DSpace at VNU: Conlutions for the fourier with geometric variable and applications

MATHEMATISCHE www.mn-journal.org Edited by R Mennicken (Regensburg) in co-operation with F Finster (Regensburg), F Gesztesy (Columbia-Missouri), K Hulek (Hannover), F Klopp (Villetaneuse), R Schilling (Dresden) NACHRICHTEN Founded in 1948 by Erhard Schmidt T N I R P RE Math Nachr 283, No 12, 1758 – 1770 (2010) / DOI 10.1002/mana.200710192 Convolutions for the Fourier transforms with geometric variables and applications Bui Thi Giang1 , Nguyen Van Mau2 , and Nguyen Minh Tuan∗3 Dept of Basic Science, Institute of Cryptography Science, 141 Chien Thang str., Thanh Tri dist., Hanoi, Vietnam Dept of Mathematical Analysis, University of Hanoi, 334 Nguyen Trai str., Thanh Xuan dist., Hanoi, Vietnam Dept of Math., Univ of Edu, Ha Noi National Univ., G7 Build., 144 Xuan Thuy Rd., Cau Giay dist., Ha Noi Vietnam Received November 2007, revised 11 November 2009, accepted 11 November 2009 Published online 29 October 2010 Key words Convolution, generalized convolution, factorization identity, Gaussian function, Hermite function MSC (2000) Primary: 43A32, 44A35; Secondary: 44-99, 44A15 This paper gives a general formulation of convolutions for arbitrary linear operators from a linear space to a commutative algebra, constructs three convolutions for the Fourier transforms with geometric variables and four generalized convolutions for the Fourier-cosine, Fourier-sine transforms With respect to applications, by using the constructed convolutions normed rings on L1 (Rn ) are constructed, and explicit solutions of integral equations of convolution type are obtained c 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim Introduction and summary of results The theory of convolutions of integral transforms has been studied for a long time, and is applied to many fields of mathematics (see [4,15–18]) The generalized convolutions for integral transforms and their applications were first studied by Churchill in 1941, then the idea of construction of convolutions was formulated by Vilenkin in 1958 (see [7, 9, 33]) In 1967, the construction methods for generalized convolutions of arbitrary integral transforms were proposed by Kakichev, and in 1990 the concept of generalized convolutions for linear operators was introduced by the same author (see [19, 20]) In 1997, some convolutions for integral transforms were obtained, and in 1998 the generalized convolutions for the Fourier-cosine and Fourier-sine integral transforms were presented (see [21, 22]) In recent years, many papers devoted to those transforms have been published containing convolutions, generalized convolutions, polyconvolutions and their applications (see [5, 6, 10–12, 25–27, 31, 32]) Generally speaking, each of the convolutions is a new transform which has become an object of study (see [1, 19]) In our view, the integral transforms of Fourier type deserve special interest The main purpose of this paper is to construct generalized convolutions for the Fourier transforms with geometric variables: shift, similarity and inversion, and consider some applications The paper is divided into three