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Article

Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Str., 76018 Ivano-Frankivsk, Ukraine
*
Author to whom correspondence should be addressed.
Axioms 2021, 10(3), 150; https://doi.org/10.3390/axioms10030150
Submission received: 14 June 2021 / Revised: 4 July 2021 / Accepted: 5 July 2021 / Published: 7 July 2021
(This article belongs to the Special Issue Analytic Functions and Nonlinear Functional Analysis)

Abstract

:
In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

1. Introduction and Preliminaries

Let X be an infinite dimensional complex Banach space. A continuous function f : X C is said to be an entire analytic function (or just an entire function) if its restriction on any finite dimensional subspace is analytic. If an entire function f satisfies f ( λ x ) = λ n f ( x ) for every x X and λ C , then f is called an n-homogeneous polynomial. It is well known that for an n-homogeneous polynomial f there exists a unique symmetric n-linear mapping B : X n C associated with f such that f ( x ) = B ( x , , x ) . Each zero-homogeneous polynomial is a constant. A finite sum of homogeneous polynomials is a polynomial. The space of all entire analytic functions on X is denoted by H ( X ) , the space of all polynomials on X is denoted by P ( X ) and the space of all n-homogeneous polynomials on X is denoted by P ( n X ) . For every entire function f there exists a sequence of continuous n-homogeneous polynomials { f n } n = 1 (so-called Taylor polynomials) such that
f ( x ) = n = 0 f n ( x )
and the series converges for every x X . The Taylor series expansion (1) uniformly converges on the open ball r B centered at zero with radius r = ϱ 0 ( f ) , where
ϱ 0 ( f ) = 1 lim sup n f n 1 / n .
The radius r = ϱ 0 ( f ) is called the radius of uniform convergence of f at zero or the radius of boundedness of f at zero because the ball r B is the largest open ball at zero such that f is bounded on every closed subset of it. If ϱ 0 ( f ) = , then f is bounded on all bounded subsets of X and is called a function of bounded type. The algebra of all functions of bounded type on X is denoted by H b ( X ) . Functions in H ( X ) \ H b ( X ) are called entire functions of unbounded type. Note that ϱ 0 ( f ) > 0 for every f H ( X ) .
It is well-known that every infinite dimensional Banach space X admits entire functions of unbounded type. For example, for a given weak*-null sequence ϕ n X * , ϕ = 1 , which always exists (see p. 157 [1]), the function
f ( x ) = n = 1 ϕ n n ( x )
is an entire function of unbounded type on X . We say that a sequence of functions g n on X (not necessary linear) is weak*-null if g n ( x ) 0 as n for every x X .
Entire functions of unbounded type were studied by many authors. In [2] Aron constructed an entire function f on a Banach space X such that for every r > 0 there is a point x 0 X such that f is unbounded on the ball of radius r , centered at x 0 . In [3,4] Ansemil, Aron, and Ponte constructed entire functions f on a Banach space which are bounded on any given finite collection of balls and unbounded on another given finite collection of balls. The set H ( X ) \ H b ( X ) is not linear and is not closed under multiplication of functions. However, Lopez–Salazar Codes in [5] show that for every infinite-dimensional Banach space X the set H ( X ) \ H b ( X ) contains an infinite-dimensional linear space (without zero) and even an infinite-dimensional algebra (without zero).
Let P = { P 1 , P 2 , , P n , } be a sequence of polynomials on X . We denote by P P ( X ) the smallest unital algebra containing all polynomials in P . Let H b P ( X ) be the closure of P P ( X ) in H b ( X ) with respect to the metrizable topology of the uniform convergence on bounded subsets of X , and H P ( X ) is the subalgebra of all entire functions f on X such that their Taylor polynomials f n are in P P ( X ) . The algebras H b P ( X ) and P P ( X ) were investigated in [6,7]. A typical example of P P ( X ) is the algebra of symmetric polynomials. Let S be a group of isometric operators on a Banach space X . A function f on X is S-symmetric if it is invariant with respect to the action of S . Symmetric polynomials and analytic functions on Banach spaces with respect to various groups were studied in [8,9,10,11,12,13,14,15,16,17,18].
Symmetric entire functions of unbounded type on 1 were studied in [19]. In [20] the authors considered the question: Let P 0 ( n X ) be subspaces of P ( n X ) , n N . Under which conditions is there a function f = n = 0 f n H ( X ) \ H b ( X ) such that f n P 0 ( n X ) ? In this paper we show that some natural subspaces P 0 ( n X ) do not support entire functions of unbounded type. In particular, we show that there are no symmetric entire functions of unbounded type on L [ a ; b ] .
In Section 2 we propose some conditions under which P P ( X ) supports entire functions of unbounded type and construct some counterexamples. In Section 3 we show that if X is an infinite dimensional Banach space, then H ( x ) \ H b ( X ) can be represented as a union of infinite dimensional linear subspaces (without the origin). Furthermore, we show that for some cases H b P ( X ) \ H P ( X ) contains infinite dimensional algebras (without the origin). Some results of this paper were announced in [21].
We refer the reader to the books of Dineen [1] and Mujica [22] for extensive studies of analytic functions on Banach spaces.

