Generalized convolutions for the Fourier integral transforms and applications Текст научной статьи по специальности «Математика»

УДК 517.5

Generalized Convolutions for the Fourier Integral Transforms and Applications

Bui Thi Giang

Department of Basic Science Institute of Cryptography Science 141 Chien Thang Str., Hanoi

Vietnam

Nguyen Minh Tuan*

Department of Mathematical Analysis University of Hanoi 334 Nguyen Trai Str., Hanoi

Vietnam

Received 15.09.2008, received in revised form 20.10.2008, accepted 15.11.2008 This paper provides some generalized convolutions for the Fourier integral transforms and treats the applications. Namely, there are six generalized convolutions with weight-function for the Fourier integral transforms. As for applications, the normed ring structures on L1(Rd) are constructed, and the explicit solution in L1(Rd) of the integral equations with the mixed Toeplitz-Hankel kernel are obtained.

Keywords: generalized convolution, normed ring, integral equation of convolution type.

Introduction

The Fourier transform and its inverse transform are defined as:

(Ff )(x) = -i-r i e-i<x'y> f (y)dy, (F-lf )(x) = -i-d f ei<x'y> f (y)dy, (2n) 2 J (2n)2 J

Rd Rd

where <x, y> denotes the scalar product of x,y G Rd, and f (x) is a function (real or complex) defined on Rd. The integral transform

(f * g)(x) = 7-—d i f (x - y)g(y) dy

F

(2n)d

is called the Fourier convolution of the two functions f and g, and it is applied in many fields of mathematics.

In 1940, Churchill gave an idea of the generalized convolutions of integral transforms, and found an application for solving boundary value problems (see Churchill [1]). In 1958, Vilenkin formulated the convolution of the integral transforms in the specific space of integrable functions (see Vilenkin [2]). In 1967, the designated methods for convolutions and generalized convolutions of the integral transforms were proposed by Kakichev, and in 1990 a concept of generalized convolutions of the linear operators was first introduced (see [3, 4]). In 1998, the generalized convolution of the Fourier-cosine and Fourier-sine transforms was presented (see [5]).

* e-mail: nguyentuan@vnu.edu.vn © Siberian Federal University. All rights reserved

In recent years, many papers were devoted to the well-known integral transforms for given convolutions, generalized convolutions, polyconvolutions and their applications (see [6]—[12]). However, there are not so many generalized convolutions for the integral transforms. Generally speaking, each of convolutions is a new transform which can become an object of study. In our view, the generalized convolutions and their applications deserve the interestthat they have attracted.

The main purpose of this paper is to construct some generalized convolutions with weight for the Fourier integral transforms.

The paper is divided into three sections and organized as follows.

In Section 1, there are six generalized convolutions with the weight being one of the functions e-i<x,h>, ei<x,h>, cog xh, gjn xh for the Fourier integral transforms. We call h the shift in the convolution transform. One interesting fact possessed by the factorization identities of those convolutions is that the shift in the convolution expression is only moved into the weight-function in the right-hand-side. We think this is the main reason for the solvability of the integral equations with different shifts as equation (2.2).

In Subsection 2.1, the linear space L1(Rd), equipped with each of the convolution multiplications, becomes the normed ring. In Subsection 2.2, using the convolutions in Section 1 we investigate the integral equations with the mixed Toeplitz-Hankel kernel having shifts and obtain explicit solutions in L1(Rd) of those equations.

1. Generalized Convolutions

The concept of generalized convolutions with weight is a nice idea focusing on the so-called factorization identity. We now deal with this concept.

Let U1,U2,U3 be linear spaces over the field of scalars K, and let V be a commutative algebra over K. Suppose that K1 G L(U1,V), K2 G L(U2,V), K3 G L(U3,V) are linear operators from U1,U2,U3 to V respectively. Let 0 denote an element in algebra V. The following definition is a formulation of the idea of convolutions and generalized convolutions (see [5]).