sections and organized as follows Section consists of three subsections The general formulation of convolutions is stated in Subsection 2.1 In Subsection 2.2, there are six convolutions for the transforms of Fourier type sorted according to those transforms with the geometric variables: shift, similarity, inversion In Subsection 2.3, there are four generalized convolutions of the Fourier-cosine and Fourier-sine transforms As usual, there exist different convolutions for the same transform, and a transform may be a convolution for different transforms Section deals with applications of the constructed convolutions In Subsection 3.1, the linear space L1 (Rn ), equipped with each of the convolutions, becomes a normed ring All normed rings in this subsection have no unit, most of them are commutative and could be used in the theories of Banach algebra Subsection 3.2 contains the most important results of this section where Fredholm integral equations of the first and second ∗ Corresponding author: e-mail: tuannm@hus.edu.vn, nguyentuan@vnu.edu.vn, Phone: +84 37548092, Fax: +84 37548092 c 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim Math Nachr 283, No 12 (2010) / www.mn-journal.com 1759 kind are considered simultaneously By using the constructed convolutions, we provide necessary and sufficient conditions of the solvability of the integral equations of convolution type and obtain explicit solutions Observing the procedures for obtaining the solutions of the first equation in Subsection 3.2, there is the perhaps surprising fact: although the non-injective transforms Tc , Ts are applied to Equation (3.2), the obtained function is still the solution of this equation The comparison with previously published works concerning integral equations of convolution type is provided at the end of this paper Convolutions 2.1 General definition of convolutions The concept of generalized convolutions with weight is a nice idea based on the so-called factorization identity Let U1 , U2 , U3 be linear spaces on the field of scalars K, and let V be a commutative algebra on K Suppose that K1 ∈ L(U1 , V ), K2 ∈ L(U2 , V ), K3 ∈ L(U3 , V ) are linear operators from U1 , U2 , U3 to V, respectively Let δ denote an element in the algebra V Definition 2.1 (See also [5, 19, 20]) A bilinear map ∗ : U1 × U2 :−→ U3 is called a convolution with weight-element δ for K3 , K1 , K2 (in that order) if K3 (∗(f, g)) = δK1 (f )K2 (g) for any f ∈ U1 , g ∈ U2 We call K3 (∗(f, g)) = δK1 (f )K2 (g) the factorization identity of the convolution In the sequel, we write ∗(f, g) := f δ ∗ K3 ,K1 ,K2 g If δ is the unit of V, we say briefly the convolution for K3 , K1 , K2 If U1 = U2 = U3 δ and K1 = K2 = K3 , the convolution is denoted simply f ∗ g, and f ∗ g if δ is the unit of V As the notation K1 f δ ∗ K1 δ K3 ,K1 ,K2 g already defines the factorization identity K3 f ∗ g K1 = δK1 (f )K2 (g), it is sufficient to formulate the convolution expressions in the theorems In the next sections, we consider U = U1 = U2 = U3 = L1 (Rn ) with the Lebesgue integral, V is the algebra of all measurable functions (real or complex) defined on Rn For x, y ∈ Rn , let x, y denote