2. Conditions of the Unboundedness

Proposition 1.
Let { Q 1 , Q 2 , , Q n , } be a weak*-null sequence of polynomials on X such that Q n = 1 and deg Q 1 deg Q 2 . Then for every strictly increasing sequence of positive integers { k n } , the function
f ( x ) = n = 1 Q n k n ( x )
is a function of unbounded type.
Proof. 
Let x X . Since Q n ( x ) 0 as n , there exists a number m such that | Q n ( x ) | ε for some 0 < ε < 1 , all n > m . Hence the series n = 1 | Q n | n k converges. Thus
f ( x ) = n = 1 Q n k n ( x )
is well-defined for every x X . On the other hand, Q n k n = 1 and so ϱ 0 ( f ) = 1 . Therefore, f is an entire function of unbounded type. □
Let us notice the condition “ { Q n } is a weak*-null sequence of polynomials on X such that Q n = 1 ” is not sufficient to claim that n = 1 Q n ( x ) is a function of unbounded type. For example, if deg Q n = deg Q m for all n , m N , then the series may be divergent.
Throughout the paper we will use the notations a 1 / n for the principal root of a and a n for the multi-valued root function of a .
Proposition 2.
Let { Q 1 , Q 2 , , Q n , } be a sequence of polynomials on X such that Q n = 1 and deg Q 1 < deg Q 2 < . The function
f ( x ) = n = 1 Q n ( x )
is of unbounded type if and only if for every x X
| Q n ( x ) | 1 deg Q n 0 as n .
Proof. 
Let us suppose that { Q n } n = 1 satisfies (3). Then for every 0 < ε < 1 there is a number n 0 N such that for every n > n 0 , | Q n ( x ) | 1 deg Q n < ε . Thus,
n = n 0 + 1 Q n ( x ) n = n 0 + 1 | Q n ( x ) | 1 1 ε < .
Hence, f ( x ) is well-defined for every x X . On the other hand, ϱ 0 ( f ) = 1 and so f is an entire function of unbounded type.
Conversely, let f H ( X ) \ H b ( X ) . Then for every x 0 X , x 0 = 1 the series
f ( λ x 0 ) = n = 1 Q n ( λ x 0 ) = n = 1 λ deg Q n Q n ( x 0 )
converges for every λ C . This implies
| Q n ( x 0 ) | 1 deg Q n 0 as n
and so
| λ | | Q n ( x 0 ) | 1 deg Q n = | Q n ( λ x 0 ) | 1 deg Q n 0 as n .
Since, every vector x X can be represented by x = λ x 0 , x 0 = 1 , λ C , the proposition is proved. □
In [20] the following theorem was proved.
Theorem 1.
Let us suppose that there is a dense subset Ω X and a sequence of polynomials P n P ( n X ) , lim sup n P n 1 / n = c , 0 < c < such that for every z Ω there exists m N with the property that for every y X ,
B P n ( z , , z k , y , , y n k ) = 0
for all k > m and n > k , where B P n is the symmetric n-linear mapping associated with P n . Then
g ( x ) = n = 1 P n ( x ) H ( X ) \ H b ( X ) .
It is not difficult to show that if a sequence of polynomials P n satisfies the conditions of Theorem 1, then it is weak*-null. From Proposition 2 it follows that if P n satisfies the conditions of Theorem 1, then | P n | 1 / n is weak*-null.
For a given sequence of polynomials P = { P 1 , P 2 , , P n , } on X , P n = 1 , deg P n = n , n N we denote by P a multi-valued map from X to defined by
P ( x ) = P 1 ( x ) , P 2 ( x ) , , P n ( x ) n , .
