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3 pages, 150 KiB

Open AccessEditorial

Theory of Functions and Applications

by Inna Kal’chuk

Axioms 2024, 13(3), 168; https://doi.org/10.3390/axioms13030168 - 05 Mar 2024

Viewed by 668

Abstract

In this editorial, we present “Theory of Functions and Applications”, a Special Issue of Axioms [...] Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

Research

Jump to: Editorial

17 pages, 325 KiB

Open AccessArticle

On the Realization of Exact Upper Bounds of the Best Approximations on the Classes H^1,1 by Favard Sums

by Dmytro Bushev and Inna Kal’chuk

Axioms 2023, 12(8), 763; https://doi.org/10.3390/axioms12080763 - 02 Aug 2023

Viewed by 745

Abstract

In this paper, we find the sets of all extremal functions for approximations of the Hölder classes of

H^{1}

2

π

-periodic functions of one variable by the Favard sums, which coincide with the set of all extremal functions realizing the exact [...] Read more.

In this paper, we find the sets of all extremal functions for approximations of the Hölder classes of

H^{1}

2

π

-periodic functions of one variable by the Favard sums, which coincide with the set of all extremal functions realizing the exact upper bounds of the best approximations of this class by trigonometric polynomials. In addition, we obtain the sets of all of extremal functions for approximations of the class

H^{1}

by linear methods of summation of Fourier series. Furthermore, we receive the set of all extremal functions for the class

H^{1}

in the Korneichuk–Stechkin lemma and its analogue, the Stepanets lemma, for the Hölder class

H^{1, 1}

functions of two variables being 2

π

-periodic in each variable. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

14 pages, 309 KiB

Open AccessArticle

Joint Discrete Universality in the Selberg–Steuding Class

by Roma Kačinskaitė, Antanas Laurinčikas and Brigita Žemaitienė

Axioms 2023, 12(7), 674; https://doi.org/10.3390/axioms12070674 - 08 Jul 2023

Viewed by 529

Abstract

In the paper, we consider the approximation of analytic functions by shifts from the wide class

\tilde{S}

of L-functions. This class was introduced by A. Selberg, supplemented by J. Steuding, and is defined axiomatically. We prove the so-called joint discrete universality [...] Read more.

In the paper, we consider the approximation of analytic functions by shifts from the wide class

\tilde{S}

of L-functions. This class was introduced by A. Selberg, supplemented by J. Steuding, and is defined axiomatically. We prove the so-called joint discrete universality theorem for the function

L (s) \in \tilde{S}

. Using the linear independence over

Q

of the multiset

\{(h_{j} log p : p \in P), j = 1, \dots, r; 2 π\}

for positive

h_{j}

, we obtain that there are many infinite shifts

(L (s + i k h_{1}), \dots, L (s + i k h_{r}))

,

k = 0, 1, \dots

, approximating every collection

(f_{1} (s), \dots, f_{r} (s))

of analytic non-vanishing functions defined in the strip

{s \in C : σ_{L} < σ < 1}

, where

σ_{L}

is a degree of the function

L (s)

. For the proof, the probabilistic approach based on weak convergence of probability measures in the space of analytic functions is applied. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

17 pages, 374 KiB

Open AccessArticle

Scalar-on-Function Relative Error Regression for Weak Dependent Case

by Zouaoui Chikr Elmezouar, Fatimah Alshahrani, Ibrahim M. Almanjahie, Zoulikha Kaid, Ali Laksaci and Mustapha Rachdi

Axioms 2023, 12(7), 613; https://doi.org/10.3390/axioms12070613 - 21 Jun 2023

Viewed by 856

Abstract

Analyzing the co-variability between the Hilbert regressor and the scalar output variable is crucial in functional statistics. In this contribution, the kernel smoothing of the Relative Error Regression (RE-regression) is used to resolve this problem. Precisely, we use the relative square error to [...] Read more.

