Certain fundamental properties of generalized natural transform in generalized spaces

This paper considers the definition and the properties of the generalized natural transform on sets of generalized functions. Convolution products, convolution theorems, and spaces of Boehmians are described in a form of auxiliary results. The constructed spaces of Boehmians are achieved and fulfilled by pursuing a deep analysis on a set of delta sequences and axioms which have mitigated the construction of the generalized spaces. Such results are exploited in emphasizing the virtual definition of the generalized natural transform on the addressed sets of Boehmians. The constructed spaces, inspired from their general nature, generalize the space of integrable functions of Srivastava et al. (Acta Math. Sci. 35B:1386–1400, 2015) and, subsequently, the extended operator with its good qualitative behavior generalizes the classical natural transform. Various continuous embeddings of potential interests are introduced and discussed between the space of integrable functions and the space of integrable Boehmians. On another aspect as well, several characteristics of the extended operator and its inversion formula are discussed.

On Pointwise Product Vector Measure Duality

This article is devoted to the study of pointwise product vector measure duality. The properties of Hilbert function space of integrable functions and pointwise sections of measurable sets are considered through the application of integral representation of product vector measures, inner product functions and products of measurable sets.

DIFFERENCE EQUATIONS IN A MULTIDIMENSIONAL SPACE

One considers a general difference equation in a multidimensional space with continuous coefficients in the space of integrable functions. Necessary and sufficient conditions for a Fredholm property are obtained with a help of the Fourier transform and the Riemann boundary value problem. For simplest cases solvability conditions and formula of a general solution for the difference equation are given.

LINEAR INTEGRAL VOLTERRA EQUATION OF THE FIRST KIND IN THE BANACH SPACE

We construct solutions to a singular Volterra integral equation of the first kind with а polynomial kernel in the space of integrable functions whose values be-long to a Banach space.