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The problem of the integrability of Mellin transforms is presented. Sufficient Lipschitz conditions are given to solve this problem. These results are inspired by well-known works of Titchmarsh in classical Fourier harmonic analysis. Some results on the integrability of Mellin transforms of the Mellin convolutions are also given. [ABSTRACT FROM AUTHOR]

Usually, convolution refers to Laplace convolution in the literature, but Mellin convolutions can yield very ueful results. This aspect is illustrated in the coming sections. This study deals with Mellin convolutions of products and ratios. Functions belonging to the pathway family of functions are considered. Several types of integral representations, their equivalent representations in terms of G and H-functions, and their equivalent computable series representations are examined in this study.Mathematics Subject Classification 2010: 26A33, 44A10, 33C60, 35J10.

In this paper, we study the convergence properties of certain semi‐discrete exponential‐type sampling series in a multidimensional frame. In particular, we obtain an asymptotic formula of Voronovskaya type, which gives a precise order of approximation in the space of continuous functions, and we give some particular example illustrating the theory. Applications to the study of the seismic waves are illustrated. [ABSTRACT FROM AUTHOR]

Some fifteen G-functions are given two integral representations and a series form each through an interesting idea of constructing statistical distributions of products and ratios of statistically independently distributed real positive scalar random variables which are equivalent to computing the Mellin convolutions of products and ratios involving two functions. These integral representations and the explicit series forms may not be available in the literature. For special values of the parameters some integral representations and series forms are available in the literature for some of the G-functions considered in this paper. But the representations given in this paper are for general parameters. [ABSTRACT FROM AUTHOR]

A real scalar variable integral is known in the literature by different names in different disciplines. It is basically a Bessel integral called specifically Krätzel integral. An integral transform with this Krätzel function as kernel is known as Krätzel transform. This article examines some mathematical properties of Krätzel integral, its connection to Mellin convolutions and statistical distributions, its computable representations, and its extensions to multivariate and matrix-variate cases, in both the real and complex domains. An extension in the pathway family of functions is also explored.

Mellin convolution equations acting in Bessel potential spaces are considered. The study is based upon two results. The first one concerns the interaction of Mellin convolutions and Bessel potential operators (BPOs). In contrast to the Fourier convolutions, BPOs and Mellin convolutions do not commute and we derive an explicit formula for the corresponding commutator in the case of Mellin convolutions with meromorphic symbols. These results are used in the lifting of the Mellin convolution operators acting on Bessel potential spaces up to operators on Lebesgue spaces. The operators arising belong to an algebra generated by Mellin and Fourier convolutions acting on Lp-spaces. Fredholm conditions and index formulae for such operators have been obtained earlier by one of the authors and are employed here. The results of the present work have numerous applications in boundary value problems for partial differential equations, in particular, for equations in domains with angular points. [ABSTRACT FROM AUTHOR]

Starting from Gauss hypergeometric function, which Ramanujan studied in detail, limiting forms are considered and pathways are constructed to go from a Gauss hypergeometric series to confluent hypergeometric series to binomial series to Bessel series and finally to exponential series. The path from binomial to exponential is discussed in detail. It is shown that this path is connected to reaction-rate probability integrals, non-extensive statistical mechanics, Tsallis statistics, superstatistics, Kratzel integrals, inverse Gaussian density Bayesian procedures and generalizations of these. Products of binomials forms are shown to be connected to Mellin convolutions, densities of products and ratios and finally to fractional integrals. Some aspects of the corresponding theory of functions of matrix argument are also discussed. [ABSTRACT FROM AUTHOR]

In this article we give an overview of the recent developments in the area of fractional integrals and fractional derivatives. A new definition is given by this author in terms of Mellin convolutions of ratios and products in the case of real scalar variables and M-convolutions of ratios and products in the case of matrix variables, where one of the functions is a type-1 beta type so that all the definitions available in the literature for fractional integrals can be brought under one definition. Once the fractional integrals are defined, fractional derivatives can be defined as certain fractional integrals so that the results coming from fractional derivatives can describe global activities compared to integer order derivatives which can describe only local activities at a point. When fractional derivatives are defined as certain fractional integrals then these derivatives cover not only given points of interest but also their neighborhoods so that fractional derivatives become more useful in practical applications. An ideal situation may be a local activity but in reality the real-life situation may be in the neighborhood of the ideal case. The new definition is also extended to real matrix-variate case as well as to complex matrix-variate case. Thus, for the first time, fractional calculus of functions of complex variables is also given through the new definition. [ABSTRACT FROM AUTHOR]

Abstract: We present a method for deriving asymptotic expansions of integrals of the form for small x based on analytic continuation. The expansion is given in terms of two asymptotic sequences, the coefficients of both sequences being Mellin transforms of h and . Many known and unknown asymptotic expansions of important integral transforms are derived trivially from the approach presented here. This paper reconsiders earlier work of McClure and Wong [Explicit error terms for asymptotic expansions of Stieltjes transforms, J. Inst. Math. Appl. 22 (1978) 129–145; Exact remainders for asymptotic expansions of fractional integrals, J. Inst. Math. Appl. 24 (1979) 139–147] and Asymptotic approximations of integrals, Academic Press, New York, 1989. Chaps. 5, where elements of distribution theory are used, and Wong [Explicit error terms for asymptotic expansions of Mellin convolutions, J. Math. Anal. Appl. 72(2) (1979) 740–756], where, as in the present paper, the asymptotic expansions are obtained without the use of distributions. In this paper we re-derive the expansions given in Wong [Explicit error terms for asymptotic expansions of Mellin convolutions, J. Math. Anal. Appl. 72(2) (1979) 740–756] by using a different approach and we obtain new results which are not present in Wong [Explicit error terms for asymptotic expansions of Mellin convolutions, J. Math. Anal. Appl. 72(2) (1979) 740–756]: a proof of the asymptotic character of the expansions and accurate error bounds. [Copyright &y& Elsevier]

A real scalar variable integral is known in the literature by different names in different disciplines. It is basically a Bessel integral called specifically Krätzel integral. An integral transform with this Krätzel function as kernel is known as Krätzel transform. This article examines some mathematical properties of Krätzel integral, its connection to Mellin convolutions and statistical distributions, its computable representations, and its extensions to multivariate and matrix-variate cases, in both the real and complex domains. An extension in the pathway family of functions is also explored. [ABSTRACT FROM AUTHOR]