Let I n be polynomials on defined by I n ( z ) = z n n , z = ( z 1 , , z n , ) , n N . The algebra, generated by polynomials { I n } was considered in [7]. Let us fix some evident properties of P .
Proposition 3.
For every sequence of polynomials P = { P n } n = 1 on X , P n = 1 , deg P n = n , n N the following statements hold:
1
The range P ( X ) of X under P is in .
2
P maps the ball r B X into the ball r B , r 0 .
3
If z is in the range of P ( x ) , then P n ( x ) = I n ( z ) .
Proof. 
Since P n = 1 , | P n ( x ) | 1 / n x for every x X . So if z n P n ( x ) n , then z = ( z 1 , , z n , ) and z x X . In addition, I n ( z ) = z n n = P n ( x ) .
Lemma 1.
Let f H ( ) . Then the restriction f 0 of f to c 0 belongs to H b ( c 0 ) .
Proof. 
According to the Aron–Berner result [23], a function f 0 H ( c 0 ) can be extended to an analytic function f on if and only if f 0 H b ( c 0 ) .
We say that the polynomial algebra P P ( X ) supports analytic functions of unbounded type if there exists a function f H P ( X ) \ H b P ( X ) .
Theorem 2.
Let P = { P n } n = 1 be as in Proposition 3.
1
If P maps X onto , then the algebra P P ( X ) does not support analytic functions of unbounded type.
2
If P maps X to c 0 , then
n = 1 P n ( x ) H ( X ) \ H b ( X ) .
3
If there is x X such that P ( x ) \ c 0 , then
n = 1 P n ( x ) H ( X ) .
Proof. 
(1) Let us prove first that P I ( ) does not support analytic functions of unbounded type on , where P I ( ) is P P ( ) for P = I = { I n } n = 1 . Suppose that
f ( z ) = n = 0 f n ( z ) H ( ) \ H b ( ) , and f n P I ( ) .
Since each f n is an algebraic combination of I 1 , I 2 , , I n and I k ( z ) = z k k , we have that f n ( z ) depends of finitely many coordinates z 1 , z 2 , , z n . Thus, there is s = ( s 1 , s 2 , , s n , 0 , 0 ) such that f n = | f n ( s ) | . In other words, the norm of f n in is equal to the norm of the restriction of f n on c 0 . Let f 0 be the restriction of f on c 0 . Hence we have that
ϱ 0 ( f ) = ϱ 0 ( f 0 ) .
So if f is a function of unbounded type, then f 0 is a function of unbounded type. However, it contradicts Lemma 1.
Let f now be an arbitrary function of unbounded type in H P ( X ) . Since all polynomials f n belong to P P ( X ) , there are polynomials q n of n complex variables, n N such that
f n ( x ) = q n ( P 1 ( x ) , P 2 ( x ) , , P n ( x ) ) = q n ( I 1 ( z ) , I 2 ( z ) , , I n ( z ) ) = : Q n ( z )
for every z P ( x ) . Clearly every Q n is an n-homogeneous continuous polynomial. Since P maps bounded sets to bounded sets,
g ( z ) = n = 0 Q n ( z )
must be a function of unbounded type. However, it is impossible because of the first part of the proof.
(2) If z = ( z 1 , z 2 , ) P ( x ) , then
f ( x ) = n = 1 P n ( x ) = n = 1 z n n = n = 1 I n ( z ) H ( c 0 ) .
Thus f is well defined on X and so belongs to H ( X ) . Since P n = 1 , ϱ 0 ( f ) = 1 . Hence, f H ( X ) \ H b ( X ) .
(3) If P ( x ) c 0 for some fixed x X , then there exists a constant c > 0 and a subsequence n k N such that | P n k n k ( x ) | > c for all k . Let us consider the following function of one complex variable
