Entire Functions Of Exponential Type Research Articles

Extremal Polynomials and Entire Functions of Exponential Type

In this paper, we discuss asymptotic relations for the approximation of left| xright| ^{alpha },alpha >0 in L_{infty }left[ -,1,1right] by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind. The limiting process reveals an entire function of exponential type for which we can present an explicit formula.

Turán’s and Fejér’s Extremal Problems for Jacobi Transform

We give the solution of the Turan extremal problem for compact supported functions on the half-line with nonnegative Jacobi transform and the dual Fejer extremal problem for even nonnegative entire functions of exponential type that are Jacobi transforms. We prove the uniqueness of the extremal functions. The Markov quadrature formula on the half-line at zeros of the modified Jacobi function is used for the proof of these results.

Average Dimension of Shift Spaces

Recently the author obtained a series of sharp estimates of convolution classes by spaces of shifts of a single function. Those estimates generalize the well-known classical inequalities of Favard, Akhiezer and Krein. In the present paper we compute the average dimension of shift spaces. It appears that this dimension coincides with the average dimension of the spaces of entire functions of exponential type and of equidistant splines.

Difference Equation for Functions Analytic Outside Two Squares

We consider a totality of two squares built on primitive periods 1 and i and “sufficiently close to each other“. In a vicinity of this set we investigate four-element difference equation with constant coefficients, whose linear shifts are generating transforms of the corresponding doubly periodic group and the inverse transforms. We seek a solution in a class of functions, which are analytic outside this set and vanish at infinity. The equation is applicable to the moments problem for entire functions of exponential type.

Askey–Wilson operator on entire functions of exponential type

In this paper, we first establish a series representation formula for the Askey–Wilson operator applied on entire functions of exponential type and then demonstrate its power in discovering summation formulas, some known and some new.

Generalized Logan’s Problem for Entire Functions of Exponential Type and Optimal Argument in Jackson’s Inequality in L2(ℝ3)

We study Jackson’s inequality between the best approximation of a function f ∈ L2(ℝ3) by entire functions of exponential spherical type and its generalized modulus of continuity. We prove Jackson’s inequality with the exact constant and the optimal argument in the modulus of continuity. In particular, Jackson’s inequality with the optimal parameters is obtained for classical modulus of continuity of order r and Thue–Morse modulus of continuity of order r ∈ ℕ. These results are based on the solution of the generalized Logan problem for entire functions of exponential type. For it we construct a new quadrature formulas for entire functions of exponential type.

Bohman Extremal Problem for the Jacobi Transform

We give a solution to the Bohman extremal problem for nonnegative even entire functions of exponential type that are Jacobi transforms of compactly supported functions. We prove that the extremal function is unique. The Gauss quadrature formula on the half-line over zeros of the Jacobi function is used.

Approximation by Entire Functions on a Countable Union of Segments on the Real Axis: 3. Further Generalization

In this paper, an approximation of functions of extensive classes set on a countable unit of segments of a real axis using the entire functions of exponential type is considered. The higher the type of the approximating function is, the higher the rate of approximation near segment ends can be made, compared with their inner points. The general approximation scale, which is nonuniform over its segments, depending on the type of the entire function, is similar to the scale set out for the first time in the study of the approximation of the function by polynomials. For cases with one segment and its approximation by polynomials, this scale has allowed us to connect the so-called direct theorems, which state a possible rate of smooth function approximation by polynomials, and the inverse theorems, which give the smoothness of a function approximated by polynomials at a given rate. The approximations by entire functions on a countable unit of segments for the case of Holder spaces have been studied by the authors in two preceding papers. This paper significantly expands the class of spaces for the functions, which are used to plot an approximation that engages the entire functions with the required properties.

Plancherel–Pólya Inequality for Entire Functions of Exponential Type in L2(ℝ)

We investigate the Plancherel–Polya inequality $$\sum {_{k \in \mathbb{Z}}} |f(k){|^2} \leqslant {c_2}\left( \sigma \right)||f||_{{L^2}\left( \mathbb{R} \right)}^2$$ on the set of entire functions f of exponential type at most σ whose restrictions to the real line belong to the space L2(ℝ). We prove that c2(σ) = [σ/π] for σ > 0 and describe the extremal functions.

Nikol’skii Inequality Between the Uniform Norm and Integral Norm with Bessel Weight for Entire Functions of Exponential Type on the Half-Line

We study a Nikol’skii type inequality for even entire functions of given exponential type between the uniform norm on the half-line [0,∞) and the norm (∫ 0 ∞ |f(x)| q x2α+1dx)1/q of the space L q ((0,∞), x2α+1) with the Bessel weight for 1 ≤ q −1/2. An extremal function is characterized. In particular, we prove that the uniform norm of an extremal function is attained only at the end point x = 0 of the half-line. To prove these results, we use the Bessel generalized translation.

