\in \mathbb{N}},{\xi}_{k})\mapsto ({\left({r}_{m}\left(k\right)\right)}_{m\in \mathbb{N}},{\xi}_{k}^{\prime})$,
$k\in \mathbb{N}$ such that, for all
$k\in \mathbb{N}$, we have
By concatenation, we conclude that
$k\mapsto \parallel {\mathit{f}}_{k}{\parallel}_{\u2605}$,
$k\in \mathbb{N}$ is computable as well. For all
$k\in \mathbb{N}$, we define
Then,
${r}_{<}\left(k\right)$ is a rational number, and the mapping
$k\mapsto {r}_{<}\left(k\right)$,
$k\in \mathbb{N}$, is computable. For all
$k\in \mathbb{N}$, we further have
${r}_{<}\left(k\right)<{lim}_{m\to \infty}{r}_{m}\left(k\right)<{r}_{<}\left(k\right)+1$ by construction, and thus,
$k\in \mathsf{\Omega}\iff {r}_{<}\left(k\right)>0$ by the requirements of Lemma 1, Lemma 2, respectively. We define
and observe that, we have
$g={\mathbb{1}}_{\mathsf{\Omega}}$. By the SMNTheorem (c.f.
Section 2),
$(k\mapsto g(k\left)\right)$ is computable, i.e.,
g is a
$\mu $recursive function. Accordingly,
$\mathsf{\Omega}$ is recursive, which contradicts the prerequisite of
$\mathsf{\Omega}$ being
nonrecursive. □
Theorem 4. We have ${\mathfrak{F}}_{\u2605}\succ {\mathfrak{X}}_{\u2605}$. In the sense of Section 2, the inequality is strict. Proof. Denote by
${\mathrm{Id}}_{\mathfrak{X},\mathfrak{F}}^{\u2605}:{\mathfrak{X}}_{\u2605}\to {\mathfrak{F}}_{\u2605}$ and
${\mathrm{Id}}_{\mathfrak{F},\mathfrak{X}}^{\u2605}:{\mathfrak{F}}_{\u2605}\to {\mathfrak{X}}_{\u2605}$ the relevant identity mappings in the sense of
Section 2. That is, we have
for all
$\mathit{f}\in \mathcal{CB}(\u2605)$. We divide the proof in two parts: first, we prove that
${\mathrm{Id}}_{\mathfrak{X},\mathfrak{F}}^{\u2605}$ is
not computable; second, we prove that
${\mathrm{Id}}_{\mathfrak{F},\mathfrak{X}}^{\u2605}$ is computable.
In essence, the first part is a corollary of Theorems 2 and 3, which follows by contradiction. Assume
${\mathrm{Id}}_{\mathfrak{X},\mathfrak{F}}^{\u2605}$ is computable. By Theorem 2, the mapping
${\parallel \xb7\parallel}_{\u2605}:{\mathfrak{F}}_{\u2605}\to \mathfrak{R}$,
$(\mathit{f},{\mathsf{\Lambda}}_{\mathfrak{F}},{[\xb7]}_{\mathfrak{F}}^{\u2605})\mapsto {(\parallel \mathit{f}\parallel}_{\u2605},{\mathsf{\Lambda}}_{\mathfrak{R}},{[\xb7]}_{\mathfrak{R}})$, is computable. Hence, by concatenation, we obtain the computable mapping
contradicting Theorem 3. Thus,
${\mathrm{Id}}_{\mathfrak{X},\mathfrak{F}}^{\u2605}$ cannot be computable.
The second part is a consequence of the continuity of the sampling operator
${\mathit{S}}_{\u2605}:\mathcal{B}(\u2605)\to \mathcal{l}(\u2605)$. Particularly, there exist constants
${C}_{\u2605}\in \{q\in \mathbb{Q}:{log}_{2}\left(q\right)\in \mathbb{Z}\}$ such that
$\parallel {\mathit{S}}_{\u2605}{\mathit{f}\parallel}_{\u2605}<{C}_{\u2605}{\parallel \mathit{f}\parallel}_{\u2605}$ holds true for all
$\mathit{f}\in \mathcal{B}(\u2605)$. Let
${[{\left({s}_{n,m}^{\prime}\right)}_{n,m\in \mathbb{N}},{\xi}^{\prime}]}_{\mathfrak{F}}^{\u2605}={\mathit{T}}_{\u2605}\mathit{x}$ be arbitrary. For all
$M\in \mathbb{N}$, we have
Define
${K}_{\u2605}:={log}_{2}{C}_{\u2605}$,
$\xi :\mathbb{N}\to \mathbb{N},M\mapsto {\xi}^{\prime}(M+{K}_{\u2605})$, and
${\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}}:={\left({s}_{n,m+{K}_{\u2605}}^{\prime}\right)}_{n,m\in \mathbb{N}}$, and observe that
${\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}}$ is a computable double sequence of rationalcomplex numbers and
$\xi $ is a
$\mu $recursive function. For all
$M\in \mathbb{N}$, we have
Thus, the pair
$({\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}},\xi )$ determines
$\mathit{x}$ in the sense of Definition 5, and we have
${[{\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}},\xi ]}_{\mathfrak{X}}^{\u2605}={\mathit{T}}_{\u2605}\mathit{x}$. Further, by the SMNTheorem (c.f.
