Entire Functions Of Exponential Type Research Articles

Completeness of the systems of Bessel functions of index $-5/2$

Let $L^2((0;1);x^4 dx)$ be the weighted Lebesgue space of all measurable functions $f:(0;1)\rightarrow\mathbb C$, satisfying $\int_{0}^1 t^4 |f(t)|^2\, dt<+\infty$. Let $J_{-5/2}$ be the Bessel function of the first kind of index $-5/2$ and $(\rho_k)_{k\in\mathbb N}$ be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system $\big\{\rho_k^2\sqrt{x\rho_k}J_{-5/2}(x\rho_k):k\in\mathbb N\big\}$ in the space $L^2((0;1);x^4 dx)$ are found in terms of an entire function with the set of zeros coinciding with the sequence $(\rho_k)_{k\in\mathbb N}$. In this case, we study an integral representation of some class $E_{4,+}$ of even entire functions of exponential type $\sigma\le 1$. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than $-1$, due to B. Vynnyts'kyi, V. Dilnyi, O. Shavala and the author.

The Sylvester problem and uniqueness sets in classes of entire functions

In this paper, we study the problem of finding, by a chosen sequence of complex numbers tending to infinity, the widest possible class of entire functions in a given scale for which this sequence is a uniqueness set. Within the framework of this general problem, we establish uniqueness theorems in various classes of entire functions, distinguished by restrictions on the type and indicator under a refined order. In particular, we complement the previously proven uniqueness theorem, using the concept of the Sylvester circle of the indicator diagram of an entire function of exponential type. We discuss the accuracy of the results obtained and their connection with known facts.

Bernstein Inequality for the Riesz Derivative of Order $$0<\alpha<1$$ of Entire Functions of Exponential Type in the Uniform Norm

Bernstein Inequality for the Riesz Derivative of Order $$0<\alpha<1$$ of Entire Functions of Exponential Type in the Uniform Norm

Subharmonic additions to the Beurling–Malliavin Theorems. II. On the completeness radius

The Beurling–Malliavin Theorem on the multiplier, considered in a subharmonic framework in the first part of our work, already in its original classical version within the framework of entire functions of exponential type, allowed in the 1960s to completely solve the problem of the radius of completeness of exponential system in the form of a remarkable criterion and exclusively in geometric terms for the exponents of this exponential system without any additional restrictions on the relative position of these exponents. The exact formulations of the Beurling– Malliavin Theorem on the radius of completeness in the introduction are somewhat adapted as a problem of the possible minimum growth along the real axis R of subharmonic functions with given constraints on their Riesz distributions of masses. In this largely overview part of the paper, we discuss the Beurling–Malliavin Theorem on the radius of completeness, along with its somewhat more general subharmonic manifestations. Thus, our results from 2014-16 allow us to obtain significantly more subtle results with respect to the Beurling – Malliavin Theorem itself on the radius of completeness with a defect excess of no more than 1 or 2 for exponents in classical rigid Banach spaces of continuous functions on a segment of fixed length or functions with integrable module in the p-th degree on such segments.

Bernstein Inequality for the Riesz Derivative of Fractional Order Less Than Unity of Entire Functions of Exponential Type

Bernstein Inequality for the Riesz Derivative of Fractional Order Less Than Unity of Entire Functions of Exponential Type

On Constants in the Bernstein–Szegő Inequality for the Weyl Derivative of Order Less Than Unity of Trigonometric Polynomials and Entire Functions of Exponential Type in the Uniform Norm

On Constants in the Bernstein–Szegő Inequality for the Weyl Derivative of Order Less Than Unity of Trigonometric Polynomials and Entire Functions of Exponential Type in the Uniform Norm

Integral Inequalities for Entire Functions of Exponential Type in Morrey Spaces

Integral Inequalities for Entire Functions of Exponential Type in Morrey Spaces

Representation of analytic functions in bounded convex domains on the complex plane

The paper is devoted to entire functions of exponential type and regular growth. Exceptional sets are investigated outside of which these functions have estimates from below that asymptotically coincide with their estimates from above. An explicit construction of an exceptional set, which consists of disks with centers at zeros of the entire function, is indicated. The concept of a properly balanced set is introduced, which is a natural generalization of the concept of a regular set by B. Ya. Levin. It is proved that the zero set of an entire function is properly balanced if and only if each function analytic in the interior of the conjugate diagram of the entire function in question and continuous up to the boundary is represented by a series of exponential monomials whose exponents are zeros of this entire function. This result generalizes the classical result of A. F. Leont′ev on the representation of analytic functions in a convex domain to the case of a multiple zero set.

BERNSTEIN INEQUALITY FOR RIESZ DERIVATIVE OF FRACTIONAL ORDER LESS THAN 1 OF ENTIRE FUNCTION OF EXPONENTIAL TYPE

We consider Bernstein inequality for the Riesz derivative of order \(0 \alpha 1\) of entire functions of exponential type in the uniform norm on the real line. The interpolation formula for this operator is obtained; this formula has non-equidistant nodes. By means of this formula, the sharp Bernstein inequality is obtained for all \(0 \alpha 1\), more precisely, the extremal entire function and the exact constant are written out.

