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.

Boundary controllability for a 1D degenerate parabolic equation with a Robin boundary condition

AbstractIn this paper, we prove the null controllability of a one-dimensional degenerate parabolic equation with a weighted Robin boundary condition at the left endpoint, where the potential has a singularity. We use some results from the singular Sturm–Liouville theory to show the well-posedness of our system. We obtain a spectral decomposition of a degenerate parabolic operator with Robin conditions at the endpoints, we use Fourier–Dini expansions and the moment method introduced by Fattorini and Russell to prove the null controllability and to obtain an upper estimate of the cost of controllability. We also get a lower estimate of the cost of controllability by using a representation theorem for analytic functions of exponential type.

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.

Sectorial Paley–Wiener theorem

We consider the problem of restoring a two‐dimensional convex source of gravity based on values of its potential function on a union of half‐planes. With that aim we prove a Paley–Wiener type theorem for functions of exponential type in a sector. In particular, we find the maximal set of analytic continuation whose complement is convex inside a sector. Our result serves as a sectorial analog of G. Polya's indicator theorem. It corrects, simplifies, and extends a result of M. Morimoto. It also improves a related result of M. Dzhrbashyan and A. Avetisyan.

Analytical solutions for time-fractional diffusion equation with heat absorption in spherical domains

Under the time-variable Dirichlet condition, the time-fractional diffusion equation with heat absorption in a sphere is taken into consideration. The time-fractional derivative with the power-law kernel is used in the generalized Cattaneo constitutive equation of the thermal flux. The Laplace transform and a suitable transformation of the independent variable and function are used to determine the analytical solution of the problem in the Laplace domain. To obtain the temperature distribution in the real domain, the inverse Laplace transforms of two functions of exponential type are obtained. These formulae are new in the literature. The particular cases of the classical Cattaneo law of heat conduction and of the classical Fourier's law are obtained from the solutions corresponding to the time-fractional generalized Cattaneo law.

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.

A Computational Approach to Exponential-Type Variable-Order Fractional Differential Equations

We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterization of such operators is performed in the Laplace domain, it is necessary to resort to accurate numerical methods to derive the corresponding behaviours in the time domain. In this regard, we develop a computational procedure to solve variable-order fractional differential equations of this novel class. Furthermore, we provide some numerical experiments to show the effectiveness of the proposed technique.

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.

Reconstruction of Nusselt number in RANS computations with wall function approach

Two novel methods for reconstruction of Nusselt number (Nu) are proposed when the wall function approach is used in the RANS framework for defining wall boundary conditions for high Reynolds number flows. The work is motivated by the observation that the velocity and temperature fields are often well predicted when the wall function formulations are used, while at the same time Nu is not. The reason for discrepancy in predictions of flow fields and temperature, and prediction of Nu, is because Nu estimation is solely dependent on the wall function, while the variables obtained from the transport equations are determined also by the convection and pressure force in the momentum transport. The method based on the correction factor Ψ∗ originates from the simplified analytical wall function approach of Popovac and Hanjalić (2007) and modifies the formulation of so-called wall viscosity. The method based on a wall shape function of exponential type approximates temperature variation in the near-wall region. Once the shape function’s coefficients are calculated using values of temperature in three near-wall consecutive cells, the wall values of temperature or temperature gradient, needed for calculation of Nu, can be determined. The methods are tested for four standard benchmark cases, backward-facing step, impinging jet, differentially heated square cavity, and Rayleigh-Bénard convection. The method based on the correction factor significantly improved Nu predictions for the cases where the velocity field is strongly impacted by pressure force. The method based on the wall shape function improves predictions of Nu in all tested cases, with somewhat less accurate predictions of the maximum value of Nu compared to the correction factor method.

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.

Entire Bivariate Functions of Exponential Type II

Let $f(z_{1},z_{2})$ be a bivariate entire function and $C$ be a positive constant. If $f(z_{1},z_{2})$ satisfies the following inequality for non-negative integer $M$, for all non-negative integers $k,$ $l$ such that $k+l\in\{0, 1, 2, \ldots, M\}$, for some integer $p\ge 1$ and for all $(z_{1},z_{2})=(r_{1}e^{\mathbf{i}\theta_{1}},r_{2}e^{\mathbf{i}\theta_{2}})$ with $r_1$ and $r_2$ sufficiently large:\begin{gather*}\sum_{i+j=0}^{M}\frac{\left(\int_{0}^{2\pi}\int_{0}^{2\pi}|f^{(i+k,j+l)}(r_{1}e^{\mathbf{i}\theta_{1}},r_{2}e^{\mathbf{i}\theta_{2}})|^{p}d\theta_{1}d\theta_{2}\right)^{\frac{1}{p}}}{i!j!}\ge \\\ge \sum_{i+j=M+1}^{\infty}\frac{\left(\int_{0}^{2\pi}\int_{0}^{2\pi}|f^{(i+k,j+l)}(r_{1}e^{\mathbf{i}\theta_{1}},r_{2}e^{\mathbf{i}\theta_{2}})|^{p}d\theta_{1}d\theta_{2}\right)^{\frac{1}{p}}}{i!j!},\end{gather*}then $f(z_{1},z_{2})$ is of exponential type not exceeding\[2+2\log\Big(1+\frac{1}{C}\Big)+\log[(2M)!/M!].\]If this condition is replaced by related conditions, then also $f$ is of exponential type.