Derivation of an integral of Boros and Moll via convolution of Student t-densities

We show that the evaluation of an integral considered by Boros and Moll is a special case of a convolution result for Student t-densities obtained by the authors in 2008.     

On integral convolution operators

We study integral convolutions defined on functions ofn variables in symmetric spaces and obtain new additive estimates for the mean value of the nonincreasing permutation(K ^* f) ^** (t) of the absolute value of the integral convolution on the interval [0,t] for anyt>0. As an example of application of the estimates obtained, we prove the boundedness of the integral convolution operator acting from the intersection of symmetric spaces into the Marcinkiewicz space. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 551–555, October, 1999.     

A class of convolution integral equations

A new class of convolution integral equations whose kernels involve an H-function of several variables, which is defined by a multiple contour integral of the Mellin-Barnes type, is solved. It is also indicated how the main theorem can be specialized to derive a number of (known or new) results on convolution integral equations involving simpler special functions of interest in problems of applied mathematics and mathematical physics.     

Factorization of integral transformations of convolution type

We give a survey and recount methods of representing integral transforms of convolution type as compositions of other integral transforms, in particular direct and inverse Laplace transforms. Tables of representations are given for a wide class of integral transforms.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 21, pp. 3–41, 1983.     

The general solution of a non-linear integral equation of convolution type

All continuous, non-negative solutions of a non-linear convolution equation are explicitly determined.

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Zusammenfassung Alle stetigen nichtnegativen Lösungen einer nichtlinearen Konvolutionsgleichung werden explizit bestimmt.

     

A class of convolution integral equations involving a generalized polynomial set

The aim of this paper is to derive a solution of a certain class of convolution integral equation of Fredholm type whose kernel involves a generalized polynomial set. Our main result is believed to be general and unified in nature. A number of (known or new) results follow as special cases, simply by specializing the coefficients and parameters involved in the generalized polynomial set. For the sake of illustration, some special cases are mentioned briefly.     

On the eigenproblem for convolution integral equations

Summary This paper concerns the eigenproblem for convolution integral equations whose kernels can be expressed as finite or infinite Fourier transforms of integrable functions. A procedure which closely parallels previous work on displacement integral equations is derived and the problem of existence is treated. Approximations are obtained for both the eigenvalues and the eigenfunctions.The results of this paper are taken from the author's doctoral dissertation at the University of New Mexico. The research was supported by the United States Atomic Energy Commission.     

Remarks on the solvability of a convolution integral equation on a finite interval

We present some results on the solvability of an integral equation of the second kind with a difference kernel on a finite interval, construct a counterexample to an assertion, earlier believed to have been proved, on the solvability of this equation, and pose an open problem.     

On solving a system of singular Volterra integral equations of convolution type

This paper presents approximate analytical solutions for a system of singular Volterra integral equations of convolution type by using the fractional differential transform method. The solutions are calculated in the form of convergent series with easily computable terms and also the exact solutions can be achieved by well-known series solutions. Several examples are given to demonstrate reliability and performance of the presented method.     

An estimate for a double stieltjes integral

Department of Mathematics, The University of Alberta, Edmonton, Alberta, Canada. Published in Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 3, pp. 583–589, July–September, 1989.     

On multi-dimensional integral equations of convolution type with shift

The criterion of invertibility or Fredholmness of some multi-dimensional integral equations with Carleman type shifts are given. The investigation is based on some Banach space approach to equations with an involutive operator. A modified version of this approach is also presented in the paper.This approach is applied to multi-dimensional convolution type equations when the kernels may be integrable or of singular Calderon-Zygmund-Mikhlin type and shift generated by a linear transformation in the Euclidean space satisfying the generalized Carleman condition. The convolution type equations are also specially considered in the two-dimensional case in a sector on the plane symmetric with respect to one of the axes and the corresponding reflection shift. Another application deals with multi-dimensional equations with homogeneous kernels and the shift .