Analytic Functions Of Several Variables Research Articles

A multidimensional Hermite-Gauss sampling formula for analytic functions of several variables

A multidimensional Hermite-Gauss sampling formula for analytic functions of several variables

Countably Generated Algebras of Analytic Functions on Banach Spaces

In the paper, we study various countably generated algebras of entire analytic functions on complex Banach spaces and their homomorphisms. Countably generated algebras often appear as algebras of symmetric analytic functions on Banach spaces with respect to a group of symmetries, and are interesting for their possible applications. Some conditions of the existence of topological isomorphisms between such algebras are obtained. We construct a class of countably generated algebras, where all normalized algebraic bases are equivalent. On the other hand, we find non-isomorphic classes of such algebras. In addition, we establish the conditions of the hypercyclicity of derivations in countably generated algebras of entire analytic functions of the bounded type. We use methods from the theory of analytic functions of several variables, the theory of commutative Fréchet algebras, and the theory of linear dynamical systems.

Dominating Sets in Bergman Spaces on Strongly Pseudoconvex Domains

We obtain local estimates, also called propagation of smallness or Remez-type inequalities, for analytic functions in several variables. Using Carleman estimates, we obtain a three sphere-type inequality, where the outer two spheres can be any sets satisfying a boundary separation property, and the inner sphere can be any set of positive Lebesgue measure. We apply this local result to characterize the dominating sets for Bergman spaces on strongly pseudoconvex domains in terms of a density condition or a testing condition on the reproducing kernels. Our methods also yield a sufficient condition for arbitrary domains and lower-dimensional sets.

Wiman’s Type Inequality in Multiple-Circular Domain

In the paper we prove for the first time an analogue of the Wiman inequality in the class of analytic functions f∈A0p(G) in an arbitrary complete Reinhard domain G⊂Cp, p∈N represented by the power series of the form f(z)=f(z1,⋯,zp)=∑‖n‖=0+∞anzn with the domain of convergence G. We have proven the following statement: If f∈Ap(G) and h∈Hp, then for a given ε=(ε1,…,εp)∈R+p and arbitrary δ>0 there exists a set E⊂|G| such that ∫E∩Δεh(r)dr1⋯drpr1⋯rp<+∞ and for all r∈Δε∖E we have Mf(r)≤μf(r)(h(r))p+12lnp2+δh(r)lnp2+δ{μf(r)h(r)}∏j=1p(lnerjεj)p−12+δ. Note, that this assertion at p=1,G=C,h(r)≡const implies the classical Wiman–Valiron theorem for entire functions and at p=1, the G=D:={z∈C:|z|<1},h(r)≡1/(1−r) theorem about the Kővari-type inequality for analytic functions in the unit disc D; p>1 implies some Wiman’s type inequalities for analytic functions of several variables in Cn×Dk, n,k∈Z+,n+k∈N.

On the convergence of multidimensional S-fractions with independent variables

The paper investigates the convergence problem of a special class of branched continued fractions, i.e. the multidimensional S-fractions with independent variables, consisting of \[\sum_{i_1=1}^N\frac{c_{i(1)}z_{i_1}}{1}{\atop+}\sum_{i_2=1}^{i_1}\frac{c_{i(2)}z_{i_2}}{1}{\atop+} \sum_{i_3=1}^{i_2}\frac{c_{i(3)}z_{i_3}}{1}{\atop+}\cdots,\] which are multidimensional generalizations of S-fractions (Stieltjes fractions). These branched continued fractions are used, in particular, for approximation of the analytic functions of several variables given by multiple power series. For multidimensional S-fractions with independent variables we have established a convergence criterion in the domain \[H=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|\arg(z_k+1)|<\pi,\; 1\le k\le N\right\}\] as well as the estimates of the rate of convergence in the open polydisc \[Q=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|z_k|<1,\;1\le k\le N\right\}\] and in a closure of the domain $Q.$

Polynomial Approximation of Anisotropic Analytic Functions of Several Variables

Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on Taylor expansions, and study their approximation by finite dimensional polynomial spaces $${{\mathcal {P}}}_\Lambda $$ described by lower sets $$\Lambda $$ . Given a budget n for the dimension of $${{\mathcal {P}}}_\Lambda $$ , we prove that certain lower sets $$\Lambda _n$$ , with cardinality n, provide a certifiable approximation error that is in a certain sense optimal, and that these lower sets have a simple definition in terms of simplices. Our main goal is to obtain approximation results when the number of variables d is large and even infinite, and so we concentrate almost exclusively on the case $$d=\infty $$ . We also emphasize obtaining results which hold for the full range $$n\ge 1$$ , rather than asymptotic results that only hold for n sufficiently large. In applications, one typically wants n small to comply with computational budgets.