Mittag–Leffler expansions for inverse spectral problems with mixed data

We consider the inverse spectral problem for the Sturm–Liouville problem in the interval with the Dirichlet boundary conditions at both ends. We provide a method for the unique reconstruction of the potential that is known a priori on the subinterval , where m and r being positive integers and satisfying , by using the spectral data set consisting of all the eigenvalues and a subset of the norming constants. The method is based on the Mittag–Leffler decomposition which can help us to decompose an entire function of exponential type into two functions of smaller types. This decomposition allows the use of Lagrange interpolation in order to reconstruct the potential in the Sturm–Liouville problem. We also provide necessary and sufficient conditions related to the existence issues.

Carleson and sampling measures on Bernstein spaces on Siegel CR manifolds

AbstractIn this paper, we introduce and study Carleson and sampling measures on Bernstein spaces on a class of quadratic CR manifold called Siegel CR manifolds. These are spaces of entire functions of exponential type whose restrictions to the given Siegel CR manifold are ‐integrable with respect to a natural measure. For these spaces, we prove necessary and sufficient conditions for a Radon measure to be a Carleson or a sampling measure. We also provide sufficient conditions for sampling sequences.

Hermite interpolation theorems for band-limited functions of the linear canonical transforms with equidistant samples

We establish convergence analysis for Hermite-type interpolations for L^{2} ( mathbb {R})-entire functions of exponential type whose linear canonical transforms (LCT) are compactly supported. The results bridges the theoretical gap in implementing the derivative sampling theorems for band-limited signals in the LCT domain. Both complex analysis and real analysis techniques are established to derive the convergence analysis. The truncation error is also investigated and rigorous estimates for it are given. Nevertheless, the convergence rate is O(1/sqrt{N}), which is slow. Consequently the work on regularization techniques is required.

Exceptional sets

In this paper, we study sequences of complex numbers of the first order. Multiple terms are allowed for such sequences. We also consider complex sequences with a finite maximum density. We construct special coverings of multiple sets {λk,nk} consisting of circles centered at points λk of special radii. In particular, we construct coverings are with connected components of a relatively small diameter, as well as coverings that are C0-sets. These coverings act as exceptional sets for entire functions of exponential type. Outside these sets, we obtain a representation of the logarithm of the modulus of an entire function. Previously, a similar representation was obtained by B. Ya. Levin outside the exceptional set, with respect to which only its existence is asserted. In contrast to this, in this paper we present a simple effective construction of an exceptional set. We construct bases of the invariant subspace of analytic functions in a convex domain. They consist of linear combinations of eigenfunctions and associated functions (exponential monomials) of the differentiation operator divided into relatively small groups.

Integral representation of superoscillations via complex Borel measures and their convergence

In the last decade there has been a growing interest in superoscillations in various fields of mathematics, physics and engineering. However, while in applications as optics the local oscillatory behaviour is the important property, some convergence to a plane wave is the standard characterizing feature of a superoscillating function in mathematics and quantum mechanics. Also there exists a certain discrepancy between the representation of superoscillations either as generalized Fourier series, as certain integrals or via special functions. The aim of this work is to close these gaps and give a general definition of superoscillations, covering the well-known examples in the existing literature. Superoscillations will be defined as sequences of holomorphic functions, which admit integral representations with respect to complex Borel measures and converge to a plane wave in the space A 1 ( C ) \mathcal {A}_1(\mathbb {C}) of entire functions of exponential type.

On generalized K-functionals in L_p for 0

We show that the Peetre K-functional between the space L_p with 0<p<1 and the corresponding smooth function space W_p^psi generated by the Weyl-type differential operator psi (D), where psi is a homogeneous function of any positive order, is identically zero. The proof of the main results is based on the properties of the de la Vallée Poussin kernels and the quadrature formulas for trigonometric polynomials and entire functions of exponential type.

Logan–Hermite Extremal Problems for Entire Functions of Exponential Type

Logan–Hermite Extremal Problems for Entire Functions of Exponential Type

Произведение Дюамеля в пространствах целых функций экспоненциального типа

Let Ω be a polydomain in CN containing the point 0; H(Ω) be the space of all holomorphic functions on Ω with the compact open topology. In the dual H(Ω)&amp;#x27; of H(Ω) a multiplication ⊛ is introduced and investigated. It is defined by the convolution rule with the help of shifts which are associated with the system of partial backward shift operators. The realizations of the algebra (H(Ω)&amp;#x27;,⊛) with the help of the Cauchy and the Laplace transformations and its representation in H(Ω) are obtained. By means of the Laplace transformation the introduced multiplication ⊛ is realized as the many-dimensional Duhamel product in the appropriate space of entire functions of exponential type.

Logan-Hermite Extremal Problems for Entire Functions of Exponential Type

Logan-Hermite Extremal Problems for Entire Functions of Exponential Type

Nikol’skii-type inequalities for entire functions of exponential type in Lorentz–Zygmund spaces

Nikol’skii-type inequalities for entire functions of exponential type on \({\mathbb{R}}^{n}\) for the Lorentz–Zygmund spaces are obtained. Some new limiting cases are examined. Application to Besov–type spaces of logarithmic smoothness is given.

Bernstein spaces on Siegel CR manifolds

In this paper we introduce and study Bernstein spaces on a class of quadratic CR manifolds, which we call Siegel CR manifolds. These are spaces of entire functions of exponential type whose restrictions to a given Siegel CR submanifold are \(L^p\)-integrable with respect to a natural measure. For these spaces, among other results, we prove the Plancherel–Pólya inequality, a Bernstein inequality and a sufficient condition for a sequence to be sampling.