scholarly journals Symmetric polynomials on the Cartesian power of the real Banach space $L_\infty[0,1]$

2020 ◽  
Vol 53 (2) ◽  
pp. 192-205
Author(s):  
T. Vasylyshyn ◽  
A. Zagorodnyuk
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Analytic Functions ◽  
Symmetric Functions ◽  
Real Banach Space ◽  
Symmetric Polynomials ◽  
The Real ◽  
Cartesian Power ◽  
The Family ◽  
Algebraic Basis

We construct an algebraic basis of the algebra of symmetric (invariant under composition of the variable with any measure preserving bijection of $[0,1]$) continuous polynomials on the $n$th Cartesian power of the real Banachspace $L_^{(\mathbb{R})}\infty[0,1]$ of Lebesgue measurable essentially bounded real valued functions on $[0,1].$ Also we describe the spectrum of the Fr\'{e}chet algebra $A_s(L_^{(\mathbb{R})}\infty[0,1])$ of symmetric real-valued functions on the space $L_^{(\mathbb{R})}\infty[0,1]$, which is the completion of the algebra of symmetric continuous real-valued polynomials on  $L_^{(\mathbb{R})}\infty[0,1]$ with respect to the family of norms of uniform convergence of complexifications of polynomials. We show that $A_s(L_^{(\mathbb{R})}\infty[0,1])$ contains not only analytic functions. Results of the paper can be used for investigations of algebras of symmetric functions on the $n$th Cartesian power of the Banach space $L_^{(\mathbb{R})}\infty[0,1]$.

scholarly journals Algebras of symmetric analytic functions on Cartesian powers of Lebesgue integrable in a power $p\in [1,+\infty)$ functions

2021 ◽  
Vol 13 (2) ◽  
pp. 340-351
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Banach Space ◽  
Analytic Functions ◽  
Entire Functions ◽  
Complex Banach Space ◽  
Cartesian Power ◽  
Entire Complex ◽  
Continuous Complex ◽  
Fréchet Algebra ◽  
Algebraic Basis ◽  
Complex Valued

The work is devoted to the study of Fréchet algebras of symmetric (invariant under the composition of every of components of its argument with any measure preserving bijection of the domain of components of the argument) analytic functions on Cartesian powers of complex Banach spaces of Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ and on the semi-axis. We show that the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,1]$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ is isomorphic to the Fréchet algebra of all analytic entire functions on $\mathbb C^m,$ where $m$ is the cardinality of the algebraic basis of the algebra of all symmetric continuous complex-valued polynomials on this Cartesian power. The analogical result for the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,+\infty)$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the semi-axis $[0,+\infty)$ is proved.

scholarly journals Symmetric functions on spaces $\ell_p(\mathbb{{R}}^n)$ and $\ell_p(\mathbb{{C}}^n)$

2020 ◽  
Vol 12 (1) ◽  
pp. 5-16
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Linear Combination ◽  
Banach Spaces ◽  
Symmetric Functions ◽  
Symmetric Polynomial ◽  
Complex Numbers ◽  
Symmetric Polynomials ◽  
Algebraic Basis

This work is devoted to study algebras of continuous symmetric, that is, invariant with respect to permutations of coordinates of its argument, polynomials and $*$-polynomials on Banach spaces $\ell_p(\mathbb{R}^n)$ and $\ell_p(\mathbb{C}^n)$ of $p$-power summable sequences of $n$-dimensional vectors of real and complex numbers resp., where $1\leq p < +\infty.$ We construct the subset of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$ such that every continuous symmetric polynomial on the space $\ell_p(\mathbb{R}^n)$ can be uniquely represented as a linear combination of products of elements of this set. In other words, we construct an algebraic basis of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n).$ Using this result, we construct an algebraic basis of the algebra of all continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n).$ Results of the paper can be used for investigations of algebras, generated by continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n),$ and algebras, generated by continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n).$

scholarly journals The polarization constant of finite dimensional complex spaces is one

2021 ◽  
pp. 1-19
Author(s):  
VERÓNICA DIMANT ◽  
DANIEL GALICER ◽  
JORGE TOMÁS RODRÍGUEZ
Keyword(s):  
Banach Space ◽  
Analytic Functions ◽  
Homogeneous Polynomial ◽  
Complex Space ◽  
Necessary Conditions ◽  
Best Constant ◽  
The Real ◽  
Complex Spaces ◽  
Finite Dimensional ◽  
Image Position

Abstract The polarization constant of a Banach space X is defined as \[{\text{c}}(X){\text{ }}{\text{ }}\mathop {\lim }\limits_{k \to \infty } {\text{ }}\sup {\text{c}}{(k,X)^{\frac{1}{k}}},\] where \[{\text{c}}(k,X)\] stands for the best constant \[C > 0\] such that \[\mathop P\limits^ \vee \leqslant CP\] for every k-homogeneous polynomial \[P \in \mathcal{P}{(^k}X)\] . We show that if X is a finite dimensional complex space then \[{\text{c}}(X) = 1\] . We derive some consequences of this fact regarding the convergence of analytic functions on such spaces. The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak’s complexification procedure. We also study some other properties connected with polarization. Namely, we provide necessary conditions related to the geometry of X for \[c(2,X) = 1\] to hold. Additionally we link polarization constants with certain estimates of the nuclear norm of the product of polynomials.

