Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases

This paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on a Banach space X with a symmetric basis. We obtain some positive results for the case when X admits a separating polynomial using a symmetrization operator. However, even in this case, there is a counter-example because the symmetrization operator is well defined only on a narrow, proper subspace of the space of analytic functions on X. For X=c0, we introduce ε-slice G-analytic functions that have a behavior similar to G-analytic functions at points x∈c0 such that all coordinates of x are greater than ε, and we prove a theorem on approximations of uniformly continuous functions on c0 by ε-slice G-analytic functions.

Approximation of Analytic Functions by Shifts of Certain Compositions

In the paper, we obtain universality theorems for compositions of some classes of operators in multidimensional space of analytic functions with a collection of periodic zeta-functions. The used shifts of periodic zeta-functions involve the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function.

Power series of several variables with condition of logarithmical convexity

We obtain a new version of Hardy theorem about power series of several variables reciprocal to the power series with positive coefficients. We prove that if the sequence {as} = as1,s2,...,sn, ||s|| ≥ K satisfies condition of logarithmically convexity and the first coefficient a0 is sufficiently large then reciprocal power series has only negative coefficients {bs} = bs1,s2,...,sn, except b0,0,...,0 for any K. The classical Hardy theorem corresponds to the case K = 0, n = 1. Such results are useful in Nevanlinna - Pick theory. For example, if function k(x, y) can be represented as power series Σn≥0 an(x-y)n, an > 0, and reciprocal function 1/k(x,y) can be represented as power series Σn≥0 bn(x-y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna-Pick property. The reproducing kernel 1/1-x-y of the classical Hardy space H2(D) is a prime example for our theorems.

Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc

For any $\alpha \in \mathbb {R},$ we consider the weighted Taylor shift operators $T_{\alpha }$ acting on the space of analytic functions in the unit disc given by $T_{\alpha }:H(\mathbb {D})\rightarrow H(\mathbb {D}),$ $ \begin{align*}f(z)=\sum_{k\geq 0}a_{k}z^{k}\mapsto T_{\alpha}(f)(z)=a_1+\sum_{k\geq 1}\Big(1+\frac{1}{k}\Big)^{\alpha}a_{k+1}z^{k}.\end{align*}$ We establish the optimal growth of frequently hypercyclic functions for $T_\alpha $ in terms of $L^p$ averages, $1\leq p\leq +\infty $ . This allows us to highlight a critical exponent.

Asymptotic lower bound for the radius of spatial analyticity to solutions of KdV equation

It is shown that the uniform radius of spatial analyticity [Formula: see text] of solutions at time [Formula: see text] to the KdV equation cannot decay faster than [Formula: see text] as [Formula: see text] given initial data that is analytic with fixed radius [Formula: see text]. This improves a recent result of Selberg and da Silva, where they proved a decay rate of [Formula: see text] for arbitrarily small positive [Formula: see text]. The main ingredients in the proof are almost conservation law for the solution to the KdV equation in space of analytic functions and space-time dyadic bilinear [Formula: see text] estimates associated with the KdV equation.

Characterizations of Bloch-Type Spaces of Harmonic Mappings

We study the Banach space BHα (α>0) of the harmonic mappings h on the open unit disk D satisfying the condition supz∈D⁡(1-z2)α(hzz+hz¯z)<∞, where hz and hz¯ denote the first complex partial derivatives of h. We show that several properties that are valid for the space of analytic functions known as the α-Bloch space extend to BHα. In particular, we prove that for α>0 the mappings in BHα can be characterized in terms of a Lipschitz condition relative to the metric defined by dH,α(z,w)=sup⁡{hz-hw:h∈BHα,hBHα≤1}. When α>1, the harmonic α-Bloch space can be viewed as the harmonic growth space of order α-1, while for 0<α<1, BHα is the space of harmonic mappings that are Lipschitz of order 1-α.

A Weighted Universality Theorem for L-Functions of Elliptic Curves

In the paper, a short survey on universality results for L-functions of elliptic curves over the field of rational numbers is given and weighted universality theorem is proven. All stated universality theorems are of continuous type. The proof of the universality for L-functions of elliptic curves is based on a limit theorem in the sense of weak convergence of probability measures in the space of analytic functions.

ON DISCRETE VALUE DISTRIBUTION OF CERTAIN COMPOSITIONS

In the paper, we obtain universality theorems and a lower estimate for the number of zeros for the composition F ζ (s, α; a, b) , where F is an operator in the space of analytic functions satisfying the Lipschitz type condition, and ζ (s, α; a, b) is a collection consisting of periodic and periodic Hurwitz zeta-functions.