Functions analytic in the unit ball having bounded $L$-index in a direction

We propose a generalization of a concept of bounded index for analytic functions in the unit ball. Use of directional derivatives gives us a possibility to deduce the necessary and sufficient conditions of boundedness of $L$-index in a direction for analytic functions of several variables, namely, we obtain an analog of Hayman's theorem and a logarithmic criteria for this class. The criteria describe the behavior of the directional logarithmic derivative outside the zero set and a uniform distribution of zeros in some sense. The criteria are useful for studying analytic solutions of partial differential equations and estimating their growth. We present a scheme of this application.

Multiplier projections on spaces of real analytic functions in several variables

Let be open with . We characterize the sets having the following property: for every real analytic function f on with Taylor expansion at zero, the series is also the Taylor expansion at zero of some real analytic function on . This result gives a characterization of the idempotents in the algebra of Hadamard-type operators on the space of all real analytic functions , i.e. operators with all monomials being eigenvectors. In many cases, we also describe the multiplicative functionals on and the (continuous) algebra homomorphisms . We show that the algebra is never locally m-convex and in many cases it is not a Q-algebra.

Interpolation of holomorphic functions and surjectivity of Taylor coefficient multipliers

We prove criteria for global analytic solvability for some Euler type partial differential equations — a class of linear partial differential equations with variable (polynomial) coefficients. This is based on a very general Mellin type theory developed in the present paper which is valid for analytic functionals (in particular, distributions of compact support) substituting the classical Mellin transform. As a consequence, we get a characterization of moment sequences of analytic functionals by interpolation of holomorphic functions satisfying some restrictive growth conditions. This allows to study the so-called Hadamard type multipliers on real analytic functions of several variables, i.e., operators for which monomials are eigenvectors. We characterize multiplier sequences (i.e., sequences of eigenvalues of multipliers) by interpolation and describe the image of some multipliers, especially, the image of the principal examples of Hadamard multipliers: Euler type partial differential operators. We also get some results showing that the image remains unchanged after some perturbation of the original Hadamard type operator.

Hadamard type operators on spaces of real analytic functions in several variables

We consider multipliers on the space of real analytic functions of several variables A ( Ω ) , Ω ⊂ R d open, i.e., linear continuous operators for which all monomials are eigenvectors. If zero belongs to Ω these operators are just multipliers on the sequences of Taylor coefficients at zero. In particular, Euler differential operators of arbitrary order are multipliers. We represent all multipliers via a kind of multiplicative convolution with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals. Moreover, we represent multipliers via suitable holomorphic functions with Laurent coefficients equal to the eigenvalues of the operator. We identify in some standard cases what topology should be put on the suitable space of analytic functionals in order that the above mentioned isomorphism becomes a topological one when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on A ( Ω ) . We also characterize in the same cases when the discovered topology coincides with the classical topology of bounded convergence on the space of analytic functionals. We provide several examples of multipliers and show surjectivity results for multipliers on A ( Ω ) if Ω ⊂ R + d .

Generators for Rings of Compactly Supported Distributions

Let \({{\tt C}}\) denote a closed convex cone in \({\mathbb R^d}\) with apex at 0. We denote by \({\mathcal E'({\tt C})}\) the set of distributions on \({\mathbb R^d}\) having compact support contained in \({{\tt C}}\). Then \({\mathcal E'({\tt C})}\) is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on \({\widehat{f}_1,\dots, \widehat{f}_n}\) for \({f_1,\dots ,f_n \in \mathcal E'({\tt C})}\) to generate the ring \({\mathcal E'({\tt C})}\). (Here \({\widehat{\;\cdot\;}}\) denotes Fourier-Laplace transformation.) This result is an application of a general result on rings of analytic functions of several variables by Lars Hörmander. En route we answer an open question posed by Yutaka Yamamoto.

Invariant subspaces of finite codimension in Banach spaces of analytic functions

We give a characterization of invariant subspaces of finite codimension in Banach spaces of vector-valued analytic functions in several variables, where invariant refers to invariance under multiplication by any polynomial. We obtain very weak conditions under which our characterization applies, that unifies and improves a number of previous results. In the vector-valued case, the results are new even for one complex variable. As a concrete application in several variables, we consider the Bergman space on a strictly pseudo-convex domain, and we improve previous results (assuming C ∞ -boundary) to the case of C 2 -boundary.

