Two related extremal problems for entire functions of several variables

We establish a relationship between the Logan problem for functions whose Fourier transform is supported in a centrally symmetric convex closed subset of ℝ ^m and whose mean value on ℝ ^m is nonnegative and the Chernykh problem on the optimal point for the Jackson inequality inL ₂(ℝ ^m ), which relates the best approximation of a function by the class of entire functions of exponential type to the first modulus of continuity. Both problems are solved exactly in several cases. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 336–350, September, 1999.     

Stieltjes moments of entire functions of exponential type

The lacunary homogeneous moment problem

On Bernstein's inequality for entire functions of exponential type

It was shown by S.N. Bernstein that if f is an entire function of exponential type τ such that |f(x)|?M for −∞<x<∞, then |f^′(x)|?Mτ for −∞<x<∞. If p is a polynomial of degree at most n with |p(z)|?M for |z|=1, then f(z):=p(eiz) is an entire function of exponential type n with |f(x)|?M on the real axis. Hence, by the just mentioned inequality for functions of exponential type, |p^′(z)|?Mn for |z|=1. Lately, many papers have been written on polynomials p that satisfy the condition znp(1/z)≡p(z). They do form an intriguing class. If a polynomial p satisfies this condition, then f(z):=p(eiz) is an entire function of exponential type n that satisfies the condition f(z)≡einzf(−z). This led Govil [N.K. Govil, Lp inequalities for entire functions of exponential type, Math. Inequal. Appl. 6 (2003) 445-452] to consider entire functions f of exponential type satisfying f(z)≡eiτzf(−z) and find estimates for their derivatives. In the present paper we present some additional observations about such functions.     

An extremal problem for even positive definite entire functions of exponential type

We consider an extremal problem for even positive definite entire functions of exponential type with zero mean with power weight on the semiaxis. This problem is related to the multidimensional Jackson-Stechkin theorem in the space L ₂(?ⁿ).     

On some criteria for completely regular growth of entire functions of exponential type

This paper sharpens the author’s previous results concerning the completely regular growth of an entire function of exponential type all of whose zeros are simple, forming a sequence Λ = {λ_k} _k=1 ^∞ . For a function with real zeros, we write the growth regularity conditions (on the real axis and on the entire plane) in terms of lower bounds only for the absolute value of the derivative at the points λ_k. We also obtain an analog of Krein’s theorem concerning the functions whose inverse can be expanded in the corresponding series of simple fractions.     

Extremum Problem for Periodic Functions Supported in a Ball

We consider the Turan n-dimensional extremum problem of finding the value of A_n(hB ⁿ ) which is equal to the maximum zero Fourier coefficient of periodic functions f supported in the Euclidean ball hB ⁿ of radius h, having nonnegative Fourier coefficients, and satisfying the condition f(0)= 1. This problem originates from applications to number theory. The case of A₁([–h,h]) was studied by S. B. Stechkin. For A_n(hB ⁿ we obtain an asymptotic series as h 0 whose leading term is found by solving an n-dimensional extremum problem for entire functions of exponential type.     

Sharp error estimates for the numerical differentiation formulas on the classes of entire functions of exponential type

We establish sharp error estimates for some numerical di.erentiation formulas on the classes of entire functions of exponential type. The estimates strengthen some classical sharp inequalities of approximation theory.     

Plancherel–Polya-type inequalities for entire functions of exponential type in

The goal of the paper is to prove generalizations of the classical Plancherel–Polya inequalities in which point-wise sampling of functions (δ-distributions) is replaced by more general compactly supported distributions on . As an application it is shown that a function , 1p∞, which is an entire function of exponential type is uniquely determined by a set of numbers {Ψ_j(f)}, , where {Ψ_j}, , is a countable sequence of compactly supported distributions. In the case p=2 a reconstruction method of a Paley–Wiener function f from a sequence of samples {Ψ_j(f)}, , is given. This method is a generalization of the classical result of Duffin–Schaeffer about exponential frames on intervals.     

A new property of entire functions of exponential type not vanishing in a half-plane and applications

Let H be the open upper half-plane, and H its closure. If ? is a non-constant transcendental entire function of exponential type such that |?(x)| ≥ μ > 0 on the real axis, then it may happen that |?(z)| is not greater than μ anywhere in H even if ?(z) in H. We show that if ? is an entire function of exponential type not vanishing in H such that inf _x∈?|?(x)| = μ > 0, then inf _{z∈ H} |?(z)| = μ if and only if h_? (π/2) := lim sup _y →∞y ^? log|?(iy) ≥ 0, and that |?(z)| = mu for some z ∈ H only if ? is a constant. This result, which can be seen as a ‘minimum modulus principle’ for entire functions of exponential type not vanishing in a half-plane, helps us to obtain generalizations of two inequalities of Boas, and one of Turán.     

Application of the Borel transform to the study of the spectrum of integral equations whose kernels are entire functions of exponential type

Using the Borel transform, we study the spectrum of a class of non-compact integral operators whose kernels are of exponential type and square integrable on the real line. Our method also enables us to obtain an interesting characterization of a well-known integral equation involving the Bessel function

     

Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type

In many applications one seeks to recover an entire function of exponential type from its non-uniformly spaced samples. Whereas the mathematical theory usually addresses the question of when such a function in can be recovered, numerical methods operate with a finite-dimensional model. The numerical reconstruction or approximation of the original function amounts to the solution of a large linear system. We show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite-dimensional reconstruction problem. This legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results. The main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values. From another point of view our approach provides a new method for proving sampling theorems.

     

Bessel generalized translations and some problems of approximation theory for functions on the half-line

Approximation problems for functions on the half-line [0,+∞) in a weighted L _p -metric are studied with the use of Bessel generalized translation. A direct theorem of Jackson type is proven for the modulus of smoothness of arbitrary order which is constructed on the basis of Bessel generalized translation. Equivalence is stated between the modulus of smoothness and the K-functional constructed by the Sobolev space corresponding to the Bessel differential operator. A particular class of entire functions of exponential type is used for approximation. The problems under consideration are studied mostly by means of Fourier-Bessel harmonic analysis.     

Difference schemes of exponential type for singularly perturbed Volterra integro-differential problems

Extended one-step schemes of exponential type are introduced for solving singularly perturbed Volterra integro-differential problems. These schemes are of order (m + 1), m = 0, 1, 2, …, when the perturbation parameter, ε, is fixed. These schemes have the property that if ε is of order h they reduced to first order of accuracy and optimal when ε → 0. Stability analysis of these schemes are presented. Numerical results and comparisons with other schemes are presented.     

Approximation of bandlimited functions by finite exponential sums

For a compact set K in ℝ ⁿ , let B ² _K be the set of all functions f ∈ L ²(ℝ²) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B ² _K by finite exponential sums