Entire Functions Of Exponential Type Research Articles

Sharp Constants of Approximation Theory. I. Multivariate Bernstein–Nikolskii Type Inequalities

We prove limit relations between the sharp constants in the multivariate Bernstein-Nikolskii type inequalities for trigonometric polynomials and entire functions of exponential type with the spectrum in a centrally symmetric convex body.

Суммарно-разностное уравнение для аналитических функций, порожденное шестиугольником, и его приложения

Let D be a hexagon having a pair of parallel sides, equal in length, and L be a half of its boundary. We study solvability of a seven-element sum-difference equation equation in the class of functions holomorphic outside L and vanishing at infinity. Their boundary values satisfy the Holder condition on every compact not containing the nodes. At the nodes they have at most logarithmic singularities. To regularize the equation, on the boundary of the hexagon we introduce a Carleman shift having jump discontinuity at the vertices. We seek a solution in the form of Cauchy type integral over L with an unknown density. We find conditions providing equivalence of such regularization. We also consider a particular case for which the corresponding Fredholm equation is solvable. We give some applications to the moment problem for entire functions of exponential type (EFETs). In particular, we construct a system of EFETs biorthogonal, with a piece-wise quasipolynomial weight, to a system of three degree functions on three rays. For such EFETs, the conjugate diagram is an octagon. We note that various generalizations of our investigations are possible due to the large arbitrariness in the choice of set L.

Sharp constants of approximation theory. III. Certain polynomial inequalities of different metrics on convex sets

Let V⊂Rm be a centrally symmetric convex body and let V∗⊂Rm be its polar. We prove limit relations between the sharp constants in the multivariate Markov–Bernstein–Nikolskii type inequalities for algebraic polynomials on V∗ and the corresponding constants for entire functions of exponential type with the spectrum in V.

Interpolation Through Approximation in a Bernstein Space

Let Bσ be the Bernstein space of entire functions of exponential type at most σ bounded on the real axis. Consider a sequence Λ = {zn}n∈ℤ, zn = xn + iyn, such that xn+1 − xn ≥ l > 0 and |yn| ≤ L, n ∈ ℤ. Using approximation by functions from Bσ, we prove that for any bounded sequence A = {an}n∈ℤ, |an| ≤ M, n ∈ ℤ, there exists a function f ∈ Bσ with σ ≤ σ0(l,L) such that f|Λ = A.

Summary Equation for Functions, which are Holomorphic Outside Two Triangles, and Its Applications

A method is proposed for regularizing a six-element linear total equation in the class of odd functions that are holomorphic outside two regular triangles and having zero with multiplicity at least three at infinity. The applications to the problem of moments for entire functions of exponential type are indicated.

Corrigendum to “Abel Summability of Entire Functions of Exponential Type” [Progress in Approximation Theory, P. Nevai and A. Pinkus, Eds. (1991) 117–139

Corrigendum to “Abel Summability of Entire Functions of Exponential Type” [Progress in Approximation Theory, P. Nevai and A. Pinkus, Eds. (1991) 117–139

Representation of Analytic Functions by Series of Exponential Monomials in Convex Domains and Its Applications

In this paper lower bounds for entire functions of exponential type and regular growth, zero sets of which have zero condensation indices, are obtained. In this case, the exceptional set consists of pairwise disjoint disks centered at zeroes. Sufficient conditions for radii of these circles are indicated. We also obtain a result on representation of analytic functions in the closure of a bounded convex domain (as well as analytic functions in domain and continuous up to the boundary) by series of exponential monomials. This result extends the classical result of A.F. Leont’ev to the case of multiple zero set of entire function. The obtained result is applied to the problem on distribution of singular points of a sum of series of exponential monomials at the boundary of its convergence domain.