Bernstein–Walsh Type Theorems for Real Analytic Functions in Several Variables

The aim of this paper is to extend the classical maximal convergence theory of Bernstein and Walsh for holomorphic functions in the complex plane to real analytic functions in ℝN. In particular, we investigate the polynomial approximation behavior for functions F:L→ℂ, L={(Re z,Im z):z∈K}, of the structure \(F=g\overline{h}\), where g and h are holomorphic in a neighborhood of a compact set K⊂ℂN. To this end the maximal convergence number ρ(Sc,f) for continuous functions f defined on a compact set Sc⊂ℂN is connected to a maximal convergence number ρ(Sr,F) for continuous functions F defined on a compact set Sr⊂ℝN. We prove that ρ(L,F)=min {ρ(K,h)),ρ(K,g)} for functions \(F=g\overline{h}\) if K is either a closed Euclidean ball or a closed polydisc. Furthermore, we show that min {ρ(K,h)),ρ(K,g)}≤ρ(L,F) if K is regular in the sense of pluripotential theory and equality does not hold in general. Our results are based on the theory of the pluricomplex Green’s function with pole at infinity and Lundin’s formula for Siciak’s extremal function Φ. A properly chosen transformation of Joukowski type plays an important role.

Uniqueness of polynomial canonical representations

Let $P(z)$ and $Q(y)$ be polynomials of the same degree $k \geq 1$ in the complex variables $z$ and $y$, respectively. In this extended abstract we study the non-linear functional equation $P(z)=Q(y(z))$, where $y(z)$ is restricted to be analytic in a neighborhood of $z=0$. We provide sufficient conditions to ensure that all the roots of $Q(y)$ are contained within the range of $y(z)$ as well as to have $y(z)=z$ as the unique analytic solution of the non-linear equation. Our results are motivated from uniqueness considerations of polynomial canonical representations of the phase or amplitude terms of oscillatory integrals encountered in the asymptotic analysis of the coefficients of mixed powers and multivariable generating functions via saddle-point methods. Uniqueness shall prove important for developing algorithms to determine the Taylor coefficients of the terms appearing in these representations. The uniqueness of Levinson's polynomial canonical representations of analytic functions in several variables follows as a corollary of our one-complex variables results.

Linear dependence of quotients of analytic functions of several variables with the least subcollection of generalized Wronskians

We study linear dependence in the case of quotients of analytic functions in several variables (real or complex). We identify the least subcollection of generalized Wronskians whose identical vanishing is sufficient for linear dependence. Our proof admits a straight-forward algebraic generalization and also constitutes an alternative proof of the previously known result that the identical vanishing of the whole collection of generalized Wronskians implies linear dependence. Motivated by the structure of this proof, we introduce a method for calculating the space of linear relations. We conclude with some reflections about this method that may be promising from a computational point of view.

L2-Estimates and Existence Theorems for Generalized Analytic Functions in Several Variables

We consider the generalized Cauchy–Riemann system with nonlinear terms in an arbitrary domain of the complex space. Under some natural conditions on the coefficients and compatibility conditions, we prove solvability of this system in the space of locally square integrable functions.

Banach envelopes of vector valued H p spaces

In the paper we describe the Banach envelopes of Hardy spaces of analytic functions of several variables on polydiscs taking values in quasi-Banach spaces.

On positivity of analytic matrix functions in polydisks

We study analytic functions of several variables in the Korányi class such that, when applied to a contractive tuple of doubly commuting matrices, a positive definite matrix results. A criterion is given for a finite set to have the property that every measure supported on the set yields, via an integral formula, a function as above.

On the Weierstrass Preparation Theorem

An analytic function of several variables is considered. It is assumed that the function vanishes at some point. According to the Weierstrass preparation theorem, in the neighborhood of this point the function can be represented as a product of a nonvanishing analytic function and a polynomial in one of the variables. The coefficients of the polynomial are analytic functions of the remaining variables. In this paper we construct a method for finding the nonvanishing function and the coefficients of the polynomial in the form of Taylor series whose coefficients are found from an explicit recursive procedure using the derivatives of the initial function. As an application, an explicit formula describing a bifurcation diagram locally up to second-order terms is derived for the case of a double root.

The singular values of the imbedding operators of some classes of analytic functions of several variables

We discuss three results. The first exhibits the order of decrease of the s-values as a function of the C R-dimension of a compact set on which we approximate the class of analytic functions being studied. The second is an asymptotic formula for the case when the domain of analyticity and the compact set are Reinhart domains. The third is the computation of the s-values of a special operator that is of interest for approximation theory on one-dimensional manifolds. Bibliography: 6 titles.