Definition 1.1. A bilinear map * : U1 x U2 :—► U3 is called convolution with weight-element 0 for K3,K1,K2 (that in order) if K3(*(f,g)) = 0K1(f)K2(g) for any f G U1,g G U2. We call K3(*(f,g)) = 0K1(f )K2(g) the factorization identity of the convolution.

The image *(f, g) is denoted by f * g. If 0 is the unit of V, we say briefly the convolution

for K3, K1, K2. If U1 = U2 = U3 and K1 = K2 = K3, the convolution is denoted simply f * g,

Ki

and f * g if 0 is the unit of V. We think the factorization identity plays the key role in the

Ki

convolution.

In what follows, we write F = F-1. For any h G Rd, put a(x) = e-l<x,h>, ¡3(x) = ei<x,h>, y(x) = cosxh, S(x) = sinxh. Note that (Ff)(x) = (Ff)(x) = (Ff)(-x). Theorem 1.1 presents some generalized convolutions with weight-function for the Fourier integral transforms F, F .

Theorem 1.1. If f,g G L1(Rd), then each of the integral transforms (1.1), (1.2), (1.3), (1.4), (1.5), (1.6) below defines the generalized convolution followed by its factorization identity:

(f a g)(x) = I f (x - y - h)g(y)dy, (1.1)

F (2n)2 J

F(f* g)(x

r

(f * g)(x

F

F(f * g)(x

F

(f a , g)(x

F,F,F

F(f g)(x F,F,F

(f _ * g)(x

F,F,F

F(f , ß g)(x

F,F,F

(f * g)(x

F

F(f * g)(x

F

(f * g)(x

F

a(x)(Ff )(x)(Fg)(x).

1

(2n)d

f (x - y - h)g(y)dy,

ß(x)(Ff )(x)(Fg)(x).

1

(2n)d

f (x + y - h)g(y)dy,

= a(x)(Ff )(x)(Fg)(x).

(2n)d

f (x + y - h)g(y)dy,

ß(x)(Ff )(x)(Fg)(x).

1

2(2n)d

Y(x)(Ff )(x)(Fg)(x).

2(2n)d

(1.2)

(1.3)

(1.4)

J[f (x - y - h)+ f (x - y + h)]g(y)dy, (1.5)

■ J[f (x - y - h) - f (x - y + h)]g(y)dy, (1.6)

F(f * g)(x) = S(x)(Ff)(x)(Fg)(x).

F

Proof of convolution (1.1). Obviously, the integral transform (1.1) is Fourier convolution taken at point x — h. Since, f * g G L1(Rd). We now prove the factorization identity. We have

F

a(x)(Ff )(x)(Fg)(x) =

—i<x,h>

(2n)d

—i<x,u> —i<x,v>

e-i<x,V> f (u)g(v)dudv =

1

'(2n)d 1

(2n)d

—i<x,u+v+h>

f (u)g(v)dudv

J J e—i<x,t>f (t - y - h)g(y)dtdy = F(f a g)(x).

It is easy to see that the convolutions (1.2), (1.3), (1.4) are the immediate consequences of the convolution (1.1).

1/

Proof of convolution (1.5). We prove f * g G L1(Rd). We have

F

J(f F g)(x)\dx < Mu)l lf(x — u — h)l +lf(x — u + h)l

\g(u)\du I\f (x — u — h)\dx + '\f (x — u + h)\dx

dudx =

2(2n)2

(2n)d

\g(u)\du j\f (x)\dx <

1

1

We prove the factorization identity. We have

Y(x)(Ff )(x)(Fg)(x) = ^^ J J e-i<x,u> e-i<x,v> f (u)g(v)dudv =

Rd Rd

1

[e-i<x,u+v-h> + e-i<x,u+v+h>]f (u)g(v)dudv = 2{2n)d f fe-i<x't>[f (t - y - h)+ f (t - y + h)]g(y)dtdy = F(f F g)(x).

