The Lambert transform over distributions of compact support, $$L^1$$ L 1 -functions a
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Annals of Functional Analysis https://doi.org/10.1007/s43034-020-00103-8 ORIGINAL PAPER
The Lambert transform over distributions of compact support, L1‑functions and Boehmian spaces Benito J. González1 · Emilio R. Negrín1 · R. Roopkumar2 Received: 4 August 2020 / Accepted: 26 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract In this paper, we study the Lambert transform over distributions of compact support on (0, ∞) . We obtain an inversion formula for this transform and we prove a Parseval-type relation for the Lambert transform of functions in L1 ((0, ∞)) . We also extend this transform to Boehmian spaces. Keywords Lambert transform · distributions of compact support · L1-functions · Parseval-type relation · Mellin-type convolutions · Boehmian spaces Mathematics Subject Classification 46E30 · 47G07 · 44A05
1 Introduction and preliminaries The Lambert transform was introduced by Widder [21] inspired by the series of type ∞ ∑ n=1
an
xn 1 − xn
studied by Lambert in investigations on the theory of numbers. Widder replace n by the continuous variable t, x by e−x and the series by an integral and thus one has Communicated by Sorina Barza. * Benito J. González [email protected] Emilio R. Negrín [email protected] R. Roopkumar [email protected] 1
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de La Laguna (ULL), Campus de Anchieta, 38271 La Laguna, Tenerife, Spain
2
Department of Mathematics, Central University of Tamil Nadu, Thiruvarur 610101, India
Vol.:(0123456789)
B. J. González
F(x) =
∫0
∞
f (t)
1 dt. ext − 1
(1)
Since the denominator of the integrand vanishes at t = 0 , it is convenient to replace f(t) by tf(t) and thus introduce the bounded kernel extxt−1. In this paper, we consider the Lambert transform of a function f in L1 ((0, ∞)) by means of
F(x) =
∫0
∞
f (t)
xt dt, ext − 1
x > 0.
(2)
The Lambert transform has been the subject of research in several works. Goldberg [2] extended the Lambert transform by means of
F(x) =
∫0
∞
f (t)
∞ ∑ k=0
ak e−kxt dt,
x > 0,
(3)
for a certain class of coefficients ak . For ak = 1 , for all k = 1, 2, … , the generalized Lambert transform (3) agrees with the Lambert transform (1). Other outstanding references on the Lambert transform are [1, 7, 14, 22], amongst others. Raina and Srivastava [18] introduced a generalization of the Lambert transform related with the generalized Riemann zeta function. The study of the Lambert transform over spaces of distributions has been the subject of several papers. In [13], Negrín extended the Lambert transform to the space E� ((0, ∞)) of distributions of compact support on (0, ∞) given by ⟩ ⟨ 1 , x > 0, f ∈ E� ((0, ∞)). F(x) = f (t), xt (4) e −1 Hayek et al. [3] extended the generalized Lambert transform to the context of compactly supported distributions on (0, ∞) given by ⟨ ⟩ ∞ ∑ f (t), ak e−kxt , x > 0. k=0
Roopkumar and Negrín [17] exhibit the exchange formula for the generalized Lambert transform. Continuity for this transform from E� ((0, ∞)) into E(
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