Properties of the Simpson discrete Fourier transform

Abstract

The Simpson discrete Fourier transform (SDFT) and its inverse are transformations relating the time and frequency domains. In this paper we state and prove the important properties of shift, circular convolution, conjugation, time reversal and Plancherel's theorem. In addition, we provide an alternative representation of the duality property.     

On the spectrum of a discrete Laplacian on ℤ with finitely supported potential

We study spectral properties of the discrete Laplacian L?=??Δ?+?V on ? with finitely supported potential V. We give sufficient and necessary conditions for L to satisfy that the number of negative (resp. positive) eigenvalues is equal to one of the points x on which V(x) is negative (resp. positive). In addition, we prove that L has at least one discrete eigenvalue. If ∑ _x∈? V(x)?=?0, then L has both negative and positive discrete eigenvalues.     

Family of Pearson discrete distributions generated by the univariate hypergeometric function 3F2(α1, α2, α3; γ1, γ2; λ)

In this work we present a study of the Pearson discrete distributions generated by the hypergeometric function ₃F₂(α₁, α₂, α₃;γ₁, γ₂; λ), a univariate extension of the Gaussian hypergeometric function, through a constructive methodology. We start from the polynomial coefficients of the difference equation that lead to such a function as a solution. Immediately after, we obtain the generating probability function and the differential equation that it satisfies, valid for any admissible values of the parameters. We also obtain the differential equations that satisfy the cumulants generating function, moments generating function and characteristic function, From this point on, we obtain a relation in recurrences between the moments about the origin, allowing us to create an equation system for estimating the parameters by the moment method. We also establish a classification of all possible distributions of such type and conclude with a summation theorem that allows us study some distributions belonging to this family. © 1997 by John Wiley & Sons, Ltd.     

Szego-Type Limit Theorems for Generalized Discrete Convolution Operators

We study the asymptotic behavior of the averaged f-trace of a truncated generalized multidimensional discrete convolution operator as the truncation domain expands. By definition, the averaged f-trace of a finite-dimensional operator A is equal to , where n is the dimension of the space in which the operator A acts, the set of numbers γ_k, k = 1,..., n, is the complete collection of eigenvalues of the operator A, counting multiplicity; a generalized discrete convolution is an operator from the closure of the algebra generated by discrete convolution operators and by operators of multiplication by functions admitting a continuous continuation onto the sphere at infinity.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 265–277.Original Russian Text Copyright © 2005 by I. B. Simonenko.     

Singular integrals on the discrete Besov space B1 0,1 (Z)

Summary We show the boundedness of singular integral operators on the discrete Besov space B₁^0,1 (Z). For this purpose, we introduce discrete special molecules on B₁^0,1 (Z).     

Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L²(Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation.     

Superconvergence of tetrahedral quadratic finite elements for a variable coefficient elliptic equation

For a variable coefficient elliptic boundary value problem in three dimensions, using the properties of the bubble function and the element cancelation technique, we derive the weak estimate of the first type for tetrahedral quadratic elements. In addition, the estimate for the W^1,1‐seminorm of the discrete derivative Green's function is also given. Finally, we show that the derivatives of the finite element solution u_h and the corresponding interpolant Π₂u are superclose in the pointwise sense of the L^∞‐norm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Estimating the spectral radius of a real matrix by discrete Lyapunov equation

For any real matrix A, this paper is concerned with the estimation of the spectral radius of A. The relationship between the weighted norm and the discrete Lyapunov equation of the matrix A is obtained. On the basis of the relationship, an iterative algorithm is presented to obtain the spectral radius of A and to estimate the solution of the corresponding linear discrete system. Several numerical examples are given to show that the iterative algorithm is effective.     

Pointwise supercloseness of pentahedral finite elements

This article derives the weak estimate of the first type for pentahedral finite elements over uniform partitions of the domain for the Poisson equation. The estimate for the W^1,1‐seminorm of the discrete derivative Green's function is also given. Using these two estimates, we obtain the pointwise supercloseness of derivatives of the pentahedral finite element approximation and the interpolant to the true solution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A discrete Fourier transform based on Simpson's rule

Fourier analysis plays a vital role in the analysis of continuous‐time signals. In many cases, we are forced to approximate the Fourier coefficients based on a sampling of the time signal. Hence, the need for a discrete transformation into the frequency domain giving rise to the classical discrete Fourier transform. In this paper, we present a transformation that arises naturally if one approximates the Fourier coefficients of a continuous‐time signal numerically using the Simpson quadrature rule. This results in a decomposition of the discrete signal into two sequences of equal length. We show that the periodic discrete time signal can be reconstructed completely from its discrete spectrum using an inverse transform. We also present many properties satisfied by this transform. Copyright © 2012 John Wiley & Sons, Ltd.     

