Convolution theorem for fractional cosine‐sine transform and its application

Fractional cosine transform (FRCT) and fractional sine transform (FRST), which are closely related to the fractional Fourier transform (FRFT), are useful mathematical and optical tool for signal processing. Many properties for these transforms are well investigated, but the convolution theorems are still to be determined. In this paper, we derive convolution theorems for the fractional cosine transform (FRCT) and fractional sine transform (FRST) based on the four novel convolution operations. And then, a potential application for these two transforms on designing multiplicative filter is presented. Copyright © 2016 John Wiley & Sons, Ltd.     

On the Hartley superconvolution

In this paper we introduce some superconvolutions related to the Hartley transform, and apply them to solve an integral equation of Toeplitz plus Hankel type.     

A discrete convolution involving Fourier sine and cosine series and its applications

ABSTRACT

In this paper, we introduce a discrete convolution involving both the Fourier sine and cosine series. We study Young's type inequality and a discrete transform related to this convolution and solve in closed form a class of discrete Toeplitz plus Hankel equations.     

On the stability of the generalized sine functional equations

The aim of this paper is to study the stability problem of the generalized sine functional equations as follows：
g（x）f（y）=f（x＋y/2）^2-f（x-y/2）^2 f（x）g（y）=f（x＋y/2）^2-f（x-y/2）^2,g（x）g（y）=f（x＋y/2）^-f（x-y/2）^2
Namely, we have generalized the Hyers Ulam stability of the （pexiderized） sine functional equation.     

On the inversion of the convolution and Laplace transform

We present a new inversion formula for the classical, finite, and asymptotic Laplace transform of continuous or generalized functions . The inversion is given as a limit of a sequence of finite linear combinations of exponential functions whose construction requires only the values of evaluated on a Müntz set of real numbers. The inversion sequence converges in the strongest possible sense. The limit is uniform if is continuous, it is in if , and converges in an appropriate norm or Fréchet topology for generalized functions . As a corollary we obtain a new constructive inversion procedure for the convolution transform ; i.e., for given and we construct a sequence of continuous functions such that .

     

New fractional integral formulas and kinetic model associated with the hypergeometric superhyperbolic sine function

In the paper, we consider some fractional integral formulas in terms of the Riemann–Liouville, Erdélyi–Kober type, and Weyl fractional integral operators and present the general fractional kinetic model involving the hypergeometric superhyperbolic sine function via the Gauss hypergeometric series.     

On a generalized Hankel type convolution of generalized functions

The classical generalized Hankel type convolution are defined and extended to a class of generalized functions. Algebraic properties of the convolution are explained and the existence and significance of an identity element are discussed.     

Supercharacters and the discrete Fourier,cosine, and sine transforms

Using supercharacter theory, we identify the matrices that are diagonalized by the discrete cosine and discrete sine transforms, respectively. Our method affords a combinatorial interpretation for the matrix entries.     

The generalized localization for multiple Fourier integrals

In this paper we investigate almost-everywhere convergence properties of the Bochner-Riesz means of N-fold Fourier integrals under summation over domains bounded by the level surfaces of the elliptic polynomials. It is proved that if the order of the Bochner-Riesz means s?(N−1)(1/p−1/2), then the Bochner-Riesz means of a function f∈Lp(RN), 1?p?2 converge to zero almost-everywhere on RN?supp(f).     

On solvability of singular integral-differential equations with convolution

In this paper, we study a class of singular integral-different equations of convolution type with Cauchy kernel. By means of the classical boundary value theory, of the theory of Fourier analysis, and of the principle of analytic continuation, we transform the equations into the Riemann-Hilbert problems with discontinuous coefficients and obtain the general solutions and conditions of solvability in class $\{0\}$. Thus, the result in this paper generalizes the classical theory of integral equations and boundary value problems.     

A convolution for the Hankel–Kontorovich–Lebedev transformation

The analysis of a kernel involving the product of three Bessel functions motivates the introduction of the translation operator and the convolution associated to the Hankel–Kontorovich–Lebedev tranformation, first in a classical framework, and then in certain spaces of generalized functions. The main properties of this convolution are investigated, the more important operational rules are obtained and some applications are shown. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Demystification of the geometric Fourier transforms and resulting convolution theorems

T. Qian As it will turn out in this paper, the recent hype about most of the Clifford–Fourier transforms is not thoroughly worth the pain. Almost everyone that has a real application is separable, and these transforms can be decomposed into a sum of real valued transforms with constant multivecor factors. This fact makes their interpretation, their analysis, and their implementation almost trivial. Copyright © 2015 John Wiley & Sons, Ltd.     

On the theory of convolution equations of the third kind

The autoconvolution equation of the third kind with coefficient of general power type is dealt with by the method of weighted norms developed for equations with coefficients of linear and integer power type in recent joint work of the author with L. Berg, J. Janno, and B. Hofmann. For this equation two existence theorems and a uniqueness theorem are proved. Further, as an auxiliary equation a linear singular integral equation of Abel is treated anew and the existence of solutions to a related class of linear Volterra equations of the third kind is derived.     

The boundaries of the solutions of the linear Volterra integral equations with convolution kernel

Some boundaries about the solution of the linear Volterra integral equations of the second type with unit source term and positive monotonically increasing convolution kernel were obtained in Ling, 1978 and 1982. A method enabling the expansion of the boundary of the solution function of an equation in this type was developed in I. Özdemir and Ö. F. Temizer, 2002.

In this paper, by using the method in Özdemir and Temizer, it is shown that the boundary of the solution function of an equation in the same form can also be expanded under different conditions than those that they used.

     

On the theory of convolution equations of the third kind, II

Existence results of Part I of the paper are generalized to two types of autoconvolution equations of the third kind having free terms with nonzero values at x=0 like the well-known Bernstein-Doetsch equation for the Jacobian theta zero functions. Also uniqueness results for the linear convolution equations in Part I of the paper are extended to more general function spaces. Further, a special class of integro-differential equations with autoconvolution integral and two classes of the linear singular Abel-Volterra equations are dealt with.     

Some remarks concerning the Fourier transform in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> (ℝ)

Two useful estimates are proved for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus.     

On Fourier transform of generalized Brownian functionals

Let $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>B</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ and $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ be the spaces of generalized Brownian functionals of the white noises ? and ?, respectively. A Fourier transform from $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>B</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ into $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ is defined by ??(?) = ∫$\text{S}$¹: exp[?i ∫_$\text{R}$?(t) ?(t) dt]: ${}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}\text{?(}\text{B}\text{?}\text{) dμ(}\text{B}\text{?}$), where : :${}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}$ denotes the renormalization with respect to ? and μ is the standard Gaussian measure on the space $\text{S}$¹ of tempered distributions. It is proved that the Fourier transform carries ?(t)-differentiation into multiplication by i?(t). The integral representation and the action of?? as a generalized Brownian functional are obtained. Some examples of Fourier transform are given.     

On convolution in weighted ‐spaces

A distributional generalization of Young's inequality was stated by L. Schwartz. It asserts that convolution yields a continuous bilinear map if A generalization to the weighted ‐spaces in the form was given by N. Ortner and the author in 1989. By means of interpolation theory, we improve this result with respect to the image space under certain restrictions on μ and ν. This implies limit relations in for the Poisson kernel and yields a solution of the Dirichlet problem for the half‐space with boundary values in the space By this we generalize a former result of J. Alvarez, M. Guzmán‐Partida and S. Pérez‐Esteva referring to the special case of