γ ( t ) = n = 1 P n ( t x ) = n = 1 t n P n ( x ) .
The radius of convergence of this series satisfies ϱ 0 ( γ ( t ) ) 1 / c , so if t 0 > 1 / c , then the series
n = 1 P n ( t x )
diverges. Thus it does not belong to H ( X ) .
Remark 1.
Formally, we do not assume in Theorem 2 that X is infinite dimensional. However, any finite dimensional space does not admit entire functions of unbounded type. Thus, if P ( X ) c 0 , then X must be infinite dimensional.
Example 1.
In [19] (see also [20]) it is shown that if P n ( x ) = n ! G n ( x ) , where
G n ( x ) = k 1 < k 2 < < k n x k 1 x k n , x = ( x 1 , x 2 , ) 1 ,
then
f ( x ) = n = 1 P n ( x )
is an entire function of unbounded type on 1 . It is known [13] that G n = 1 / n ! . Thus, Theorem 2 implies that P ( x ) c 0 for every x 1 . The algebra P P ( 1 ) coincides with the algebra of all symmetric polynomials on 1 (see e.g., [12]) and admits another algebraic basis of homogeneous polynomials
F n ( x ) = k = 1 x k n , n = 1 , 2 , .
It is easy to see that 1 F n = 1 and F n ( e 1 ) = 1 for every n N , where e 1 = ( 1 , 0 , 0 , ) . Hence, ( F n ( e 1 ) n ) n = 1 c 0 . However, as we observed, the algebra of symmetric polynomials supports entire functions of unbounded type. Therefore, if c 0 P ( X ) , then n = 1 P n H ( X ) , but P P ( X ) may still support entire functions of unbounded type.
Note that the existence of an isomorphism of H b P ( X ) and H b I ( ) does not imply that P P ( X ) does not support analytic functions of unbounded type.
Example 2.
It is known [7] that there is an isomorphism J : H b I ( c 0 ) H b I ( ) such that J : I n I n but P I ( c 0 ) supports analytic functions of unbounded type. For example
f ( x ) = n = 0 I n ( x ) = n = 0 x n n H ( c 0 ) \ H b ( c 0 ) .
In other words, the isomorphism J can not be extended to an isomorphism between H ( c 0 ) and H ( ) .
Let us recall that a function on L [ 0 , 1 ] is symmetric if it is invariant with respect to measuring and measure preserving automorphisms of the interval [ 0 , 1 ] . The polynomials
R n ( x ) = [ 0 , 1 ] ( x ( t ) ) n d t , x ( t ) L [ 0 , 1 ] , n N
form an algebraic basis in the space of all symmetric polynomials on L [ 0 , 1 ] Thus, the algebra of symmetric polynomials P s ( L [ 0 , 1 ] ) is a partial case of P P ( X ) if X = L [ 0 , 1 ] and P n = R n . In [20] the authors asked: Does an entire symmetric analytic function of unbounded type exist on L [ 0 , 1 ] ? Now we have a negative answer to this question.
Corollary 1.
All entire symmetric functions on L [ 0 , 1 ] are functions of bounded type.
Proof. 
Let
f ( x ) = n = 0 f n ( x )
be an entire symmetric function on L [ 0 , 1 ] . Then each Taylor’s polynomial f n must be symmetric and so can be represented as an algebraic combination of polynomials R 1 , , R n . In [14] is proved that the map
x R 1 ( x ) , R 2 ( x ) , , R n ( x ) n ,
is onto . Thus, by Theorem 2, the algebra of symmetric polynomials on L [ 0 ; 1 ] does not support entire functions of unbounded type. Hence f ( x ) is of bounded type. □