Section 2), the mapping
$({\left({s}_{n,m}^{\prime}\right)}_{n,m\in \mathbb{N}},{\xi}^{\prime})\mapsto ({\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}},\xi )$ is computable, which concludes the proof. □
For the languages
${\mathfrak{F}}_{\u2605}$ and
${\mathfrak{X}}_{\u2605}$, Theorem 4 corresponds to the Case 1 of the distinction made in
Section 2: descriptions in the language
${\mathfrak{F}}_{\u2605}$ contain more information than descriptions in the language
${\mathfrak{X}}_{\u2605}$. In
Section 2, we also indicated a link between the relationship of
${\mathfrak{F}}_{\u2605}$ and
${\mathfrak{X}}_{\u2605}$, i.e., the inequality
${\mathfrak{F}}_{\u2605}\succ {\mathfrak{X}}_{\u2605}$, and the computability of the operators
${\mathit{S}}_{\u2605}$ and
${\mathit{T}}_{\u2605}$. In turn, whether
${\mathit{S}}_{\u2605}$ and
${\mathit{T}}_{\u2605}$ are computable is the formal rephrasing of whether the generalized Shannon equivalence holds true on the algorithmic level. Consider the set
i.e.,
${\mathit{S}}_{\u2605}\left(\mathcal{CB}(\u2605)\right)$ consists of those sequences
$\mathit{x}\in \mathcal{C}\mathcal{l}(\u2605)$ that equate to some computable signal
$\mathit{f}\in \mathcal{CB}(\u2605)$ under the action of
${\mathit{T}}_{\u2605}$. Naturally, we can define a machinereadable language
${\mathfrak{l}}_{\u2605}$ for
${\mathit{S}}_{\u2605}\left(\mathcal{CB}(\u2605)\right)$ according to
in which case the identity mappings (in the sense of
Section 2)
${\mathrm{Id}}_{\mathfrak{X},\mathfrak{F}}^{\u2605}:{\mathfrak{X}}_{\u2605}\to {\mathfrak{F}}_{\u2605}$ and
${\mathrm{Id}}_{\mathfrak{F},\mathfrak{X}}^{\u2605}:{\mathfrak{F}}_{\u2605}\to {\mathfrak{X}}_{\u2605}$ become the interpolation operator
${\mathit{T}}_{\u2605}:{\mathfrak{l}}_{\u2605}\to {\mathfrak{F}}_{\u2605}$ and sampling operator
${\mathit{S}}_{\u2605}:{\mathfrak{F}}_{\u2605}\to {\mathfrak{l}}_{\u2605}$, respectively. Then, according to Theorem 4,
${\mathit{S}}_{\u2605}$ is computable, while
${\mathit{T}}_{\u2605}$ is
not. As indicated in
Section 2, this is due to the discontinuity of
${\mathit{T}}_{\u2605}$ with regard to the relevant norm. In other words, the generalized Shannon equivalence between
$\mathcal{B}(\u2605)$ and
$\mathcal{l}(\u2605)$ does
not hold true on an algorithmic level! Analytically, if
holds true, both
$({\left({s}_{n,m}\right)}_{n,m\in \mathbb{N}},\xi )$ and
$({\left({s}_{n,m}^{\prime}\right)}_{n,m\in \mathbb{N}},{\xi}^{\prime})$ uniquely determine all mathematically welldefined properties of
$\mathit{f}$, including
${\parallel \mathit{f}\parallel}_{\u2605}$. Algorithmically, as Theorems 2–4 show, this is
not the case. With respect to the requirements summarized above, we conclude our analysis as follows:
The generalized Shannon equivalence between $\mathcal{B}(\u2605)$ and $\mathcal{l}(\u2605)$ does not hold true on an algorithmic level. In particular, we observe the following:Regarding ${\parallel \mathit{f}\parallel}_{\u2605}$ as relevant property, discretetime descriptions of signals $\mathit{f}\in \mathcal{CB}(\u2605)$ are an insufficient representative of analog information on digital hardware; Regarding ${\parallel \mathit{f}\parallel}_{\u2605}$ as relevant property, continuoustime descriptions of signals $\mathit{f}\in \mathcal{CB}(\u2605)$ are a sufficient representative of analog information on digital hardware; Any computation on the basis of discretetime descriptions can be processed on the basis of continuoustime descriptions as well, i.e., continuoustime descriptions capture more information (in the sense of the distinction made in Section 2) than discretetime descriptions.