scholarly journals A note on approximation of continuous functions on normed spaces

2020 ◽  
Vol 12 (1) ◽  
pp. 107-110
Author(s):  
M.A. Mytrofanov ◽  
A.V. Ravsky
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Normed Space ◽  
Analytic Functions ◽  
Real Banach Space ◽  
Complex Banach Space ◽  
Continuous Functions ◽  
Normed Spaces ◽  
Approximation Of Continuous Functions

Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions, which are analytic on open subsets of $X$. Also we prove that each continuous function to a complex Banach space from a complex separable normed space, admitting a separating $*$-polynomial, can be uniformly approximated by $*$-analytic functions.

On a Multivalued Differential Problem

2003 ◽  
Vol 13 (07) ◽  
pp. 1877-1882 ◽  
Author(s):  
Andrzej Smajdor
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Real Banach Space ◽  
Convex Compact ◽  
Closed Convex Cone ◽  
Continuous Linear ◽  
Differential Problem ◽  
Linear Set ◽  
Nonempty Convex ◽  
The Family

Let K be a closed convex cone with the nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. Assume that two continuous linear set-valued functions G, Ψ : K → cc(K) are given. The following problem is considered: [Formula: see text] for t ≥ 0 and x ∈ K, where DtΦ(t, x) denotes the Hukuhara derivative of Φ(t, x). with respect to t.

Extremal Problems for the Classes SR-p and TR-p

1990 ◽  
Vol 42 (4) ◽  
pp. 619-645
Author(s):  
Walter Hengartner ◽  
Wojciech Szapiel
Keyword(s):  
Uniform Convergence ◽  
Linear Space ◽  
Unit Disk ◽  
Real Axis ◽  
Analytic Functions ◽  
Dual Space ◽  
Extremal Problems ◽  
Space Of Analytic Functions ◽  
The Real

Let H(D) be the linear space of analytic functions on a domain D of ℂ endowed with the topology of locally uniform convergence and let H‘(D) be the topological dual space of H(D). For domains D which are symmetric with respect to the real axis we use the notation Furthermore, denote by S the set of all univalent mappings f defined on the unit disk Δ which are normalized by f (0) = 0 and f‘(0) =1.

SOME PROPERTIES OF ELEMENTARY SYMMETRIC POLYNOMIALS ON THE CARTESIAN SQUARE OF THE COMPLEX BANACH SPACE L∞ [ 0,1 ]

2018 ◽  
pp. 9-16
Author(s):  
T. V. Vasylyshyn
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Symmetric Polynomials ◽  
Bounded Functions ◽  
Elementary Symmetric Polynomials ◽  
Algebraic Basis

We construct the element of the Cartesian square of the complex Banach space L ∞ [ 0,1 ] of all Lebesgue measurable essentially bounded functions on [ 0,1 ] by the predefined values of elementary symmetric polynomials on this element. Results of this work can be applied to the investigation of an algebraic basis of the algebra of continuous symmetric polynomials on the Cartesian square of the complex Banach space L ∞ [ 0,1 ] .

scholarly journals Lipschitz symmetric functions on Banach spaces with symmetric bases

2021 ◽  
Vol 13 (3) ◽  
pp. 727-733
Author(s):  
M.V. Martsinkiv ◽  
S.I. Vasylyshyn ◽  
T.V. Vasylyshyn ◽  
A.V. Zagorodnyuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Symmetric Functions ◽  
Large Family ◽  
Symmetric Polynomials ◽  
Symmetric Basis ◽  
Several Variables ◽  
Symmetric Bases ◽  
Unbounded Subset

We investigate Lipschitz symmetric functions on a Banach space $X$ with a symmetric basis. We consider power symmetric polynomials on $\ell_1$ and show that they are Lipschitz on the unbounded subset consisting of vectors $x\in \ell_1$ such that $|x_n|\le 1.$ Using functions $\max$ and $\min$ and tropical polynomials of several variables, we constructed a large family of Lipschitz symmetric functions on the Banach space $c_0$ which can be described as a semiring of compositions of tropical polynomials over $c_0$.

Commutativity of set-valued cosine families

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Andrzej Smajdor ◽  
Wilhelmina Smajdor
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Real Banach Space ◽  
Nonempty Interior ◽  
Convex Compact ◽  
Closed Convex Cone ◽  
Cosine Family ◽  
Cosine Families ◽  
Nonempty Convex ◽  
The Family

AbstractLet K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. If {F t: t ≥ 0} is a regular cosine family of continuous additive set-valued functions F t: K → cc(K) such that x ∈ F t(x) for t ≥ 0 and x ∈ K, then $F_t \circ F_s (x) = F_s \circ F_t (x)fors,t \geqslant 0andx \in K$.

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