2(2n)d 1

Proof of convolution (1.6). The proof of ft g G L1(Rd) is similar to that of convolution (1.5).

F

So, we prove the factorization identity. We have

sin xh

(2n)d

S(x)(Ff )(x)(F g)(x) = S^xd j J e-i<x'u>e-i<x'v> f (u)g(v)dudv =

[e-i<x'u+v+h> - e-i<x'u+v-h>]f (u)g(v)dudv

2(2n)d

Rd Rd

= r^f f e-i<x,t> [f (t - y - h) - f (t - y + h)]g(y)dtdy =

Rd Rd

= F(f F g)(x).

□

Example. Let d = 1. Put u(x) := 1/nx. It is well-known that the Hilbert transform of a function (or signal) v(x) is given by

(Hv)(x) = p.v. J u(x - y)v(y)dy,

provided that this integral exists as Cauchy's principal value. This is precisely the convolution of v with the tempered distribution p.v. u(x).

xh Now we put r(x) := —==-, s(x) := —==-. By the convolution transforms

w vrn(x2 - h2) vrn(x2 - h2)

(1.5) and (1.6) we have the informal identities:

+^

g(y)dy = p.v. r(x - y)g(y)dy,

2\/2n

2\j2/K

11

+

.x - y - h x - y + h.

.x - y - h x - y + h.

g(y)dy = p.v. s(x - y)g(y)dy.

1

1

2. Application

2.1. Normed Ring Structures on L1(Rd)

Definition 2.1 (see Naimark [13]). A vector space V with a ring structure and a vector norm is called the normed ring if ||vw|| < ||v||||w||, for all v,w G V.

If V has a multiplicative unit element e, it is also required that ||e|| = 1.

Let X denote the linear space L1(Rd). Now we define norms for f G X. For each of all convolutions in Section 1, the norm is chosen as

If II = -\r I If (x)|dx.

(2n )2 J

Rd

Theorem 2.1. The space X, equipped with each of the convolution multiplications, becomes a normed ring with no unit.

Proof. The proof is divided into two steps.

Step 1. X has a normed ring structure. It is clear that X, equipped with each of the convolution multiplications listed above, has the ring structure. We have to prove the multiplicative inequality. We now prove that for convolution (1.5), the proof that for the others is similar. We have

/f 1 g|(x)dx ^ 2(27r)d / / |f (X - U + h)||g(u)|dxdu+

+

2(2n)d 1

j j |/ (x — u — h)||g(u)|dxdu :

|/(x + u — h)|dx I I —- |g(u)|du I = ||/||||g||.

(2n) 2 W " v ' ' ' I \ (2n)d

Hence ||/* g|| < ||/||-MgM-

Step 2. The space X has no unit. For briefness of our proof, let us use the common symbols: * for the convolutions and 70 for the weight function of a, 7, 5. Suppose that there exists an e G X such that / = / * e = e * / for every / G X. Choose 5(x) := e-2|x| g L1(Rd). We then have (F5)(x) = (F5)(x) = 5(x) (see [14, Theorem 7.6]). By 5 = 5 * e = e * 5 and the factorization identities of the convolutions, we have

Fj (5)= 70 F (5)F*(e), (2.1)

where Fj, Fk, F^ G {F, F} (note that it may be Fj = Fk = F^ = F, etc.).

By (2.1) we have 5 = 705F^(e). Due to 5(x) = 0 for every x G Rd, 70(x)(F^ e)(x) = 1 for every x G Since |Y0(x) | < 1, the last identity contradicts the Riemann-Lebesgue lemma: lim (F^ e)(x) = 0 (see Rudin [14, Theorem 7.5]). Hence, X has no unit. □

x—

2.2. Integral Equations of the Convolution Type

Consider the integral equation with the mixed Toeplitz-Hankel kernel having shifts

A^(x) + 1 d Ak1(x + y - h1) + k2(x - y - = P(x) (2.2)

(2n)2 J

Rd

where A G C is predetermined, ki,k2,p are given, the shifts hi,h-2 G Rd, and y(x) is to be determined.