Wavelet approximations on closed surfaces and their application to boundary-value problems of potential theory

Wavelets on closed surfaces in Euclidean space ℝ³ are introduced starting from a scale discrete wavelet transform for potentials harmonic down to a spherical boundary. Essential tools for approximation are integration formulas relating an integral over the sphere to suitable linear combinations of function values (resp. normal derivatives) on the closed surface under consideration. A scale discrete version of multiresolution is described for potential functions harmonic outside the closed surface and regular at infinity. Furthermore, an exact fully discrete wavelet approximation is developed in case of band-limited wavelets. Finally, the role of wavelets is discussed in three problems, namely (i) the representation of a function on a closed surface from discretely given data, (ii) the (discrete) solution of the exterior Dirichlet problem, and (iii) the (discrete) solution of the exterior Neumann problem. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Initial-value problems for linear partial difference equations

The method presented in [4] for the solution of linear difference equations in a single variable is extended to some equations in two variables. Every linear combination of a given functionf and of its partial differences can be obtained by the discrete convolution product off by a suitable functionl (which depends on the considered linear combination), and we want to solve in a convolutional form difference equations in the whole plane. However, the convolution of two functions may not be possible if their supports contain half straight lines with opposite directions. To avoid this, we take four sets of functions corresponding to the quadrants such thatl belong to every set, every set endowed with the convolution and with the usual addition is a ring, and there is an inverse ofl in each of the four rings. This is attained by taking, for each ring, a set of functions whose supports belong to suitable cones. After choosing such rings, a very natural initial-value first-order Cauchy Problem (in partial differences) is reduced to a convolutional form. This is done either by a direct method or by introducing the forward difference functions _i f(i=1,2) in a general way depending on the shape of the support off so that Laplace-like formulas with initial and final values) hold. Applications to difference equations in the whole plane and to partial differential problems are made.     

Discrete semi-stable distributions

The purpose of this paper is to introduce and study the concepts of discrete semi-stability and geometric semi-stability for distributions with support inZ ₊. We offer several properties, including characterizations, of discrete semi-stable distributions. We establish that these distributions posses the property of infinite divisibility and that their probability generating functions admit canonical representations that are analogous to those of their continuous counterparts. Properties of discrete geometric semi-stable distributions are deduced from the results obtained for discrete semi-stability. Several limit theorems are established and some examples are constructed.     

Iterative learning control for linear discrete delay systems via discrete matrix delayed exponential function approach

ABSTRACT

This paper proposes iterative learning control (ILC) for linear discrete delay systems with randomly varying trial lengths without knowing prior information on the probability distribution of random iteration length. Based on matrix delayed exponential function approach, an explicit solution to the linear discrete delay controlled systems is used to generate a sequence of outputs that approximate the desired reference by adopting two ILC update laws in the presence of randomly iteration-varying lengths. A new and direct mathematical technique is explored to deal with ILC for linear discrete delay systems. Two illustrative examples are provided to verify the theoretical results.     

Pointwise supercloseness of tensor‐product block finite elements

In this article, we first introduce interpolation operator of projection type in three dimensions, from which we then derive weak estimates for tensor‐product block finite elements of degree m ≥ 1. Finally, using estimates for the discrete Green's function and the discrete derivative Green's function, we prove that both of the gradient and the function value of the finite element solution u_h and the corresponding interpolant Π_mu of projection type and degree m are superclose in the pointwise sense of the L^∞‐norm. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On Periodic Solutions of Second Order Linear Difference Equations

In this paper, we apply a new procedure initially developed in Refs. [H. El-Owaidy and H.Y. Mohamed. "On the periodic solutions for nth order difference equations". Journal of Applied Mathematics and Computation , (to appear); "The necessary and sufficient conditions of existence of periodic solutions of nonautonomous difference equations". Journal of Applied Mathematics and Computation , (to appear)] to simplify the use of Carvalho's method to the case of discrete difference equations, in order to find the periodic solutions of second order linear difference equations. We can also find the complex periodic solutions.     

Sufficient Conditions for the Oscillation of Delay Difference Equations

The most important result of this paper is a new oscillation criterion for delay difference equations. This criterion constitutes a substantial improvement of the one by Ladas et al. [J. Appl. Math. Simulation 2 (1989), 101–111] and should be looked upon as the discrete analogue of a well-known oscillation criterion for delay differential equations.