About Functional Equation, Generated by Equilateral Triangle

We considered 6-fold linear equation in the class of analytic functions outside the equilateral triangle and vanishing at infinity. We proposed a method of equivalent regularization, using the theory of boundary value problem of Carleman. Also, the paper is dedicated to applications to the problem of moments for entire functions of exponential type. In particular, we construct a system of functions biorthogonal whit piecewise-quasipolynomial weight to some power system on the three rays. The indicator of such functions is a piecewise-trigonometric function, with a period of 2π/3.

Exact errors of best approximation for complex-valued nonperiodic functions

We extend Markov’s and Nagy’s theorems on best approximation by entire functions of exponential type in the L1(R) metric to some complex-valued integrable and locally integrable functions. We use these results for finding sharp constants of best approximation in L1(R) and L∞(R) on some complex convolution classes. For classes of real-valued convolutions those constants were found by Akhiezer. As an example, we apply these results to the Schwarz-type kernel and to the corresponding convolution classes.

Interpolation Problems for Entire Functions Induced by Regular Hexagons

We consider linear equations for analytic functions in the plane with cuts along a “half” of the boundary of a hexagon. We propose a regularization method, reducing them to an equation with difference kernel. Applications are given to the moment problem for entire functions of exponential type.

On Summary Equation Generated by a Quadrangle

We investigate a four-element functional equation in a class of solutions holomorphic outside a quadrangle and vanishing at infinity. We propose equivalent regularization method of such quadrangles. We construct examples with only single condition of solvability for such an equation. We show applications to moments problem for entire functions of exponential type.

On the $\tau $-Li coefficients for automorphic $L$-functions

In this paper, we extend the Li coefficients for automorphic $L$-functions and the Li criterion for the Riemann hypothesis to yield a necessary and sufficient condition for the existence of zero-free strips for automorphic $L$-functions inside the critical strip. Next, we give an arithmetical and asymptotical formula for these coefficients. Finally, we show that there exists an entire function of exponential type that interpolates the extended Li coefficients (or the $\tau $-Li coefficients) at integer values. The results of this paper arise from ideas of the author~\cite {15}, Freitas~\cite {8}, Lagarias~\cite {10} and Odz$\breve {a}$k and Smajlovi$\grave {c}$~\cite {17}.

Entire functions of exponential type represented by pseudo-random and random Taylor series

We study the influence of the multipliers ξ(n) on the angular distribution of zeroes of the Taylor series $${F_\xi }\left( z \right) = \sum\limits_{n \geqslant 0} {\xi \left( n \right)} \frac{{{z^n}}}{{n!}}$$ . We show that the distribution of zeroes of Fξ is governed by certain autocorrelations of the sequence ξ. Using this guiding principle, we consider several examples of random and pseudo-random sequences ξ and, in particular, answer some questions posed by Chen and Littlewood in 1967. As a by-product, we show that if ξ is a stationary random integer-valued sequence, then either it is periodic, or its spectral measure has no gaps in its support. The same conclusion is true if ξ is a complex-valued stationary ergodic sequence that takes values in a uniformly discrete set.

Entire functions of exponential type not vanishing in the half-plane \(\Im z > k\), where \(k > 0\)

Let \(P (z)\) be a polynomial of degree \(n\) having no zeros in \(|z| < k\), \(k \leq 1\), and let \(Q (z) := z^n \overline{P (1/{\overline {z}})}\). It was shown by Govil that if \(\max_{|z| = 1} |P^\prime (z)|\) and \(\max_{|z| = 1} |Q^\prime (z)|\) are attained at the same point of the unit circle \(|z| = 1\), then \[\max_{|z| = 1} |P'(z)| \leq \frac{n}{1 + k^n} \max_{|z| = 1} |P(z)|.\]<br />The main result of the present article is a generalization of Govil's polynomial inequality to a class of entire functions of exponential type.<br /><br />

Poly-element equations reducible to moment problem for entire functions of class A

We consider poly-element linear functional equations in the class of analytic functions in the complex plane with cuts along certain segments of imaginary axis. The obtained results are applied for study of the Stieltjes moment problem for entire functions of exponential type in the class A.

Sharp inequalities for approximations of convolution classes on the real line as the limit case of inequalities for periodic convolutions

We establish sharp estimates for the best approximations of convolution classes by entire functions of exponential type. To obtain these estimates, we propose a new method for testing Nikol’skiĭ-type conditions which is based on kernel periodization with an arbitrarily large period and ensuing passage to the limit. As particular cases, we obtain sharp estimates for approximation of convolution classes with variation diminishing kernels and generalized Bernoulli and Poisson kernels.

Subsequences of zeros for classes of entire functions of exponential type distinguished by growth restrictions

Subsequences of zeros for classes of entire functions of exponential type distinguished by growth restrictions