Since the convolutions in Section 1 are considered in L1(Rd), given functions are assumed to be elements of L1(Rd), and unknown function will be determined there. In what follows, the function identity f (x) = g(x) means that it is valid for almost every x G Rd. However, if both functions f, g are continuous, the identity f (x) = g(x) must be valid for every x G Rd.

For any f G L1(Rd), we write f(x) := f (— x). Put

71 (x) = e-i<x'hl>, 72 (x) = e-i<x'h2>,

A(x) : = A + Y2(x)(Fk2)(x), B(x) := 71(x)(F*1)(x), (2.3) DFp(x) : = A2 + A[72(x)(Fk2)(x) + Y2(x)(F*2)(x)] +

+ Y2(x)F [k2 * k2)](x) — Y1(x)F [k1 k1)](x), (2.4) F,F,F F,F,F

Dp(x) : = A(Fp)(x) + F[(p *2 — (*1 * p)](x), (2.5)

f,f,F f,f,F

Df(x) : = A(Fp)(x) + Fp2 * p) — (p * *1)](x). (2.6)

f,f,F f,f,F

Actually,

DfP(x) := A2 + A[72(x)(Fk2)(x) + Y2(-x)(Fk2)(-x)] +

+ (Fk2)(x)(Fk2)(-x) - (Fki)(x)(Fki)(-x), (2.7)

Df(x) := A(Fp)(x) + Y2(-x)(Fk2)(-x)(Fp)(x)-

- Yi(x)(Fki)(x)(Fp)(-x), (2.8) D_p(x) := A(Fp)(-x) + Y2(x)(Fk2)(x)(Fp)(-x)-

- Yi(-x)(Fki)(-x)(Fp)(x). (2.9)

Theorem 2.2. Assume that the following conditions are fulfilled: Dp p(x) = 0 for every x G

Rd, and P G L1(Rd). Then the equation (2.2) has a solution in L1(Rd) if and only if Dp,p

F "M -D^ G L1(Rd). (2.10)

\Dp,p J

If (2.10) is satisfied, then its solution can be obtained in the explicit form

*x>=f-i ( Dprp) <x>-

Proof. Note that the shift h in the convolutions (1.1), (1.2), (1.3), (1.4) is separate. Thus from those convolutions it follows that

1 f Yi Yi

f (x + y - hi)g(y)dy = (/ *^g)(x) = (/^* g)(x)

(2n)2 j f,f,f f,f,f"

1 /(x - y - h2)g(y)dy = (/ I! g)(x) = (/ "* g)(x).

(2n) §7

R§

By the factorization identities of those convolutions we get

F ' (¿1 J f (x + y - hi)g(y)dy I = Yi(x)Ff (x)Fg(x), (2.11)

( ) Rd

F ( T^d if (x - y - h2)g(y)dy ] = 72(x)Ff (x)Fg(x), (2.12)

\ (2n)2 J I

F f (x + y - hi)g(y)dy | = 7i(x)Ff(x)Fg(x), (2.13)

F Ij^l f f (x - y - h2)g(y)dy | = 72(x)if (x)Fg(x), (2.14)

and

for any f, g G L1(

Necessity. Suppose that the equation (2.2) has a solution ^ G L1(Rd), i.e.,

A^(x) + \ d /[ki(x + y - hi) + &2(x - y - ^2)]^(y)dy = p(x).