Analyzing the co-variability between the Hilbert regressor and the scalar output variable is crucial in functional statistics. In this contribution, the kernel smoothing of the Relative Error Regression (RE-regression) is used to resolve this problem. Precisely, we use the relative square error to establish an estimator of the Hilbertian regression. As asymptotic results, the Hilbertian observations are assumed to be quasi-associated, and we demonstrate the almost complete consistency of the constructed estimator. The feasibility of this Hilbertian model as a predictor in functional time series data is discussed. Moreover, we give some practical ideas for selecting the smoothing parameter based on the bootstrap procedure. Finally, an empirical investigation is performed to examine the behavior of the RE-regression estimation and its superiority in practice. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

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26 pages, 380 KiB

Open AccessArticle

On a Sum of More Complex Product-Type Operators from Bloch-Type Spaces to the Weighted-Type Spaces

by Cheng-Shi Huang and Zhi-Jie Jiang

Axioms 2023, 12(6), 566; https://doi.org/10.3390/axioms12060566 - 07 Jun 2023

Viewed by 1193

Abstract

The aim of the present paper is to completely characterize the boundedness and compactness of a sum operator defined by some more complex products of composition, multiplication, and mth iterated radial derivative operators from Bloch-type spaces to weighted-type spaces on the unit [...] Read more.

The aim of the present paper is to completely characterize the boundedness and compactness of a sum operator defined by some more complex products of composition, multiplication, and mth iterated radial derivative operators from Bloch-type spaces to weighted-type spaces on the unit ball. In some applications, the boundedness and compactness of all products of composition, multiplication, and mth iterated radial derivative operators from Bloch-type spaces to weighted-type spaces on the unit ball are also characterized. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

14 pages, 475 KiB

Open AccessArticle

Matrix Representations for a Class of Eigenparameter Dependent Sturm–Liouville Problems with Discontinuity

by Shuang Li, Jinming Cai and Kun Li

Axioms 2023, 12(5), 479; https://doi.org/10.3390/axioms12050479 - 15 May 2023

Viewed by 718

Abstract

Matrix representations for a class of Sturm–Liouville problems with eigenparameters contained in the boundary and interface conditions were studied. Given any matrix eigenvalue problem of a certain type and an eigenparameter-dependent condition, a class of Sturm–Liouville problems with this specified condition was constructed. [...] Read more.

Matrix representations for a class of Sturm–Liouville problems with eigenparameters contained in the boundary and interface conditions were studied. Given any matrix eigenvalue problem of a certain type and an eigenparameter-dependent condition, a class of Sturm–Liouville problems with this specified condition was constructed. It has been proven that each Sturm–Liouville problem is equivalent to the given matrix eigenvalue problem. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

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11 pages, 390 KiB

Open AccessArticle

Initial Coefficients Upper Bounds for Certain Subclasses of Bi-Prestarlike Functions

by Tareq Hamadneh, Ibraheem Abu Falahah, Yazan Alaya AL-Khassawneh, Abdallah Al-Husban, Abbas Kareem Wanas and Teodor Bulboacă

Axioms 2023, 12(5), 453; https://doi.org/10.3390/axioms12050453 - 05 May 2023

Cited by 1 | Viewed by 1192

Abstract

In this article, we introduce and study the behavior of the modules of the first two coefficients for the classes

N_{Σ} (γ, λ, δ, μ; α)

and [...] Read more.

In this article, we introduce and study the behavior of the modules of the first two coefficients for the classes

N_{Σ} (γ, λ, δ, μ; α)

and

N_{Σ^{*}} (γ, λ, δ, μ; β)

of normalized holomorphic and bi-univalent functions that are connected with the prestarlike functions. We determine the upper bounds for the initial Taylor–Maclaurin coefficients

| a_{2} |

and

| a_{3} |

for the functions of each of these families, and we also point out some special cases and consequences of our main results. The study of these classes is closely connected with those of Ruscheweyh who in 1977 introduced the classes of prestarlike functions of order

μ

using a convolution operator and the proofs of our results are based on the well-known Carathédory’s inequality for the functions with real positive part in the open unit disk. Our results generalize a few of the earlier ones obtained by Li and Wang, Murugusundaramoorthy et al., Brannan and Taha, and could be useful for those that work with the geometric function theory of one-variable functions. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

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11 pages, 272 KiB

Open AccessArticle

On Hermite–Hadamard-Type Inequalities for Functions Satisfying Second-Order Differential Inequalities

by Ibtisam Aldawish, Mohamed Jleli and Bessem Samet

Axioms 2023, 12(5), 443; https://doi.org/10.3390/axioms12050443 - 29 Apr 2023

Cited by 2 | Viewed by 851

Abstract

Hermite–Hadamard inequality is a double inequality that provides an upper and lower bounds of the mean (integral) of a convex function over a certain interval. Moreover, the convexity of a function can be characterized by each of the two sides of this inequality. [...] Read more.