the scalar product, and |x|2 = x, x 2.2 2.2.1 Convolutions for the Fourier transforms with geometric variables Convolutions for the Fourier transform with shift Let h ∈ Rn be fixed Let F denote the Fourier and inverse transforms as (F f )(x) := n (2π) e−i x,y Rn f (y) dy, (F −1 f )(x) := n (2π) ei x,y Rn f (y) dy Consider the following integral transforms n (2π) (Fh−1 f )(x) := n (2π) (Fh f )(x) := e−i x+h,y f (y) dy, ei x,y+h f (y)dy Rn Rn We call Fh the Fourier transform with shift, and Fh−1 its inverse transform The inversion formula of Fh is proved by Lemma 3.7 Put γ1 (x) = e− |x| Theorem 2.2 If f, g ∈ L1 (Rn ), then each of the integral expressions (2.1), (2.2) below defines a convolution: n (2π) γ1 f ∗ g (x) := Fh (2π)n f ∗ g (x) := Fh www.mn-journal.com Rn f (x − y)g(y) dy, Rn Rn f (u)g(v)e− |x−u−v| (2.1) +i h,x−u−v du dv (2.2) c 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim 1760 Bui Thi Giang, Nguyen Van Mau, and Nguyen Minh Tuan: Convolutions for Fourier transforms P r o o f The proof of the convolution (2.1) is immediate and referred to the readers We prove the convolution (2.2) For any u, v ∈ Rn the following formulae hold n (2π) e−i x,y±u±v − 12 |y±u±v|2 Rn dy = e− |x|2 ; Rn n e− |x−u−v| dx = (2π) (2.2a) (see [24, Lemma 7.6]) We then have γ1 f ∗ g (x) dx ≤ Fh Rn (2π)n Rn Rn Rn |f (u)| |g(v)| e− |x−u−v| du dv dx < +∞ γ1 It implies that f ∗ g ∈ L1 (Rn ) We prove the factorization identity Using formula (2.2a), we obtain Fh γ1 (x)(Fh f )(x)(Fh g)(x) −|y−u−v|2 −i x,y +i ∀i = 1, , n) be fixed Write α · x = (α1 x1 , , αn xn ) for any x ∈ Rn Consider the following integral transforms (Fα f )(x) := |α| n (2π) n Fα−1 := j=1 e−i α·x,y Rn f (y) dy, αj n |α|(2π) ei α·x,y Rn (Fα f )(y) dy We call Fα the Fourier transform with similarity, and Fα−1 its inverse transform Theorem 2.3 If f, g ∈ L1 (Rn ), then each of the integral expressions (2.3), (2.4) below defines a convolution: f ∗ g (x) := Fα |α| n (2π) γ1 f ∗ g (x) := Fα n j=1 Rn f (x − y)g(y) dy, αj |α| (2π)n Rn Rn (2.3) f (u)g(v)e− |α·(x−u−v)|2 du dv (2.4) γ1 P r o o f It is easy to prove the convolution (2.3) The fact f ∗ g ∈ L1 (Rn ) for the convolution (2.4) is proved Fα immediately We prove the factorization identity of the convolution (2.4) From (2.2a) we get ( n j=1 (2π) αj ) n e−i x,α·(y−u−v) − 12 |α·(y−u−v)|2 Rn dy = e− |x|2 (2.4a) We then have c 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim www.mn-journal.com Math Nachr 283, No 12 (2010) / www.mn-journal.com 1761 γ1 (x)(Fα f )(x)(Fα g)(x) = n j=1 |α|2 (2π) |α|2 ( n j=1 = (2π) |α| = n (2π) αj 3n e−i f (u)g(v) Rn Rn x,α·(y−u−v) − 12 |α·(y−u−v)|2 dy e−i αj ) 3n 2 Rn e−i Rn ⎡ ⎣ α·x,y Rn f (u)g(v)e− |α·(y−u−v)| e−i n j=1 |α| e−i α·x,v du dv α·x,y du dv dy ⎤ αj − 12 |α·(y−u−v)|2 (2π)n Rn α·x,u Rn Rn Rn f (u)g(v)e du dv ⎦ dy γ1 = Fα f ∗ g (x) Fα The theorem is proved 2.2.3 Convolutions for the Fourier transform with inversion For any x ∈ Rn (xi = 0, ∀i = 1, , n), let us write (Fv f )(x) := ⎧ ⎨ n (2π) ⎩ e−i y, x Rn x 1 x1 , , xn := f (y) dy Consider the integral transform if xi = ∀i = 1, , n, if xi = We call Fv the Fourier transform with inversion Consider the following function e− | x | γ2 (x) = if xi = ∀i = 1, , n, if xi = It is worth saying that the function γ2 (x) is bounded and infinite differentiable on Rn Theorem 2.4 If f, g ∈ L1 (Rn ), then each of the integral transforms (2.5), (2.6) below defines a generalized convolution: n Fv (2π) γ2 f ∗ g (x) = Fv (2π)n f ∗ g (x) = Rn f (x − y)g(y) dy, (2.5) Rn Rn f (u)g(v)e− |x−u−v| du dv (2.6) γ2 P r o o f The convolution (2.5) is proved immediately For the convolution (2.6) the fact that f ∗ g ∈ L1 (Rn ) Fv is proved similarly to the proof of (2.2) We shall prove the factorization identity of the convolution (2.6) as γ2 Fv f ∗ g (x) = γ2 (x)(Fv f )(x)(Fv g)(x) Indeed, if at least one of the xi is zero, this identity is clear Fv Consider xi = 0, ∀i = 1, 2, , n Formula (2.2a) gives n (2π) e−i y−u−v, x − 12 |y−u−v|2 Rn 1 dy = e− | x | (2.7a) We then have γ2 (x)(Fv f )(x)(Fv g)(x) = = www.mn-journal.com (2π) 3n (2π) 3n Rn Rn f (u)g(v) e−i Rn Rn dy e−i 1 u, x −i v, x du dv Rn Rn y−u−v, x − 12 |y−u−v|2 f (u)g(v)e− |y−u−v| −i y, x du dv dy = c 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim 1762 Bui Thi Giang, Nguyen Van Mau, and Nguyen Minh Tuan: Convolutions for Fourier transforms = n (2π) e−i y, x Rn (2π)n Rn Rn f (u)g(v)e− |y−u−v| du dv dy γ2 = Fv f ∗ g (x) Fv The theorem is proved 2.3 Convolutions for the Fourier-cosine transform and Fourier-sine transform For x, y, z ∈ Rn , we write cos xy := cos x, y , sin xy := sin x, y , cos x(y±z) := cos x, y±z , sin x(y±z) := sin x, y ± z as there is no danger of confusion The Fourier-cosine and Fourier-sine transforms are defined by (Tc f )(x) = n (2π) Rn cos xyf (y) dy, and (Ts f )(x) = n (2π) Rn sin xyf (y) dy Theorem 2.5 If f, g ∈ L1 (Rn ), then each of the integral transforms (2.7)–(2.10) below defines a generalized convolution with weight-function γ1 for the transforms Tc , Ts : γ1 f ∗ g (x) = Tc 4(2π)n Rn Rn f (u)g(v) e− |x+u+v|2 + e− + e− f γ1 ∗ Tc ,Ts ,Ts g (x) = 4(2π)n Rn Rn f (u)g(v) −e− |x+u−v|2 |x−u+v|2 |x+u+v|2 + e− γ1 ∗ Ts ,Tc ,Ts g (x) = 4(2π)n Rn Rn f (u)g(v) −e− |x+u+v|2 + e− γ1 ∗ Ts ,Ts ,Tc g (x) = 4(2π)n Rn Rn f (u)g(v) −e− |x+u+v|2 − e− + e− (2.7) du dv |x+u−v|2 − e− |x−u−v|2 (2.8) du dv |x+u−v|2 − e− f |x−u−v|2 |x−u+v|2 + e− f + e− |x−u+v|2 + e− |x−u−v|2 (2.9) du dv |x+u−v|2 |x−u+v|2 + e− |x−u−v|2 (2.10) du dv P r o o f The first part for the convolutions (2.7)–(2.10) can be proved in the same way as in that of the convolution (2.2) Therefore, it suffices to prove the factorization identities of those convolutions P r o o f o f t h e c o n v o l u t i o n (2.7) We have γ1 (x) f (u)g(v) cos xu cos xv du dv (2π)n Rn Rn γ1 (x) = f (u)g(v) cos x(u + v) du dv 4(2π)n Rn Rn γ1 (x) + f (u)g(v) cos x(u − v) du dv 4(2π)n Rn Rn γ1 (x) + f (u)g(v) cos x(u − v) du dv 4(2π)n Rn Rn γ1 (x) + f (u)g(v) cos x(u + v) du dv 4(2π)n Rn Rn γ1 (x)(Tc f )(x)(Tc g)(x) = c 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim www.mn-journal.com Math Nachr 283, No 12 (2010) / www.mn-journal.com 1763 Using formula (2.2a), we have γ1 (x) f (u)g(v) cos x(u + v) du dv 4(2π)n Rn Rn γ1 (x) = f (u)g(v) ei x,u+v + e−i x,u+v du dv 8(2π)n Rn Rn |y+u+v|2 −i y,x − f (u)g(v) e e dy du dv = 8(2π)3n/2 Rn Rn Rn |y+u+v|2 f (u)g(v) ei y,x e− dy du dv + 3n/2 8(2π) Rn Rn Rn |y+u+v|2 cos xy f (u)g(v)e− dy du dv = 3n/2 4(2π) Rn Rn Rn (2.11) Similarly, γ1 (x) 4(2π)n = Rn 4(2π) γ1 (x) 4(2π)n 3n Rn = Rn f (u)g(v) cos x(u − v) du dv Rn Rn cos xy Rn Rn f (u)g(v)e (2.12) −|y+u−v|2 dy du dv, |y−u+v|2 dy du dv, f (u)g(v) cos x(u − v) du dv f (u)g(v)e− cos xy 3n (2.13) 4(2π) Rn Rn Rn γ1 (x) f (u)g(v) cos x(u + v) du dv 4(2π)n Rn Rn |y−u−v|2 cos xy f (u)g(v)e− dy du dv = 3n 4(2π) Rn Rn Rn (2.14) We thus have γ1 (x)(Tc f )(x)(Tc g)(x) = 4(2π)3n/2 Rn cos xy Rn Rn f (u)g(v) e− +e− |y+u+v|2 |y+u−v|2 + e− |y−u+v|2 + e− |y−u−v|2 du dv dy γ1 = Tc f ∗ g (x) Tc The convolutions (2.8)–(2.10) can be proved similarly to the proof of (2.7) The theorem is proved 3.1 Applications Normed rings on L1 (Rn ) Definition 3.1 (See [23].) A vector space V with a ring structure and a vector norm is called a normed ring if vw ≤ v w , for all v, w ∈ V If V has a multiplicative unit element e, it is also required that e = Let X denote the linear space L1 (Rn ) For the convolution (2.4), and for the others the norm of f ∈ X is defined by f = www.mn-journal.com |α| n (2π) Rn |f (x)| dx, f = n (2π) Rn |f (x)| dx, c 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim 1764 Bui Thi Giang, Nguyen Van Mau, and Nguyen Minh Tuan: Convolutions for Fourier transforms respectively Theorem 3.2 deals with normed ring structures on L1 (Rn ) that could be used in the theories of Banach algebra Theorem 3.2 X, equipped with each one of the above mentioned convolution multiplications, becomes a normed ring having no unit Moreover, 1) For the convolutions from (2.1) to (2.8), X is commutative 2) For the convolutions (2.9), (2.10), X is non-commutative P r o o f The proof for the first statement is divided into two steps Step X has a normed ring structure We will use the common symbols ∗ for the above convolutions It is clear that X has a ring structure We now prove the multiplicative inequality for convolution (2.2), the proof for the others is similar Using formula (2.2a), we have n (2π) Rn ≤ |f ∗ g|(x) dx 3n (2π) = (2π)n Rn Rn Rn Rn |f (u)| |g(v)| e− |x−u−v|2 +i h,x−u−v dx du dv Rn |f (u)| |g(v)| du dv This implies that f ∗ g ≤ f g Step X has no unit Suppose that there exists an element e ∈ X such that f ∗ e = e ∗ f = f, for any f ∈ X The factorization identities imply Tj f = γ0 Tk f T e, where γ0 is the common symbol for the weight functions 1, γ1 , γ2 , and Tj , Tk , T ∈ {Fh , Fα , Fv , Tc , Ts } (note that it may be Tj = Tk = T = Tc , etc.) |x|2 i) For the convolution (2.8), the factorization identity is γ1 (Ts f )(Ts e) = Tc f We now choose f0 (x) = e− Note that F = Tc − iTs on X Obviously, f0 ∈ X, and Ts f0 = We then have Tc f0 = which contradicts to the following fact: Tc f0 = Tc f0 − iTs f0 = F f0 = f0 = (see [24, Theorem 7.5]) ii) For the convolutions (2.9), (2.10), their factorization identities give (Ts f )(γ1 Tc e − 1) = By choosing f ∈ X such that (Ts f )(x) = for every x ∈ Rn (see [3, 30]), we get γ1 (x)(Tc e)(x) = for every x ∈ Rn But it fails because limx→∞ γ1 (x)(Tc e)(x) = (see [30, Theorem 1], [4, Theorem 31], or [24, Theorem 7.5]) iii) For the other convolutions, the