﻿The Finite Fractional Hilbert Transform


 In the present paper, we introduce the finite fractional Hilbert transform. Parseval-type identities concerning finite fractional Hilbert transform are proved. Moreover, we obtain inequality for finite fractional Hilbert transform of β— Hölder continuous functions. Applications for some functions are also provided.


Introduction
The finite Hilbert transform, when it exists, is defined as [13] T f (t) = 1 π 1 −1 f (t) t − x dx. In contrast to Hilbert transform, Cauchy's principal integral for finite Hilbert transform is over a finite interval. In literature [5,11,19,21] this finite interval is mostly assumed to be (−1, 1). Some of its properties, for example, linearity, boundedness, Parseval's formula, follows directly from the Hilbert transform.
But some other properties of finite Hilbert transform cannot be deduced from the infinite transform. As an example, the inverse of finite Hilbert transform is not unique and it depends on the function f (t). Finite Hilbert transform has been studied extensively based on of its application in the theory of aerodynamics, namely airfoil equation, image reconstruction, and for its own interest [5,16,17].
The fractional Hilbert transform has been introduced by Lohmann, Mendlovic, and Zalevsky [15]. Cusmariu and Zayed have given alternative definitions of fractional Hilbert transform [6,7,14,22]. The fractional Hilbert transform of a function f ∈ L p (1 ≤ p < ∞) is defined as [22] H The reader is referred to [12,18,20,23] for a detailed discussion on fractional Hilbert transform.
The following results play a vital role in the study of fractional Hilbert transform.
These well known results will be of use in later applications. [13] Let f ∈ L p for p > 1, and g ∈ L q such that (1/p) + (1/q) = 1. Then
Theorem 1.4 [8] If f is a β− Hölder continuous on (−1, 1) then In the present paper, we define finite fractional Hilbert transform, we consider some of its identities which will play a vital role in the study of this transform.
Few results of finite Hilbert transform in [13] are modified. Plots for functions of interest are also provided. The organization of the paper is as follows. In Section 2 we define finite fractional Hilbert transform. In Section 3 we discuss Parsevaltype identities. In Section 4 we study the boundedness of finite fractional Hilbert transform. Finally, in Section 5 plots for functions of interest are provided.

Finite Fractional Hilbert Transform
In the present paper we introduce the following definition of finite fractional cot α x − t f (x)dx, which exists a.e. for t ∈ (−1, 1).

Some Identities for Finite Fractional Hilbert Transform
The following results hold. Theorem 3.1 Let f ∈ L p and g ∈ L q such that p > 1, q > 1, and 1 p + 1 q = 1.If both f and g vanish identically outside of the interval (−1, 1), then Proof. The proof of first equation follows directly from Parseval-type identity for fractional Hilbert transform 1.1.

The second equation is analog for the finite fractional Hilbert transform of modified
Parseval-type identity given for finite Hilbert transform [13, p. 559].

Bounds for Finite Fractional Hilbert transform
Many authors have investigated inequalities for the finite Hilbert transform [9,10]. In this section, we obtain an inequality for finite fractional Hilbert transform of a β−Hölder continuous functions which is a modification of a result in [8, Then for all x, t ∈ (−1, 1).

Proof.
T , Taking modulus on both sides, we get Using eq.1.3, gives us the second inequality. Thus,

This completes the proof.
Using the triangle inequality for complex numbers, we can state and prove the following basic result which gives us bounds for finite fractional Hilbert transform.
exists for all t ∈ (−1, 1), then we have the inequality

Numerical Example
By using inequality 4.2, we define a lower bound for finite fractional Hilbert