Hermite–Hadamard inequality is a double inequality that provides an upper and lower bounds of the mean (integral) of a convex function over a certain interval. Moreover, the convexity of a function can be characterized by each of the two sides of this inequality. On the other hand, it is well known that a twice differentiable function is convex, if and only if it admits a nonnegative second-order derivative. In this paper, we obtain a characterization of a class of twice differentiable functions (including the class of convex functions) satisfying second-order differential inequalities. Some special cases are also discussed. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

20 pages, 2731 KiB

Open AccessArticle

Measure-Based Extension of Continuous Functions and p-Average-Slope-Minimizing Regression

by Roger Arnau, Jose M. Calabuig and Enrique A. Sánchez Pérez

Axioms 2023, 12(4), 359; https://doi.org/10.3390/axioms12040359 - 07 Apr 2023

Viewed by 943

Abstract

This work is inspired by some recent developments on the extension of Lipschitz real functions based on the minimization of the maximum value of the slopes of a reference set for this function. We propose a new method in which an integral p [...] Read more.

This work is inspired by some recent developments on the extension of Lipschitz real functions based on the minimization of the maximum value of the slopes of a reference set for this function. We propose a new method in which an integral p–average is optimized instead of its maximum value. We show that this is a particular case of a more general theoretical approach studied here, provided by measure-valued representations of the metric spaces involved, and a duality formula. For

p = 2

, explicit formulas are proved, which are also shown to be a particular case of a more general class of measure-based extensions, which we call ellipsoidal measure extensions. The Lipschitz-type boundedness properties of such extensions are shown. Examples and concrete applications are also given. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

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15 pages, 291 KiB

Open AccessArticle

Coinciding Mean of the Two Symmetries on the Set of Mean Functions

by Lenka Mihoković

Axioms 2023, 12(3), 238; https://doi.org/10.3390/axioms12030238 - 25 Feb 2023

Viewed by 715

Abstract

On the set

M

of mean functions, the symmetric mean of M with respect to mean

M_{0}

can be defined in several ways. The first one is related to the group structure on

M

, and the second one is defined trough [...] Read more.

On the set

M

of mean functions, the symmetric mean of M with respect to mean

M_{0}

can be defined in several ways. The first one is related to the group structure on

M

, and the second one is defined trough Gauss’ functional equation. In this paper, we provide an answer to the open question formulated by B. Farhi about the matching of these two different mappings called symmetries on the set of mean functions. Using techniques of asymptotic expansions developed by T. Burić, N. Elezović, and L. Mihoković (Vukšić), we discuss some properties of such symmetries trough connection with asymptotic expansions of means involved. As a result of coefficient comparison, a new class of means was discovered, which interpolates between harmonic, geometric, and arithmetic mean. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

13 pages, 267 KiB

Open AccessArticle

From Symmetric Functions to Partition Identities

by Mircea Merca

Axioms 2023, 12(2), 126; https://doi.org/10.3390/axioms12020126 - 28 Jan 2023

Cited by 1 | Viewed by 1076

Abstract

In this paper, we show that some classical results from q-analysis and partition theory are specializations of the fundamental relationships between complete and elementary symmetric functions. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

22 pages, 349 KiB

Open AccessArticle

New Formulas and Connections Involving Euler Polynomials

by Waleed Mohamed Abd-Elhameed and Amr Kamel Amin

Axioms 2022, 11(12), 743; https://doi.org/10.3390/axioms11120743 - 18 Dec 2022

Viewed by 1229

Abstract

The major goal of the current article is to create new formulas and connections between several well-known polynomials and the Euler polynomials. These formulas are developed using some of these polynomials’ well-known fundamental characteristics as well as those of the Euler polynomials. In [...] Read more.

The major goal of the current article is to create new formulas and connections between several well-known polynomials and the Euler polynomials. These formulas are developed using some of these polynomials’ well-known fundamental characteristics as well as those of the Euler polynomials. In terms of the Euler polynomials, new formulas for the derivatives of various symmetric and non-symmetric polynomials, including the well-known classical orthogonal polynomials, are given. This leads to the deduction of several new connection formulas between various polynomials and the Euler polynomials. As an important application, new closed forms for the definite integrals for the product of various symmetric and non-symmetric polynomials with the Euler polynomials are established based on the newly derived connection formulas. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

20 pages, 729 KiB

Open AccessArticle

Fractional Clique Collocation Technique for Numerical Simulations of Fractional-Order Brusselator Chemical Model

by Mohammad Izadi and Hari Mohan Srivastava

Axioms 2022, 11(11), 654; https://doi.org/10.3390/axioms11110654 - 18 Nov 2022

Cited by 13 | Viewed by 1327

Abstract

The primary focus of this research study is in the development of an effective hybrid matrix method to solve a class of nonlinear systems of equations of fractional order arising in the modeling of autocatalytic chemical reaction problems. The fractional operator is considered [...] Read more.

