Abstract
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.
Highlights
Introduction and preliminariesThe integral transform operators have attained their popularity due to their wide range of applications in various fields of science and engineering as, in most of cases, the physical phenomenon is converted into ordinary and partial differential equations
Along with interesting groups of integral transforms arising in literature, the natural transform NT was introduced by Khan and Khan [2] and renamed recently as the N -transform [3,4,5,6]
In addition to the shift and change of scale properties of the NT, the authors of [5] solved the unsteady fluid flow problem over a plane wall and highlighted that the transform converges to the Laplace and Sumudu transforms
Summary
We construct the Boehmian spaces B1 ≈ B(L1(R+), (C∞(R+), , ), ) and B2 ≈ B(L1(R2+), (C∞(R+), , ), ⊕) with the products (to act as ∗) and ⊕ (to act as ) that seem to comply with the delta sequences and the operator Mα,k. We refer to [7, 14, 15, 20–34] for an outright description and full details of abstract constructions of various Boehmian spaces and transform operators. We provide several systematic hypotheses to generate the space B2 of Boehmians. The following theorem includes a straightforward proof alluded to a simple integral calculus. Theorem 5 If U ∈ L1(R2+) and (δn) ∈ , U ⊕ δn → U as n → ∞
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