The Watson Fourier transform on a certain class of generalized functions
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The Watson Fourier transform on a certain class of generalized functions El Mehdi Loualid1 · Imane Berkak2 · Radouan Daher2 Received: 17 July 2019 / Accepted: 2 October 2020 © Springer-Verlag Italia S.r.l., part of Springer Nature 2020
Abstract In this paper, we first build an appropriate Boehmian space on which a classical theory of the Watson Fourier transform is extended. We provide convolution products, convolution theorems and generate their associated spaces of Boehmian. The properties of the extended Watson Fourier transform are also established. Moreover, continuity with respect to 𝛿 and Δ-convergence is discussed. Keywords Watson Fourier transform · Generalized function · Boehmian space Mathematics Subject Classification 42B35 · 44A35 · 42A38
1 Introduction and preliminaries Motivated by the Boehme’s regular operators [1], a generalized function space called Boehmian space is introduced by J. Mikusinski and P. Mikusinski [2] and two notions of convergence, called 𝛿-convergence and Δ-convergence, on a Boehmian space are introduced in [3]. The algebraic construction of Boehmians is similar to the construction of the field of quotients. The main difference is that Boehmians can be constructed even if the ring has divisors of zero. Thereafter, various Boehmian spaces have been defined and also various integral transforms have been extended on them [3–13]. If the construction is applied to a function space and the multiplication is interpreted as convolution, the construction yields a space of generalized functions. Those spaces provide a natural setting for extensions of * El Mehdi Loualid [email protected] Imane Berkak [email protected] Radouan Daher [email protected] 1
Laboratory of Engineering Sciences for Energy, National School of Applied Sciences of El Jadida, University Chouaib Doukkali, 24000 El Jadida, Morocco
2
Laboratory: Topology, Algebra, Geometry and Discrete Mathematics, Department of Mathematics and Informatics, Faculty of Sciences Aïn Chock, University of Hassan II, B.P 5366, Maarif, Casablanca, Morocco
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the Watson Fourier transform which generalizes many integrals transforms, e.g., Y transform, Hankel transform, G- and H-transforms. We cite here, as briefly as possible, some facts about harmonic analysis related to the Watson Fourier transform. For more details we refer to [2, 14, 15]. We denote by: • Lp = Lp (0, ∞) the class of measurable functions f on (0, ∞) for which ‖f ‖p < ∞, where
� ‖f ‖p =
∫0
�1
∞
�f (x)�p dx
p
, if p < ∞,
‖f ‖∞ = ess supx∈(0,∞) �f (x)�. • D(0, ∞) the space of test functions with bounded support over (0, ∞). • D� (0, ∞) the space of compactly supported distributions over (0, ∞).
From [16], we recall the definitions of Mellin transform and its inverse.
Definition 1.1 Let f(t) be a function defined on the positive real axis 0 < t < ∞ . The Mellin transformation M is the operation mapping of the function f into the function F
defined on the complex plane by the relation:
M[f ∶ s] = F(s) =
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