(2n) 2 J

Rd

Applying each of the transforms F and F in turn to both sides of this identity and using (2.11), (2.12), (2.13), (2.14), we obtain the system of two linear equations

iA(x)(F^)(x) + B(x)(FV)(x) = (Fp)(x),

\B(-x)(F^)(x) + A(-x)(FV)(x) = (Fp)(x), )

where A(x), B(x) are defined by (2.3), and (F^)(x), (_Fy)(x) are unknown functions. The determinants of (2.15) denoted by Df f(x),DF(x),Df(x) are defined by (2.4), (2.5), (2.6) respectively. Due to DF f (x) = 0 for every x G Rd we get

Df(x) „„j _ DF(x)

F^(x) = —-——, and fF^(x)

Df f (x)' Df f (x)

As F(. ^ G L1(Rd), we can apply the inversion theorem of the Fourier transform (see [14, Df, f(x)

Theorem 7.7]) to obtain y(x) = F-1 ( F | (x). The necessity is proved.

\Df, F /

Sufficiency. From (2.7), (2.8), (2.9) it follows that DF F(x) = DF f(-x), and DF(x) = Df (-x). It is easy to see that

F-i ( DFF ) (x) ^F ( Dfff ) <".

-1 ( Df ) _ „ ( DF )

Consider the function y(x) = F 1 —- (x) = F F (x). This implies that ^ G

\Df, f J \DF'F J

L1(Rd). We apply the inversion theorem of the Fourier integral transform to get (F^)(x) =

Dp(x) - D ~ (x)

and (Fip)(x) = F . We can see that the two functions (Fip)(x) and (Fip)(x)

DF,p (x)' DF,p (x)

satisfy (2.15). Thus

A(x)(Fp)(x) + B(x)(Fp)(x) = (Fp)(x). Using the factorization identities of convolutions (1.1), (1.3) we get

F

Equivalently,

Xp + (ki * p) + (k2 * p) (x) = (Fp)(x).

F,F,F F

F

XP(x) + ^TI [ki(x + y - hi) + k2(x - y - h2)]p(y)dy =(Fp)(x). (2n) 2 J -1

By the inversion theorem of the Fourier transform, we conclude that f(x) satisfies the equation (2.2) for almost every x G Rd. □

Let S denote the space of rapidly decreasing functions on Rd (see [14]).

Proposition 2.1. Let A = 0. Then

(a) Dp p = 0 for every x outside a ball with a finite radius.

(b) Assume that k1,k2, G S and p G L1(Rd). Then G L1(Rd), provided DF F = 0 for

df,F '

1

every x G Rd and Fp G L1

Proof. (a) Combining the facts that two functions yi,Y2 are continuous and bounded on Rd and the Riemann-Lebesgue lemma for the Fourier integral transform, we conclude that the function Dp p(x) is continuous on Rd and lim DFp(x) = A. (see [14, Theorem 7.5]). Now (a)

| x| —

follows from the fact that A = 0 and the continuity of Dp p(x).

(b) By the continuity of Dp p(x) and lim Dp p (x) = A = 0, there exist R > 0,e1 > 0 so

' |x|—'

that inf |Dp p(x)| > 61. Since Dp p(x) is continuous and does not vanish in the compact set

|x|>R , ,

S(0,R) = {x G Rd : |x| < R}, there exists 62 > 0 so that inf |Dp p(x)| > 62. We then have sup r-—- ^ max { —, — 1 < to. Hence the function --—- is continuous and bounded

xeRd \D F,F(x)l l61 62) idf,f (x)l

Dp - d\ n„ r^md\ „wr ^r^ra fl^of if J?- c T1

on Rd. Therefore, G L1(Rd), provided Dp G L1(Rd). We now prove that if Fp G L

df,f

then Dp G L1(Rd). Indeed, we have Y,Fk1,Fk2 G S (see [14, Theorem 7.7]). It follows that the functions y(x), (Fk\)(x), (Fk2)(x) are continuous and bounded on Rd. Now we can conclude that if Fp belongs to L1(Rd,) then so does each of the three terms in the right side of (2.8). Therefore, Dp G L1(Rd), provided that Fp G L1(Rd). □

This work is partially supported by the Central Project, grant QGTD.0809, Vietnam National University, Hanoi, Vietnam.

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