Multivariate Bernstein inequalities for entire functions of exponential type in L^{p}(mathbb{R}^{n})(0< p< 1)

In (Rahman and Schmeisser in Trans. Amer. Math. Soc. 320: 91–103, 1990), the authors prove that the classical Bernstein inequality also holds for 0< ple 1. We extend their result for a differential operator induced by polynomials and find the several equivalent conditions to the Paley–Wiener theorem. As applications of the results, we also derive the Paley–Wiener type theorems for some special compact sets generated by number sequences, generated by polynomial, convex compact sets, in which we show that the Bernstein type inequalities have concrete upper bounds.

On Invariant Subspaces of the Pommiez Operator in the Spaces of Entire Functions of Exponential Type

We describe closed invariant eigenspaces of the Pommmiez operator in the (LF)-space of entire functions of exponential type. This space is topologically equivalent (by means of the Laplace transform) to the strong dual space of all germs of functions that are analytic on a convex, locally closed subset of the complex plane.

Principal Submodules in the Module of Entire Functions, Which is Dual to the Schwarz Space, and Weak Spectral Synthesis in the Schwartz Space

We obtain a sufficient condition of the weak localizability of a principal submodule in the module of entire functions of exponential type and polynomial growth on the real line. Applications to the problem of the (weak) spectral synthesis in the Schwartz space C∞ (a; b) are discussed.

Lattice Functions for the Analysis of Analog-to-Digital Conversion

Analog-to-digital (A/D) converters are the common interface between analog signals and the domain of digital discrete-time signal processing. In essence, this domain simultaneously incorporates quantization both in amplitude and time, i.e. amplitude quantization and uniform time sampling. Thus, we view A/D conversion as a sampling process in both the time and amplitude domains based on the observation that the underlying continuous-time signals representing digital sequences can be sampled in a lattice---i.e. at points restricted to lie on a uniform grid both in time and amplitude. We refer to them as lattice functions. This is in contrast with the traditional approach based on the classical sampling theorem and quantization error analysis. The latter has been mainly addressed with the help of probabilistic models, or deterministic ones either confined to very particular scenarios or considering worst-case assumptions. In this paper, we provide a deterministic theoretical analysis and framework for the functions involved in digital discrete-time processing. We show that lattice functions possess a rich analytic structure in the context of integral-valued entire functions of exponential type. We derive set and spectral properties of this class of functions. This allows us to prove in a deterministic way and for general bandlimited functions a fundamental lower bound on the maximum frequency component introduced by quantization that is independent of the resolution of the quantizer.

On the Approximation of Functions by Polynomials and Entire Functions of Exponential Type

A brief overview of publications in approximation theory of functions known to the author and connected with scientific publications by V.~K.~Dzyadyk (1919--1998).

Functions in the Schwartz algebra that are invertible in the sense of Ehrenpreis

We consider the problem of obtaining the restrictions on the zero set of an entire function of exponential type under which this function belongs to the Schwartz algebra and invertible in the sense of Ehrenpreis.

On solvability of one nonlinear integral equation in the class of analytic functions

We consider a nonlinear integral equation of a special type that appears in the inverse spectral theory of integro-differential operators and whose unique solvability in the class of square-integrable functions is known. However, for some applied issues in order to construct effective algorithms for solving equations of this type, it is required to establish their solvability in the class of analytic functions. Assuming the free term of the nonlinear equation under consideration to be an entire function of exponential type, we prove that so is its solution. Leaning on this result we provide a constructive procedure for solving this equation in the class of square-integrable functions, which can be easily implemented numerically.

Jackson–Stechkin-Type Inequalities for the Approximation of Elements of Hilbert Spaces

We introduce new characteristics for elements of Hilbert spaces, namely, their generalized moduli of continuity ωφ(x, Lp,V ([0, δ])) and obtain new exact Jackson–Stechkin-type inequalities with these moduli of continuity for the approximation of elements of Hilbert spaces. These results include numerous well-known inequalities for the approximation of periodic functions by trigonometric polynomials, approximation of nonperiodic functions by entire functions of exponential type, similar results for almost periodic functions, etc. Some of these results are new even in these classical cases.

