The Meijer transformation of generalized functions

Using a generalized form of confluent hypergeometric function [N.Virchenko: On a generalized confluent hypergeometric function and its generalizations. Fract. Calc.Appl. Anal. 9(2006), 101-108], we introduce some new integral transforms and obtain their inversion theorems. Parseval-Goldstein type relations are established. Classical integral transforms, such as Laplace, Stieltjes .Widder-Potential follow as special cases of general transforms considered here. Some examples are given.

Journal of Mathematical Analysis and Applications, 1998

The classical theory of the Weierstrass transform is extended to a generalized function space which is the dual of a testing function space consisting of purely entire functions with certain growth conditions developed by Kenneth B. Howell. An inversion formula and characterizations for this transform are obtained. A comparative study with the existing literature is also undertaken.

2016

Abstract. The familiar Beurling theorem (an uncertainty principle), which is known for the Fourier transform pairs, has recently been proved by the author for the Kontorovich-Lebedev transform. In this paper analogs of the Beurling theorem are established for certain index transforms with respect to a parameter of the modified Bessel functions. In particular, we treat the generalized Lebedev-Skalskaya transforms, the Lebedev type transforms involving products of the Macdonald functions of different arguments and an index transform with the Nicholson kernel function. We also find inversion formulas for the Lebedev-Skalskaya operators of an arbitrary index and the Nicholson kernel transform.

Zeitschrift für Analysis und ihre Anwendungen, 2002

This paper introduces, by way of constructing, specific finite and infinite integral transforms with Bessel functions J ν and Y ν in their kernels. The infinite transform and its reciprocal look deceptively similar to the known Weber transform and its reciprocal, respectively, but fundamentally differ from them. The new transform enjoys an operational property that makes it useful for applications to some problems in differential equations with non-constant coefficients. The paper gives a characterization of the image of some spaces of square integrable functions with respect to some measure under the infinite and finite transforms.

Symmetry

The research presented in this paper deals with analytic p-valent functions related to the generalized probability distribution in the open unit disk U. Using the Hadamard product or convolution, function fs(z) is defined as involving an analytic p-valent function and generalized distribution expressed in terms of analytic p-valent functions. Neighborhood properties for functions fs(z) are established. Further, by applying a previously introduced linear transformation to fs(z) and using an extended Libera integral operator, a new generalized Libera-type operator is defined. Moreover, using the same linear transformation, subclasses of starlike, convex, close-to-convex and spiralike functions are defined and investigated in order to obtain geometrical properties that characterize the new generalized Libera-type operator. Symmetry properties are due to the involvement of the Libera integral operator and convolution transform.

In this paper a complete orthonormal family of Laguerre-Bessel functions is derived and certain spaces of testing functions and generalized functions are defined, whose members can be expressed in terms of a Fourier-Laguerre-Bessel series, which gives the inversion formula for Laguerre-Finite Hankel transform of generalized functions. The convergence of the series is interpreted in the weak distributional sense. An operational transform formula is derived which together with the inversion formula is applied in solving certain distributional differential equations.

Gazi University Journal of Science

In this paper, Parseval-Goldstein type theorems involving the G ̃n-integral transform which is modified from G_2n-integral transform [7] and the -integral transform [8] are examined. Then, theorems in this paper are shown to yield a number of new identities involving several well-known integral transforms. Using these theorems and their corollaries, a number of interesting infinite integrals of elementary and special functions are presented. Generalizations of Riemann-Liouville and Weyl fractional integral operators are also defined. Some theorems relating generalized Laplace transform, generalized Widder Potential transform, generalized Hankel transform and generalized Bessel transform are obtained. Some illustrative examples are given as applications of these theorems and their results.

1993

The index 2 F 1-transform of generalized functions

Acta Applicandae Mathematicae, 2004

Analysis of function spaces and special functions are closely related to the representation theory of Lie groups. We explain here the connection between the Laguerre functions, the Laguerre polynomials, and the Meixner-Pollacyck polynomials on the one side, and highest weight representations of Hermitian Lie groups on the other side. The representation theory is used to derive differential equations and recursion relations satisfied by those special functions.