3. Lineability of H P ( X ) \ H b P ( X )

Theorem 3.
If P P ( X ) supports analytic functions of unbounded type, then for every f H P ( X ) \ H b P ( X ) there exists an infinite dimensional linear subspace in H P ( X ) which consists (excepting zero) of analytic functions of unbounded type and contains f .
Proof. 
Let
f ( x ) = n = 0 f n ( x ) H P ( X ) \ H b P ( X ) .
Then ϱ 0 ( f ) = r for some 0 < r < . Let δ > 0 and δ < c = 1 / r . Denote by N 0 the subset of all nonnegative integers Z + such that f m 1 / m < δ . In other words, if n Z + \ N 0 , then δ f n 1 / n c , that is, for every subsequence { n k } k = 1 Z + \ N 0
r ϱ 0 k = 1 f n k 1 δ < .
Let
Z + = k = 0 N k
be a partition of the set Z + into infinite many disjoint subsets N k so that | N k | = for k > 0 and N 0 is the finite or infinite set defined above. Let N 0 = ( j 1 , j 2 , ) . We denote
N k = N k { j k } if k | N 0 | N k otherwise .
Thus N = ( N 1 , N 2 , ) is a partition of Z + into infinitely many disjoint subsets of infinite cardinality.
For any bounded sequence of numbers a = ( a 1 , a 2 , ) we assign a function
g a ( x ) = k = 1 a k j N k f j ( x ) .
For every a , a 0 the function g a is well defined on X and is of unbounded type. Indeed, for the Taylor polynomials ( g a ) n of g a we have
| ( g a ) m ( x ) | 1 / m a 1 / m | f m ( x ) | 1 / m 0 as m .
Moreover, since a 0 there is a number j such that a j 0 . Thus
lim sup m ( g a ) m 1 / m lim sup m N j ( g a ) m 1 / m = lim sup m N j | a j | 1 / m f m 1 / m δ
and so
ϱ 0 ( g a ) 1 δ < .
Since f H P ( X ) , g a H P ( X ) for every a . On the other hand, the set which depends on the choice of f , δ , and N
V f , δ , N = { g a : a }
is a linear space because g a + λ g b = g a + λ b for all a , b , λ C . Clearly f = g a for a = ( 1 , 1 , ) .
Note that the subspace V f , δ , N is not maximal. Indeed, if N is a subpartition of N , then V f , δ , N V f , δ , N . It is easy to deduce by the Zorn Lemma that there is a maximal linear subspace in H ( x ) \ H b ( X ) containing a given function of unbounded type. So we have the following corollary.
Corollary 2.
Let X be an infinite dimensional Banach space. The set H ( x ) \ H b ( X ) can be represented as a union of infinite dimensional linear subspaces (without the origin).
It is known (see [5]) that for every infinite dimensional Banach space X there are sequences { e k } k = 1 X and { φ k } k = 1 X * such that
  • lim k φ k ( x ) = 0 for every x X ,
  • φ k = 1 , k N ,
  • sup k N e k < ,
  • φ k ( e j ) = δ k j , where k , j N and δ k j is the Kronecker delta.
In Theorem 2 of [5], actually it was proved that if the functionals φ k are as above, then for every strictly increasing sequence of prime numbers { a j } j = 1 the following functions
f j = k = 1 a j k φ k k
generate an infinite dimensional algebra A such that every nonzero element h in A is an entire function of unbounded type and sup n | h ( e n ) | = . In particular, it is so if X = c 0 , { e k } k = 1 is the basis in c 0 and { φ k } k = 1 is the sequence of coordinate functionals.
Theorem 4.
Let { P n } n = 1 , P n = 1 , n N be a sequence of n-homogeneous polynomials on a Banach space X such that P ( X ) c 0 and there exists a sequence { z k } k = 1 in X such that sup k z k < and P n ( z k ) = δ n k . Then for every strictly increasing sequence of prime numbers { a j } j = 1 the functions
g j = k = 1 a j k P k
generate an infinite-dimensional algebra B such that every nonzero element in u B is an entire function of unbounded type and sup n | u ( z n ) | = .
Proof. 
Let us consider the algebra A generated by functions
f j ( x ) = k = 1 a j k φ k ( x ) k = k = 1 a j k x k k , where x = n = 1 x n e n c 0 and j N .
Note that φ k P ( z k ) k = e k for every evaluation of P ( z k ) . From Theorem 2 of [5] mentioned above, it follows that if h A and h 0 , then sup n | h ( e n ) | = . Every function u B can be represented as a finite algebraic combination of functions g j ,
u ( x ) = j 1 < < j m < N λ j 1 j m g j 1 ( x ) p j 1 g j m ( x ) p j m = j 1 < < j m < N λ j 1 j m f j 1 P ( x ) p j 1 f j m P ( x ) p j m ,
where N N , λ j 1 j m C \ { 0 } and p j k N for all j k . In other words,
u ( x ) = h P ( x ) for some h A and u ( z k ) = h ( e k ) .
Hence, sup n | u ( z n ) | = and so all functions in B \ { 0 } are unbounded on some bounded subsets. On the other hand, since P ( X ) c 0 , all functions g j by Theorem 2 are well-defined on X and so their finite algebraic combinations are well-defined on X too. Thus, B \ { 0 } H ( X ) \ H b ( X ) .
Corollary 3.
Let P n ( x ) = n ! G n ( x ) , x 1 be the basis of symmetric polynomials on 1 as in Example 1. Then for every strictly increasing sequence of prime numbers { a j } j = 1 the functions
g j = k = 1 ( 1 ) k + 1 a j k P k
generate an infinite-dimensional subalgebra in the algebra of symmetric analytic functions comprising (excepting zero) of analytic functions of unbounded type.
Proof. 
We need to construct a sequence { z n } n = 1 , biorthogonal to { P k } k = 1 . Let us define
z n = 1 ( n ! ) 1 / n ( α 1 , , α n , 0 , 0 , ) ,
where { α 1 , , α n } = 1 n are the roots of the unity. From the Vieta formulas it follows that
P n ( z n ) = n ! n ! α 1 α n = ( 1 ) n + 1
and
P k ( z n ) = n ! ( n ! ) k / n G k ( z n ) = 0 if k < n .
If k > n , then G k ( z n ) = 0 because z n has only n nonzero coordinates. Thus P k ( z n ) = δ k n . In addition, using the Stirling formula, we can estimate
z n = n ( n ! ) 1 / n n e n n n 1 / n = e < .
Thus, we can apply Theorem 4 for the sequence of polynomials { ( 1 ) k + 1 P k } k = 1 .

