Plancherel formula for the Shehu transform

. We discuss some existence conditions of the Shehu transform, provide a Plancherel formula and also relate the Shehu equicontinuity to exponential L 2 -equivanishing.


Introduction
Integral transforms have many applications in various fields of mathematical sciences and engineering such as physics, mechanics, chemestry, acoustic, etc. For example, integral transforms such as the Fourier transform and the Laplace transform are highly efficient in signal processing and solving differential equations. In [9] the authors introduced a Laplace-type integral which they called the Shehu transform. The Shehu transform of a function f : R + → C is defined by provided that this integral exists, the symbol R + stands for the set of nonnegative real numbers. This integral transform generalizes both the Laplace transform [5] and the Yang transform [10]. Many authors used the Shehu transform to solve partial or ordinary differential equations related to real life problems [4], [1], [2], [3], [7]. Authors in [6] extended the Shehu transform to distributions and measures. This paper is mainly devoted to search a Plancherel formula for the Shehu transform. The remainder of the paper is structured as follows. In Section 2 we discuss some existence conditions after replacing the first variable of the Shehu transform of a function with a complex variable and in Section 3, a Plancherel formula is given and Shehu equicontinuity and exponential L 2 -equivanishing are related.

Extension to complex variables and existence conditions
In order to obtain the Plancherel theorem in the next section for the Shehu transform, we want the first variable of S{f }(s, u) to be a complex variable. This is why we consider the Shehu transform of the function f : In what follows, we discuss some existence conditions. The symbol R e (z) denotes the real part of the complex number z. The complex vector spaces L 1 (R + ) and L 2 (R + ) are and Proof. Assume f ∈ L 2 (R + ). Set z = x + iy with R e (z) = x > 0. Then

Plancherel formula and applications
In harmonic analysis, the Plancherel theorem holds for the Fourier transform. it states that the L 2 -norms of a function in the time domain and that of its Fourier transform in the Fourier domain are equal. In other words it expresses conservation of energy for signals in the time domain and the Fourier domain. We would like to obtain the analogue of this result for the Shehu transform. To achieve our gaol we apply the Plancherel formula for the Laplace transform proved in [8]. We start with the following definition.
Hereafter is the analogue of the Plancherel formula for the Shehu transform.
Proof. Assume that f is a Laplace-Pego function of order x u . From Theorem 3.2, we have Replacing x and y by x u and y u respectively we obtain A family A of Laplace-Pego functions of common order x is said to be exponentially L 2equivanishing at x if for any given strictly positive number ε, we can find a strictly positive number T that depends only on ε and the order x, such that for any function f in A, we have The author in [8] related the concept of exponential L 2 -equivanishing to the notion of Laplace equicontinuity. We obtain the analogue result for Shehu equicontinuity as an application of the Plancherel formula. |S{f }(x + iy + η, u) − S{f }(x + iy, u)| 2 dy < ε. (3.4) The Plancherel formula If A is a family of Lapalce-Pego functions of common order α, we set We recall that f α (t) = f (t)e −αt , t ≥ 0.
Theorem 3.6. Let A be a family of Laplace-Pego functions with common order then the inverse is also true.
Proof. We follow the great lines of the proof of [8,Theorem 6]. Let ε > 0. Let η be such that (3.4) holds. There Thus A is exponentially L 2 -equivanishing at x u .
Now assume that A x u is L 2 -bounded and that A is exponentially L 2 -equivanishing at where g(t) = (e − η u t − 1)f (t). Now, using Theorem 3.3 we have Anaté K. Lakmon and Yaogan Mensah On the other hand, note that e − η u t − 1 2 < 1. It follows that Therefore, B < 2ε, and hence A is Shehu equicontinuous at x. ■

Conclusion
In this paper, some existence conditions of the Shehu integral transform of a function have been discussed, a Plancherel formula provided and Shehu equicontinuity and exponential L 2 -equivanishing are related.