factorization identity is Tk f (γ0 T e − 1) = By (2.2a), (2.4a), (2.7a) in the respective case, we can choose f ∈ X for example, f (x) = e− |x|2 so that (Tk f )(x) = to get γ0 (x)(T e)(x) = for every x ∈ Rn But it fails because limx→∞ γ0 (x)(T e)(x) = Therefore, the convolution multiplications have no unit We now prove the last conclusion 1) It is clear that X is commutative 2) Note that Ts is regarded as a linear operator from the linear spaces X to the linear space of all functions (real-valued or complex-valued) defined on Rn We may choose f ∈ ker Ts \ ker Tc , and g ∈ ker Ts (see [3, 4]) γ1 γ1 ∗ g ∈ ker Ts , but g ∗ f ∈ ker Ts Again, by (2.10) we have For the convolution (2.9) it follows f f γ1 ∗ Ts ,Tc ,Ts 3.2 g ∈ ker Ts , but g γ1 ∗ Ts ,Tc ,Ts Ts ,Tc ,Ts Ts ,Tc ,Ts f ∈ ker Ts The theorem is proved Integral equations of convolution type Consider the Fredholm integral equation λϕ(x) + Rn K(x, y)ϕ(y) dy = h(x), (3.1) where λ ∈ C Integral equations are important in many applications Problems in which integral equations are encountered include radiative energy transfer and the oscillation of a string, membrane, or axle The equations of the form (3.1) also occur while solving problems of synthesis of electrostatic and magnetic fields, and of digital signal processing (see [2, 8, 10, 11]) The purpose of this subsection is to solve Equation (3.1) in some cases of kernel K(x, y) In what follows, the given functions are assumed to be in L1 (Rn ), and a unknown function will be determined there Moreover, c 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim www.mn-journal.com Math Nachr 283, No 12 (2010) / www.mn-journal.com 1765 the function identity f (x) = g(x) means that it is valid for almost every x ∈ Rn However, if f (x) and g(x) are continuous on Rn , then the identity f (x) = g(x) means that it holds for every x ∈ Rn 3.2.1 First equation Consider the integral equation of convolution type λϕ(x) + n (2π) = q(x), Rn Rn a1 e− |x+u+v|2 + a2 e− |x+u−v|2 +a3 e− |x−u+v|2 + a4 e− |x−u−v|2 k(v)ϕ(u) du dv (3.2) where λ, a1 , a2 , a3 , a4 ∈ C are predetermined, k, q ∈ L1 (Rn ) are given, and ϕ(x) is to be determined The integral equations of convolution type with Gaussian kernels have applications in Physics, Medicine and Biology (see [8, 10, 11]) Put α1 := a1 + a2 + a3 + a4 , α2 := −a1 − a2 + a3 + a4 , β1 := −a1 + a2 + a3 − a4 , β2 := −a1 + a2 − a3 + a4 , and γ1 DTc ,Ts (x) := λ2 + 2λ(a3 + a4 )γ1 (x)(Tc k)(x) + α1 β2 γ1 (x)Tc k ∗ k (x) Tc + α2 β1 γ1 (x)Tc k γ1 ∗ Tc ,Ts ,Ts γ1 γ1 Tc Tc ,Ts ,Ts DTc (x) := Tc λq + β2 k ∗ q − β1 k DTs (x) := Ts λq + α1 k k (x), γ1 ∗ Ts ,Tc ,Ts ∗ q − α2 k (3.3) q (x), γ1 ∗ Ts ,Ts ,Tc q (x) Theorem 3.3 Assume that DTc ,Ts (x) = for every x ∈ Rn , and (3.2) has a solution in L1 (Rn ) if and only if F −1 given by ϕ(x) = F −1 DTc − iDTs DTc ,Ts DTc −iDTs DTc ,Ts (3.4) DT c DTc ,Ts D s , DT T,T ∈ L1 (Rn ) Then Equation c s ∈ L1 (Rn ) If this is in case, then the solution is (x) P r o o f By using the factorization identities of the convolutions (2.7), (2.8), we obtain |y+u+v|2 |y−u−v|2 + e− e− f (u)g(v) du dv (x) n 2(2π) Rn Rn = γ1 (x)[(Tc f )(x)(Tc g)(x) − (Ts f )(x)(Ts g)(x)], |y+u−v|2 |y+u−v|2 