4. Discussion and Conclusions

One of the main results of the paper is that not every infinitely generated algebra of polynomials on a Banach space supports entire functions of unbounded type. We found some necessary conditions and some sufficient conditions of this property but we have no conditions which are simultaneously necessary and sufficient. The mapping P allows us to reduce this question to polynomial algebras on subsets of . However, we do not know whether P I ( c ) supports entire functions of unbounded type, where c is the space of all convergent sequences. Moreover, we do not know if there exists a supersymmetric entire function of unbounded type (supersymmetric analytic functions and their properties were considered in [16]).
The theorems on linear subspaces and subalgebras are interesting in the context of the general question about linear structures in nonlinear sets [24] and are extensions of Lopez–Salazar Codes’ results [5] for more special cases.

Author Contributions

A.Z. and A.H. contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Research Foundation of Ukraine, 2020.02/0025, 0120U103996.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Zagorodnyuk, A.; Hihliuk, A. Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability. Axioms 2021, 10, 150. https://doi.org/10.3390/axioms10030150

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Zagorodnyuk A, Hihliuk A. Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability. Axioms. 2021; 10(3):150. https://doi.org/10.3390/axioms10030150

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Zagorodnyuk, Andriy, and Anna Hihliuk. 2021. "Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability" Axioms 10, no. 3: 150. https://doi.org/10.3390/axioms10030150

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