Integral Transforms of Fourier Cosine and Sine Generalized Convolution Type

Integral transforms of the form
 f ( x ) ↦ g ( x ) = ( 1 − d 2 / d x 2 ) { ∫ 0 ∞ k 1 ( y ) [ f ( | x + y − 1 | ) + f ( | x − y + 1 | ) − f ( x + y + 1 ) − f ( | x − y − 1 | ) ] d y + ∫ 0 ∞ k 2 ( y ) [ f ( x + y ) + f ( | x − y | ) ] d y } from L p ( ℝ + ) to L q ( ℝ + ) , ( 1 ≤ p ≤ 2 , p − 1 + q − 1 = 1 ) are studied. Watson's and Plancherel's
theorems are obtained.


Introduction
Let F c be the Fourier cosine transform [1] and let F s be the Fourier sine transform [1] In 1941, Churchill introduced the convolution of two functions f and g for the Fourier cosine transform f (y) g(x + y) + g |x − y| dy, x > 0, (1.3) and proved the following factorization equality [2]: in case the Toeplitz kernel k 2 (x) and the Hankel kernel k 1 (x) are the same [3,4].The general case is still open.The convolution of two functions f and g with the weight function γ(y) = sin y for the Fourier sine transform was introduced by Kakichev in [5] where the following factorization property has been established: Further properties of this convolution have been studied in [6].
Churchill was also the first author who introduced the generalized convolution for two different integral transforms.Namely, in 1941, he defined the generalized convolution of two functions f and g for the Fourier sine and cosine transforms and proved the following factorization identity [7]: It is easy to see that the integral equation with the Toeplitz-plus-Hankel kernel (1.5) can be written in the form where . So studying generalized convolutions may shed light on how to solve the integral equation with the Toeplitz-plus-Hankel kernel (1.5) in closed form.
In 1998, Kakichev and Thao proposed a constructive method for defining a generalized convolution for three arbitrary integral transforms (see [8]).For example, for the Fourier cosine and Fourier sine transforms, the following convolution has been introduced in [9]: Nguyen Xuan Thao et al. 3 For this convolution, the following factorization equality holds [9]: (1.12) Another generalized convolution with a weight function γ(y) = sin y for the Fourier cosine and sine transforms has been studied in [10] (1.13) It satisfies the factorization property [10] (1.14) In any convolution of two functions f and g, if we fix one function, say g, as the kernel, and allow the other function f vary in a certain function space, we will get an integral transform f → f * g.The most famous integral transforms constructed by that way are the Watson transforms that are related to the Mellin convolution and the Mellin transform [11] f Recently, a class of integral transforms that is related to the generalized convolution (1.11) has been introduced and investigated in [12].In this paper, we will consider a class of integral transform which has a connection with the generalized convolution (1.13), namely, the transforms of the form (1.16) We show that under certain conditions on the kernels k 1 and k 2 , transform (1.16) admits an inverse of similar form.We find conditions on the kernels k 1 and k 2 when transform (1.16) defines a bounded operator from Moreover, Watson-and Plancherel-type theorems for transforms (1.16) in L 2 (R + ) are also obtained.

A Watson-type theorem
Lemma 2.1.Let f ,g ∈ L 2 (R + ).Then for any x > 0, the following identity holds: (2.1) Proof.Let f 1 be the odd extension of f from R + to R and g 1 the even extension of g from R + to R. Then let F f 1 is an odd function while Fg 1 is an even function, where F is the Fourier integral transform The Parseval identity for the Fourier integral transform yields ( On the other hand, note that (F f 1 )(y)(Fg 1 )(y)cos(x + 1)y, (F f 1 )(y)(Fg 1 )(y)cos(x − 1)y are odd functions in y.Hence, their integrals over R vanish, and therefore, This completes the proof.We assumed that all the integrals over R are interpreted as Cauchy principal value integrals, if necessary.
is a necessary and sufficient condition to ensure that the integral transform is unitary on L 2 (R + ) and the inverse transformation has the form ( In particular, if h is an even or odd function such that (1 + y 2 )h(y) ∈ L 2 (R + ), then the following equalities hold: (2.9) Using the factorization equalities for convolutions (1.3), (1.6), we have (2.10) By virtue of the Parseval equalities for the Fourier cosine and sine transforms f L2(R+) = F c f L2(R+) = F s f L2(R+) and noting that k 1 and k 2 satisfy condition (2.5), we have (2.11) It follows that the transformation (2.6) is unitary.
Sufficiency.If transform (2.6) is unitary, then the Parseval identities for the Fourier cosine and sine transforms yield (2.14) The middle equality is possible if and only if k 1 and k 2 satisfy condition (2.5).This completes the proof of the theorem.

A Plancherel-type theorem
In order to examine the Plancherel-type theorem, we will need the following lemma.
Lemma 3.1.Let f and g be L 2 (R + ) functions, then Proof.Again, let f 1 be the odd extension of f from R + to R and g 1 (x) = g(|x|) the even extension of g from R + to R. By the Parseval equality, we have Then formula (3.1) holds.Formula (3.2) follows easily from formula (1.4) (3.4) The lemma has been proved.
Nguyen Xuan Thao et al. 9 Remark 3.3.Because of the definitions of f N and g N , these integrals are over finite intervals and therefore converge.
Proof.Applying the identities (3.1) and (3.2) in Lemma 3.1, we have From this and in view of Theorem 2.2, we conclude that g N ∈ L 2 (R + ).Let g be the transform of f under the transformation (2.6).Then Theorem 2.2 guarantees that g ∈ L 2 (R + ), g L2(R+) = f L2(R+) , and the reciprocal formula (2.7) holds.For g − g N , we have (3.9)