On the Generalized Lambert Transform over Lebesgue and Boehmian Spaces

In this paper we study the generalized Lambert transform introduced by Raina and Srivastava (Rev. Tec. Ing. Univ. Zulia, 18(3):301–304, 1995) over Lebesgue spaces. We also extend this transform to Boehmian spaces.


Introduction
Any series of the form is called a Lambert series.It is a natural generalization of the following formulas related to the number theory and are proved by Lambert [20]: where σ (n) and τ (n) represent the sum and the number of the divisors of n respectively.
In the same way as the Laplace transform is a generalization of the power series, Widder introduced the Lambert transform as a generalization of the Lambert series (1.1).
Different authors have looked into various aspects of the Lambert transform and inversion formulas.We note the following among other findings.Miller [10] studied convergence properties of Lambert transform that is needed to develop inversion formulas and introduced summability techniques for power series which use the Lambert transform [11].Widder [19] obtain an inversion formula in terms of a limit of the derivatives which involves the Möbius function.Different inversion formulae related to the inversion formula given by Widder are studied in [12,13,18].Raina and Srivastava [16] introduced a generalization of the Lambert transform related with the generalized Riemann zeta function.In Goyal and Laddha [4] the authors give the inversion formula for this generalized Lambert transform.Raina and Nahar [17] introduced a generalization of the Lambert transform related with a class of functions connected to the Hurwitz zeta function.Goldberg [2] introduced a more general kernel for the Lambert transform working with transforms as Stieltjes integrals and derived some inversion formulae.Moreover, Hayek et al. [5] studied this generalized Lambert transform on the space of distributions of compact support giving an inversion formula over this space.Ferreira and López [1] obtain asymptotic expansions of the Lambert transform for large and small values of the variable.We consider the following generalization of the Lambert transform proposed by Raina and Srivastava [16] as provided that f (t) ∈ , where denotes the class of functions f (t) which are continuous for t > 0 and satisfy the order estimates [16]: where (γ ) > −2.The parameter δ occuring in the order estimates is unrestricted.
The following inversion formula for the generalized Lambert transform (1.2) is valid for a suitable class of functions [16] as where (a, 1 − k, 1) is defined in [16] admits series as well as integral representation.Also, Moreover for a = 1 one has The case a = 1 is studied in González et al. [3].
Observe that the kernel of the transform (1.2) satisfies the relation as a sum of a geometric series.
The paper is assembled into four sections.The first section contains a introduction to the generalized Lambert transform.In Sect. 2 we analyse the generalized Lambert transform (1.2) over the spaces L 1 ((0, ∞)).We obtain L 1 − boundedness properties for this transform.We provide a Parseval-type relation for f (1.5) Finally, if (G L M) be the adjoint of the operator G L M, that is, for every f ∈ (L ∞ ((0, ∞))) and every g ∈ L 1 ((0, ∞)), then the Parseval-type relation (1.5) allows us to obtain that the operator (G L M) is the natural extension of the integral operator In Sect. 3 we analyse the Lambert transform (1.2) over the spaces L ∞ ((0, ∞)) and obatined Parseval relation.In Sect. 4 we extend the transforms (1.2) to Boehmian spaces.For it we make use of previous Boehmian space B ∨ analyzed in González [3] and we construct a new Boehmian space B .

Proposition 2.1
The generalized Lambert transform is a bounded linear operator from and F is a continuous function on (0, ∞).Moreover, the generalized Lambert transform is a continuous map from L 1 ((0, ∞)) to the Banach space of bounded continuous functions on (0, ∞).
Proof Indeed, since One has that f j (t) = f (t) Hence by the Lebesgue dominated convergence theorem we have and thus the continuity of F follows.
The next Proposition exhibits a Parseval-type relation for the generalized Lambert transform (1.2) (2.2) Similarly, we have Finally, using Fubini's theorem we obtain (2.2).
Let (G L M) be the adjoint of the operator G L M, that is,

Now we prove the connection between the operators (G L M) and
Denoting by C 0 ((0, ∞)) the Banach space with supremum norm defined by Also as for every t > 0, xt e xt −a → 0 as x → +∞, then using again dominated convergence theorem we get So, by means of Proposition 2.1, the transform G L M is a continuous linear map from

Remark 2.7
The corresponding case to q = ∞ of Proposition 2.5 is contemplated in Proposition 2.1.
Then by using the dominated convergence theorem one has Observe that where | f | ≤ M for some M > 0. Now for w > 0 a.e. on (0, ∞), x q w(x)dx.
is a bounded linear operator.
The next result exhibits a Parseval relation for the generalized Lambert transform.
x ), then the following Parseval relation holds Proof Applying Fubini's theorem in the following, we get The Fubini theorem holds here because from (3.3) one has Now, x 2 t e xt −a is bounded and continuous on (0, ∞), for each t > 0 and |a| ≤ 1, since: x 2 t e xt − a → 0, as x → 0 + , and x 2 t e xt − a → 0, for x → +∞.
Then for each t > 0: where N > 0 satisfies x 2 t e xt −a ≤ N for all x ∈ (0, ∞).

The Generalized Lambert Transform over Boehmian Spaces
In this section, we apply an appropriate Boehmian space to the generalized Lambert transform.
For the reader's convenience, we first quickly recap the concept of Boehmian space.The ring C((0, ∞)) of continuous functions on (0, ∞) with point-wise addition and the convolution defined by (φ * ψ) = t 0 φ(t − s)ψ(s)ds, is also an integral domain by the Titchmarsch theorem [6].The quotient field of the integral domain is called the Mikusiński operators, which contains the integral and differential operators as its elements.This integral domain of Mikusiński operators is applied to solve ordinary differential equations, by algebraic methods.As the convolution and it has non-zero zero divisors in C(R), it is irrelevant to talk about the quotient field of C(R).So to discuss about quotients in this sort of weaker setup J. Mikusiński and P. Mikusiński [7] introduced the notion of Boehmians spaces, which contains convolution quotients of sequences.An element of a Boehmian space is of the form , where ( f n ) is a sequence from a topological vector space G, and (φ n ) is a sequence from a commutative semigroup (S, ×) called a delta sequence satisfying the identity There are two notions of convergence, namely, δconvergence and -convergence on a Boehmian space.Most of the Boehmian spaces contains some distribution spaces as a proper subspace.For more details about the construction of a Boehmian space and the notions of convergence, we refer to [8,9,14].Numerous integral transformations have been extended to numerous Boehmian spaces since the discovery of Boehmian space.We start by thinking back to Mellin type convolutions.

Definition 4.1
The Mellin-type convolutions of suitable functions f and φ on (0, ∞) are defined by Now, we construct a Boehmian space B ∨ , with the topological vector space as L 1 ((0, ∞)), the commutative semigroup as (D((0, ∞)), ∨) where D((0, ∞)) is the Schwartz testing function space of infinitely differential functions with compact support.We also use the delta sequences used in Roopkumar and Negrín [15], which follows.
Let 1 be the class of all sequences (φ n ) from D((0, ∞)) satisfying the following conditions.

. 1 )Definition 4 . 7
Therefore, (GL M[ f n ])/(φ n ) is a quotient with respect to and hence [(G L M[ f n ])/(φ n )] ∈ B .The generalized Lambert transform of a Boehmian ( f n ) using the fact that for each t > 0, xt e xt −a → 0 as x → 0 + and | f (t)| xt e xt −a is dominated by the integrable function f (t), for all x > 0, by dominated convergence theorem, we obtain that