NEW INVERSION FORMULAS FOR THE KRÄTZEL TRANSFORMATION

We study in distributional sense by means of the kernel method an integral transform introduced by Krätzel. It is well known that the cited transform generalizes to the Laplace and Meijer transformation. Properties of analyticity, boundedness, and inversion theorems are established for the generalized transformation. 2000 Mathematics Subject Classification. Primary 44A15, 46F12.


Introduction.
In this paper, we study the following integral transform, where Z ν ρ (x) denotes the function with ρ > 0, ρ ∈ N, ν ∈ C. The K ρ ν transform is reduced to the Meijer transform when ρ = 1 and to the Laplace transform when ρ = 1 and ν = ±1/2.Zemanian realized a wide study of the Laplace and Meijer transformations, in distribution spaces (cf.[12,13]).Later, Krätzel, introduced the K ρ ν transformation, which generalizes the Meijer transform, and in a series of papers, investigated it in the classical sense (see [5,6]).In [8], Rao and Debnath investigated the K ρ ν transformation on certain spaces of distributions by means of the kernel method.Recently, the cited transformation is studied in [1] by the adjoint method on the McBride's spaces.
In [3], the study of Krätzel is continued obtaining Abelian, Tauberian theorems, and some inversion formulas in the classical sense.Moreover, we can emphasize in [2] the work developed by the authors studying the K ρ ν transform on weighted L p spaces, improving a result of [4].
Motivated by the cited papers, we accomplish a study of the K ρ ν transform by means of the kernel method on the space of distributions of compact support.
By E(I) we denote the infinitely differentiable functions φ(t), t ∈ I = (0, ∞), such that for all compact K we have for every k ∈ N. By E (I) we denote the dual space of E(I).Moreover, by D(I), D (I) denote the spaces of functions and distributions that can be found in [9,11].
Consider the following useful properties.
In [1] we see that By (1.4) and the asymptotic behaviour of Z ν ρ (x), we deduce that for certain positive constant α 1 and k ∈ N, we have for all x ∈ K, K compact and ν ∈ C.Moreover, by [5] we have that if ρ ∈ N, where ( The Mellin transform is defined by M f (s) = ∞ 0 x s−1 f (x)dx and the Mellin transform of the kernel is given by if Re s + min(0, Re ν) > 0.

The generalized
It is easy to see that Z ν ρ (xt) ∈ E(I) for x fixed, x > 0. Furthermore, if f is a locally absolutely integrable function, then the generalized transform of f is reduced to the classic K ρ ν transform.
Proposition 2.1.The operators A ρ,ν and B ρ,ν define a continuous linear mapping from E(I) into itself.

Proof. It is established without difficulty that
for every φ ∈ E(I).
We define the generalized operator A * ρ,ν on E (I) as the adjoint operator of A ρ,ν , that is, with f ∈ E (I) and φ ∈ E(I).Moreover, by Proposition 2.1, A * ρ,ν is a continuous linear mapping from E (I) into itself.Note that the same occurs for the operator B ρ,ν .
Next are established some properties of the generalized K ρ ν transformation.
It is easy to see that We must see that as h → 0, in the sense of the convergence in E(I), since the result is obtained for k = 1, by (2.5) and the continuity of f (t).
being C and α suitable constants and r ∈ N.
Proof.We know by [13, Theorem 1.8-1, pages 18-19] that there exists r ∈ N such that By the asymptotic behaviour we have for every K compact, (2.14) With the following proposition we obtain an operational formula for the generalized for x > 0 and ρ ∈ N.
Proof.By (1.6) and according to Proposition 2.1 it follows that (2.16) Now we establish an inversion theorem for the K ρ ν -transform using a similar procedure to employ by Malgonde and Saxena [7]. (2.17) We take {x r ,l } l r =0 a partition of the interval [0,N] such that d l = x r ,l − x r −1,l for each r = 1, 2,...,l.Then we can write x −s r ,l Z ν ρ x r ,l t . (2.19) in the sense of the convergence in E(I).
The following lemma is obtained from [7, Lemmas 2 and 3] with a slight modification.
By the Fubini's theorem we can interchange the order of integration and we can write (2.32) Then Lemma 2.6(i) permits us to obtain (2.33) Finally, interchanging the order of integration and using Lemma 2.6(ii) we achieve (2.34) as R → ∞.With this, Theorem 2.7 is demonstrated.By Theorem 2.7 the following uniqueness theorem is deduced.
where σ and K(s) are as in Theorem 2.7.
In the following theorem we obtain another inversion formula, in which appears a generalization of an operator of Post-Widder type, being obtained as a particular case an inversion formula for the Laplace and Meijer transformation.
Proceeding in a similar way we obtain that Then, it remains I 2 (t, k), for this, using the mean value theorem we can write

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: on E (I).Let ν ∈ C and ρ ∈ N.For every f ∈ E (I) we define the generalized K ρ ν transform by the relation