Entire functions of order 1/2 in the approximation of functions on the semiaxis

We present a theorem in the present paper on an approximation to functions of a Holder class on a countable union of segments lying on a positive ray by entire functions of order 1/2 bounded on this ray. Problems related to the approximation of entire functions on subsets of the semiaxis by using entire functions of order 1/2 are closely related to problems of approximating functions on subsets of the whole axis using entire functions of exponential type but have their own specifics. We consider segments In in this paper with lengths of order n such that the distance between In and In + 1 is also of order n. Cases of the whole semiaxis or the union of finitely many segments and a ray were considered in previous papers. As for the problem of approximating functions of the Holder class on the union of a countable set of segments on the whole axis, it turns out that the approximation rate at neighborhoods of the segment endpoints as the type of the functions increases is higher than that in a neighborhood of their midpoints.

On a Difference Equation Generated by Two “Close” Squares

We consider a set of two squares constructed for the primitive periods 1 and i and having four vertices on one straight line. In a neighborhood of this set, we study a four-element difference equation with constant coefficients. The linear shifts of this equation are the generating transformations of the corresponding doubly periodic group and their inverse transformations. The solution is sought in the class of functions that are analytic outside this set and vanish at infinity. We give some applications to the moment problem for entire functions of exponential type.

On Hamburger Moments Problem Generated by a Group With Two Limit Points

We study a linear four-element equation in the class of solutions that are holomorphic outside an isosceles trapezium and vanish at infinity. The equation is used here to investigate the Hamburger moments problem for entire functions of exponential type.

Pole Assignment in Hybrid Differential-Difference Systems

We present a general approach to the pole assignment problem for linear stationary hybrid differential-difference systems as a coefficient control problem for their characteristic equations. Various scales (classes) of linear feedback controllers are considered. Special attention is paid to the solvability of the pole assignment problem for such systems in the scale of general differential-difference controllers and in a general scale that, along with differential-difference controllers, contains integral controllers whose kernels are compactly supported functions. A general scheme for constructing such controllers is proposed based on the algebraic properties of the shift operator, the Paley–Wiener theorem on compactly supported functions, and the methods of interpolation theory in the class of entire functions of exponential type. Examples and counterexamples illustrating the results are given.

Вторая экстремальная задача Логана для преобразования Фурье по собственным функциям оператора Штурма–Лиувилля

For the cosine Fourier transform on the half-line two extremal problems were posed and solved by B. Logan in 1983. In the first problem it was necessary to find a minimal neighborhood of zero outside of which a nontrivial integrable even entire function of exponential type at most τ, having a nonnegative Fourier transform, is nonpositive. In the second problem it was necessary to find a minimal neighborhood of zero outside of which a nontrivial integrable even entire function of exponential type at most τ, having a nonnegative Fourier transform and a zero mean value, is nonnegative. The first Logan problem got the greatest development, because it turned out to be connected with the problem of the optimal argument in the modulus of continuity in the sharp Jackson inequality in the space L 2 between the value of the best approximation of function by entire functions of exponential type and its modulus of continuity. It was solved for the Fourier transform on Euclidean space and for the Dunkl transform as its generalization, for the Fourier transform over eigenfunctions of the Sturm-Liouville problem on the half-line, and the Fourier transform on the hyperboloid. The second Logan problem was solved only for the Fourier transform on Euclidean space. In the present paper, it is solved for the Fourier transform over eigenfunctions of the SturmLiouville problem on the half-line, in particular, for the Hankel and Jacobi transforms. As a consequence of these results, using the averaging method of functions over the Euclidean sphere, we obtain a solution of the second Logan problem for the Dunkl transform and the Fourier transform on the hyperboloid. General estimates are obtained using the Gauss quadrature formula over the zeros of the eigenfunctions of the Sturm-Liouville problem on the half-line, which was recently proved by the authors of the paper. In all cases, extremal functions are constructed. Their uniqueness is proved.