The aim of this article is to introduce a new definition for the Fourier transform. This new definition will be considered as one of the generalizations of the usual (classical) Fourier transform. We employ the new Katugampola derivative to obtain some properties of the Katugampola Fourier transform and find the relation between the Katugampola Fourier transform and the usual Fourier transform. The inversion formula and the convolution theorem for the Katugampola Fourier transform are considered.

It is well known that fractional calculus is a generalization of the classical integer calculus, where several types of fractional derivatives are introduced and studied such as Riemann–Liouville, Caputo, Hadamard, Weyl, and Grünwald–Letnikov; for more details, one can see [

To overcome all the difficulties raised, Khalil et al. [

Let ^{.} Then, the Katugampola derivative of

Let

Note that the Katugampola derivative satisfies product rule, quotient rule, and chain rule,, and it is consistent in its properties with the classical calculus of integer order. In addition, we have the following theorem.

Let

Let

If

Note that, for

The conformable and Katugampola derivatives have been investigated and applied to solve ordinary and partial differential equations of noninteger orders in physics, engineering, and other disciplines; some of these research works have been recently published by Anderson and Ulness [

In this research work, we are intended to introduce and study the properties of the Katugampola Fourier transform based on the Katugampola derivative.

From the literature, one can discover that several definitions of fractional Fourier transforms (not necessarily equivalent) have been introduced in recent years. They were motivated by their application to obtain solutions of the problems revealed from quantum mechanics, optics, signal processing, and others.

Negero [

Let

Let

By using Definition 3 and integration by parts, we have

But

The following Lemma is the relation between the Katugampola Fourier transform and the usual Fourier transform.

Let

We have

Then, by making the substitution

Let

The proof followed by applying the definition of the usual Fourier transform and Lemma 1.

Now, we list down some properties of the Katugampola Fourier transform in the theorem below.

Let

The proof is similar to the way as in the usual Fourier transform.

Let

We can prove this theorem by mathematical induction on

For

Now, assume that the theorem is true for a particular value of

Now, we need prove that the theorem is true for

Therefore, the theorem is true for every positive integer value of

Let

By using Lemma 1 and the definition of the Katugampola Fourier transform and changing the order of integration, we obtain the result.

Let

Then,

Here we consider the Katugampola infinite Fourier sine and cosine transforms with some of their properties. These transforms are convenient for problems over semi-infinite and some of finite intervals in a spatial variable in which the function or its derivative is prescribed on the boundary.

(Fourier sine transform). Let

(Fourier cosine transform). Let

The transforms

Let

When the physical problem is defined on a finite domain, it is generally not suitable to use transformation with infinite range of integration. In such cases, the usage of finite Fourier transform is very advantageous.

In this section we shall discuss the Katugampola finite Fourier sine and cosine transforms and some of their properties.

(Katugampola finite Fourier sine transform). Let

The inverse Fourier Katugampola sine transform is defined as follows:

(Katugampola Finite Fourier cosine transform). Let

The inverse Fourier Katugampola cosine transform is defined as follows:

Let

Let

The proof is direct from Theorem 6 by putting

The results we obtained in Corollary 4.4 are similar to the results in ([

In this paper we obtained several results that have close resemblance to the results found in classical calculus. We defined both the Katugampola infinite and finite Fourier transforms and Fourier sine and cosine transforms. Also we established some properties of these transforms which are considered as generalizations to the usual transform.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.