Fourier-Feynman transform, convolution and first variation

We examine several interesting relationships and expressions involving Fourier-Feynman transform, convolution product and first variation for functionals in the Fresnel class F(B) of an abstract Wiener space B. We also prove a translation theorem and Parseval's identity for the analytic Feynman integral. This revised version was published online in June 2006 with corrections to the Cover Date.     

Integration by parts formulas involving generalized Fourier-Feynman transforms on function space

In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form where denotes the Paley-Wiener-Zygmund stochastic integral .

     

Free convolution operators and free Hall transform

We define an extension of the polynomial calculus on a W^?$W^{?}$-probability space by introducing an algebra C{_Xi:i∈I}$\mathbb{C}\{X_i:i\in I\}$ which contains polynomials. This extension allows us to define transition operators for additive and multiplicative free convolution. It also permits us to characterize the free Segal–Bargmann transform and the free Hall transform introduced by Biane, in a manner which is closer to classical definitions. Finally, we use this extension of polynomial calculus to prove two asymptotic results on random matrices: the convergence for each fixed time, as N   tends to ∞, of the ?-distribution of the Brownian motion on the linear group _GLN(C)${\mathit{GL}}_N(\mathbb{C})$ to the ?-distribution of a free multiplicative circular Brownian motion, and the convergence of the classical Hall transform on U(N)$U(N)$ to the free Hall transform.     

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.

     

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.     

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.     

Double convolution integral equations involving a general class of multivariable polynomials and the multivariableH-functions

In this paper we have solved a double convolution integral equation whose kernel involves the product of theH-functions of several variables and a general class of multivariable polynomials. Due to general nature of the kernel, we can obtain from it, solutions of a large number of double and single convolution integral equations involving products of several classical orthogonal polynomials and simpler functions. We have also obtained here solutions of two double convolution integral equations as special cases of our main result. Exact reference of three known results, which are obtainable as particular cases of one of these special cases, have also been included.     

A new complex approach towards approximate inverse and generalized Radon transform

In this paper, we first consider the framework of Sobolev spaces and derive analytically a reconstruction algorithm by means of the residue theorem of complex analysis, the approximate inverse, Gaussian mollifier and integral equations. And we successfully extend Natterer’s results to the generalized Radon transform with non-uniform attenuation. Finally, we investigate the smoothing properties of the generalized Radon transform.     

New expressions of the modified generalized integral transform via the translation theorem with applications

In this paper, we obtain very natural basic formulas for the modified generalized integral transform (MGIT) on function space. In order to do this, we first introduce an MGIT of functionals on function space. We next establish some basic formulas with respect to the MGIT and the first variation. Finally, we obtain a new version of the Cameron–Storvick theorem via the translation theorem. Some applications are demonstrated as examples which are used in classification of nanoparticles.     

Relationships Involving Transforms and Convolutions Via the Translation Theorem

In this article, we use the translation theorem to obtain several relationships involving integral transforms and convolution products. In particular, we obtain several useful formulas involving various functionals, which arise naturally in quantum mechanics.     

A generalized Fourier transform and convolution on time scales

In this paper, we develop some important Fourier analysis tools in the context of time scales. In particular, we present a generalized Fourier transform in this context as well as a critical inversion result. This leads directly to a convolution for signals on two (possibly distinct) time scales as well as several natural classes of time scales which arise in this setting: dilated, closed under addition, and additively idempotent. We explore the properties of these time scales and demonstrate the utility of these concepts in discrete convolution, Mellin convolution, and transformations of a random variable.     

Isomorphism between the solution spaces of a discrete convolution equation and a convolution equation on the space of entire functions

In this paper, we solve the problem of reconstructing an arbitrary solution of a homogeneous convolution equation from its values at integer points of the real axis.     

Mean value property and a Berezin-type transform on the half-space

Let B be the open unit ball in Rn. Liu (2007) [6] has shown that if then the iterates of a Berezin-type transform of f converge to the Poisson extension of the boundary values of f, as k→∞. In this paper, we extend this to the half-space setting. First, we obtain the mean value property for harmonic functions on the half-space H. Based on this property, we define a Berezin-type transform BH and investigate the limit of the iterates of BH.     

The Kontorovich‐Lebedev transform and its associated pseudodifferential operator

In this paper, we obtained some useful estimates for convolution corresponding to Kontorovich‐Lebedev transform (KL‐transform) in Lebesgue space. Some continuity theorems for translation, convolution, and KL‐transform in test function space are discussed. Then an integral representation of pseudodifferential operator involving KL‐transform is found out, and its estimates in Lebesgue space is obtained. At the end, some applications of KL‐transform and its convolution are discussed.     

Operational,umbral methods,Borel transform and negative derivative operator techniques

ABSTRACT

Differintegral methods, namely those techniques using differential and integral operators on the same footing, currently exploited in calculus, provide a fairly unexhausted source of tools to be applied to a wide class of problems involving the theory of special functions and not only. The use of integral transforms of Borel type and the associated formalism will be shown to be an effective means, allowing a link between umbral and operational methods. We merge these two points of view to get a new and efficient method to obtain integrals of special functions and the summation of the associated generating functions as well.     

An integral transform associated to the Poisson integral and inversion of Flett potentials

We introduce a new weighted wavelet-like transform, generated by the Poisson integral and a “wavelet measure.” By making use of the relevant Calderón-type reproducing formula, we obtain an explicit inversion formula for the Flett potentials which are interpreted as negative fractional powers of the operator (E+Λ), where Λ=(−Δ)^1/2, Δ is the Laplacian and E is the identity operator.     

抽象函数空间的连续小波变换和微分方程

主要讨论了抽象函数的某些微分方程和相应的积分方程之间的关系;通过连续小波变换将这些微分方程能够转换为相应的积分方程;这些微分方程和相应的积分方程在弱收敛意义下是等价的.