Approximation of ultra-differentiable functions by polynomials and entire functions

This article deals with ultra-differentiable functions in the sense of A. Beurling. Quasi-analytic classes, in particular the real analytic, are included. A function f is shown to be -ultra-differentiable in the neighbourhood of a compact set K if and only if f can be approximated with a certain speed by holomorphic functions of exponential type n or polynomials of degree n for n. The speed is measured in terms of the weight .     

Approximation by harmonic functions on closed subsets of Riemann surfaces

Given continuous functionsu and ∈ on a closed subsetF of a Riemann surface, we seek a harmonic functionv on the surface (possibly with logarithmic singularities) such that |u−v|<∈ onF. Research supported in part by NSERC—Canada and FCAR—Quebec.     

Approximation by entire functions in the mean on the real line

Approximations in a locally integral norm of functions representable as a convolution on the whole line by means of Fourier operators are studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 121–125, January, 1991.     

Approximation numbers for composition operators on spaces of entire functions

We study the degree of compactness of composition operators $C_{\phi }$ acting on weighted Hilbert spaces of entire functions, which include (i) the space of entire Dirichlet series, (ii) the space of entire power series, and (iii) the Fock space (we must have $\phi \left(z\right)=az+b$, and it is known that $C_{\phi }$ is compact if and only if $\left|a\right|<1$). More precisely, the sequence $\left(a_n\right)$ of approximation numbers of $C_{\phi }$ is investigated: for (i), we give the exact formula for $\left(a_n\right)$, while for (ii) and (iii) we give upper and lower estimates for $a_n$, showing that $a_n$ behaves like ${\left|a\right|}^n$ up to a subexponential factor. In particular, $C_{\phi }$ belongs to all Schatten classes $S_p,p>0$ as soon as it is compact.     

The growth on a positive ray of entire functions with real Taylor coefficients

It is proved that if the Taylor coefficients of an entire transcendental function change sign sufficiently infrequently, then such a function increases on a positive ray in the same way as in the whole plane.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 577–588, October, 1973.     

Approximation by polynomials on star-shaped subsets of Cn

It is proved that every function holomorphic in a star-shaped domain in Cⁿ can be approximated by polynomials uniformly on compact subsets.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 55–60, July, 1973.     

Approximation of functions in\widetilde{S_1^\Gamma L} S 1 r H by entire functionsS 1 r H by entire functions

This paper is a continuation of [1], we study the approximation of the function classes , S ₁ ^Λ H in L_q metric (1≤q<∞) by entire functions whose spectrals lie in step hyperbolic crosses and obtain the asymptotic estimates of these quantities. In Memory of Professor M. T. Cheng Supported by Beijing Natural Science Foundation.     

On a system of entire functions of class A which is biorthogonal to a lacunar power system on the ray

We construct a system of entire functions of exponential type of class A biorthogonal with weight to some power system on the ray. The indicator diagram of such a function is a segment of the imaginary axis. Functions analytic in a circular lacuna are represented by biorthogonal series.     

Approximation by harmonic functions

For a compact set we construct a restoring covering for the space of real-valued functions on which can be uniformly approximated by harmonic functions. Functions from restricted to an element of this covering possess some analytic properties. In particular, every nonnegative function , equal to 0 on an open non-void set, is equal to 0 on . Moreover, when , the algebra of complex-valued functions on which can be uniformly approximated by holomorphic functions is analytic. These theorems allow us to prove that if a compact set has a nontrivial Jensen measure, then contains a nontrivial compact set with analytic algebra .

     

Approximation of analytic functions by Laguerre functions

We will solve the inhomogeneous Laguerre differential equation and apply this result to prove that if a function can be represented by a power series whose radius of convergence is larger than 1, then the function can be approximated, on the interval [0, 1), by a Laguerre function with an error bound C x for some constant C > 0.     

Approximation of analytic functions by Legendre functions

We will solve the inhomogeneous Legendre’s differential equation and apply this result to estimate the error bound occurring when an analytic function is approximated by an appropriate Legendre function.     

Approximation of analytic functions by Legendre functions

We will solve the inhomogeneous Legendre’s differential equation and apply this result to estimate the error bound occurring when an analytic function is approximated by an appropriate Legendre function.     

Approximation of analytic functions by Hermite functions

We solve the inhomogeneous Hermite equation and apply this result to estimate the error bound occurring when any analytic function is approximated by an appropriate Hermite function.     

Approximation of entire functions of exponential type by exponential sums<Superscript>*</Superscript>

Let K be a compact set in \( {{\mathbb R}^n} \). For \( 1 \leqslant p \leqslant \infty \), the Bernstein space \( B_K^p \) is the Banach space of all functions \( f \in {L^p}\left( {{{\mathbb R}^n}} \right) \)such that their Fourier transform in a distributional sense is supported on K. If \( f \in B_K^p \), then f is continuous on \( {{\mathbb R}^n} \) and has an extension onto the complex space \( {{\mathbb C}^n} \) to an entire function of exponential type K. We study the approximation of functions in \( B_K^p \) by finite τ -periodic exponential sums of the form

$ \sum\limits_m {{c_m}{e^{2\pi {\text{i}}\left( {x,m} \right)/\tau }}} $

in the \( {L^p}\left( {\tau {{\left[ { - 1/2,1/2} \right]}^n}} \right) \)-norm as τ → ∞ when K is a polytope in \( {{\mathbb R}^n} \).