The primary focus of this research study is in the development of an effective hybrid matrix method to solve a class of nonlinear systems of equations of fractional order arising in the modeling of autocatalytic chemical reaction problems. The fractional operator is considered in the sense of Liouville–Caputo. The proposed approach relies on the combination of the quasi-linearization technique and the spectral collocation strategy based on generalized clique bases. The main feature of the hybrid approach is that it converts the governing nonlinear fractional-order systems into a linear algebraic system of equations, which is solved in each iteration. In a weighted

L_{2}

norm, we prove the error and convergence analysis of the proposed algorithm. By using various model parameters in the numerical examples, we show the computational efficacy as well as the accuracy of our approach. Comparisons with existing available schemes show the high accuracy and robustness of the designed hybrid matrix collocation technique. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

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17 pages, 433 KiB

Open AccessArticle

Non-Canonical Functional Differential Equation of Fourth-Order: New Monotonic Properties and Their Applications in Oscillation Theory

by Amany Nabih, Clemente Cesarano, Osama Moaaz, Mona Anis and Elmetwally M. Elabbasy

Axioms 2022, 11(11), 636; https://doi.org/10.3390/axioms11110636 - 12 Nov 2022

Cited by 5 | Viewed by 1201

Abstract

In the present article, we iteratively deduce new monotonic properties of a class from the positive solutions of fourth-order delay differential equations. We discuss the non-canonical case in which there are possible decreasing positive solutions. Then, we find iterative criteria that exclude the [...] Read more.

In the present article, we iteratively deduce new monotonic properties of a class from the positive solutions of fourth-order delay differential equations. We discuss the non-canonical case in which there are possible decreasing positive solutions. Then, we find iterative criteria that exclude the existence of these positive decreasing solutions. Using these new criteria and based on the comparison and Riccati substitution methods, we create sufficient conditions to ensure that all solutions of the studied equation oscillate. In addition to having many applications in various scientific domains, the study of the oscillatory and non-oscillatory features of differential equation solutions is a theoretically rich field with many intriguing issues. Finally, we show the importance of the results by applying them to special cases of the studied equation. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

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19 pages, 350 KiB

Open AccessArticle

A Novel Approach in Solving Improper Integrals

by Mohammad Abu-Ghuwaleh, Rania Saadeh and Ahmad Qazza

Axioms 2022, 11(10), 572; https://doi.org/10.3390/axioms11100572 - 20 Oct 2022

Cited by 7 | Viewed by 1558

Abstract

To resolve several challenging applications in many scientific domains, general formulas of improper integrals are provided and established for use in this article. The suggested theorems can be considered generators for new improper integrals with precise solutions, without requiring complex computations. New criteria [...] Read more.

To resolve several challenging applications in many scientific domains, general formulas of improper integrals are provided and established for use in this article. The suggested theorems can be considered generators for new improper integrals with precise solutions, without requiring complex computations. New criteria for handling improper integrals are illustrated in tables to simplify the usage and the applications of the obtained outcomes. The results of this research are compared with those obtained by I.S. Gradshteyn and I.M. Ryzhik in the classical table of integrations. Some well-known theorems on improper integrals are considered to be simple cases in the context of our work. Some applications related to finding Green’s function, one-dimensional vibrating string problems, wave motion in elastic solids, and computing Fourier transforms are presented. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

5 pages, 898 KiB

Open AccessArticle

On the Distribution of Kurepa’s Function

by Nicola Fabiano, Milanka Gardašević-Filipović, Nikola Mirkov, Vesna Todorčević and Stojan Radenović

Axioms 2022, 11(8), 388; https://doi.org/10.3390/axioms11080388 - 07 Aug 2022

Cited by 2 | Viewed by 1269

Abstract

Kurepa’s function and his hypothesis have been investigated by means of numerical simulation. Particular emphasis has been given to the conjecture on its distribution, that should be one of a random uniform distribution, which has been verified for large numbers. A convergence function [...] Read more.

Kurepa’s function and his hypothesis have been investigated by means of numerical simulation. Particular emphasis has been given to the conjecture on its distribution, that should be one of a random uniform distribution, which has been verified for large numbers. A convergence function for the two has been found. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

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