Tc [e− + e− ]f (u)g(v) du dv (x) n 2(2π) Rn Rn = γ1 (x)[(Tc f )(x)(Tc g)(x) + (Ts f )(x)(Ts g)(x)] Tc By (2.11) and (2.14), (2.12), (2.13) it is easy to prove that Tc Tc Rn Rn Rn Rn f (u)g(v)e− f (u)g(v)e− |y+u+v|2 |y+u−v|2 du dv (x) = Tc du dv (x) = Tc Rn Rn Rn Rn f (u)g(v)e− f (u)g(v)e− |y−u−v|2 |y−u+v|2 du dv (x), du dv (x) We then have www.mn-journal.com c 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim 1766 Bui Thi Giang, Nguyen Van Mau, and Nguyen Minh Tuan: Convolutions for Fourier transforms Tc Tc Tc Tc |y+u+v|2 e− f (u)g(v) du dv (x) n (2π) Rn Rn = γ1 (x)[(Tc f )(x)(Tc g)(x) − (Ts f )(x)(Ts g)(x)], (3.5) |y+u−v|2 e− f (u)g(v) du dv (x) n (2π) Rn Rn = γ1 (x)[(Tc f )(x)(Tc g)(x) + (Ts f )(x)(Ts g)(x)], (3.6) |y−u+v|2 e− f (u)g(v) du dv (x) n (2π) Rn Rn = γ1 (x)[(Tc f )(x)(Tc g)(x) + (Ts f )(x)(Ts g)(x)], (3.7) |y−u−v|2 e− f (u)g(v) du dv (x) n (2π) Rn Rn = γ1 (x)[(Tc f )(x)(Tc g)(x) − (Ts f )(x)(Ts g)(x)] (3.8) Similarly, by using the convolutions (2.9), (2.10) we prove the following identities Ts Ts Ts Ts |y+u+v|2 e− f (u)g(v) du dv (x) n (2π) Rn Rn = −γ1 (x)[(Tc f )(x)(Ts g)(x) + (Ts f )(x)(Tc g)(x)], (3.9) |y+u−v|2 e− f (u)g(v) du dv (x) n (2π) Rn Rn = γ1 (x)[(Tc f )(x)(Ts g)(x) − (Ts f )(x)(Tc g)(x)], (3.10) |y−u+v|2 e− f (u)g(v) du dv (x) n (2π) Rn Rn = γ1 (x)[(Ts f )(x)(Tc g)(x) − (Tc f )(x)(Ts g)(x)], (3.11) |y−u−v|2 e− f (u)g(v) du dv (x) n (2π) Rn Rn = γ1 (x)[(Tc f )(x)(Ts g)(x) + (Ts f )(x)(Tc g)(x)] (3.12) Necessity Suppose that ϕ ∈ L1 (Rn ) is a solution of (3.2) Applying Tc , Ts to both sides of this equation and using (3.5)–(3.8), and (3.9)–(3.12), we obtain a system of two linear equations λ + α1 γ1 (x)(Tc k)(x) (Tc ϕ)(x) + β1 γ1 (x)(Ts k)(x) (Ts ϕ)(x) = (Tc q)(x), α2 γ1 (x)(Ts k)(x) (Tc ϕ)(x) + λ + β2 γ1 (x)(Tc k)(x) (Ts ϕ)(x) = (Ts q)(x), (3.13) where (Tc ϕ)(x), (Ts ϕ)(x) are unknown functions The determinants of the system (3.13): DTc ,Ts (x), DTc (x), DTs (x) have been defined in (3.3), (3.4) Since DTc ,Ts (x) = for every x ∈ Rn , we find (Tc ϕ)(x), (Ts ϕ)(x) Unfortunately, Tc and Ts have no inverse transforms Now, we use the inversion formula of the Fourier transform to obtain the function ϕ(x) Since DTc ,Ts (x) = for every x ∈ Rn , (Tc ϕ)(x) = DTc (x) , DTc ,Ts (x) (Ts ϕ)(x) = DTs (x) DTc ,Ts (x) As L1 (Rn ) is the domain of F, F −1 , we get (F ϕ)(x) = DTc (x) − iDTs (x) DTc ,Ts (x) c 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim www.mn-journal.com Math Nachr 283, No 12 (2010) / www.mn-journal.com 1767 By the Fourier inverse transform, we get ϕ(x) = F −1 DTc (x) − iDTs DTc ,Ts (x) The necessary condition is proved Sufficiency Let us first prove the following claim Claim 3.4 Suppose that f1 , f2 ∈ L1 (Rn ) If f1 (x) = f1 (−x), and if f2 (x) = −f2 (−x) for every x ∈ Rn , then F −1 (f1 − if2 ) = F (f1 + if2 ) P r o o f o f t h e c l a i m Since L1 (Rn ) is considered as a domain, F = Tc − iTs , and F −1 = Tc + iTs Clearly, f1 − if2 , f1 + if2 ∈ L1 (Rn ), and Tc f2 = Ts f1 = Therefore, F −1 (f1 − if2 ) = (Tc + iTs )(f1 − if2 ) = Tc f1 − iTc f2 + iTs f1 + Ts f2 = Tc f1 + Ts f2 , F (f1 + if2 ) = (Tc − iTs )(f1 + if2 ) = Tc f1 + iTc f2 − iTs f1 + Ts f2 = Tc f1 + Ts f2 The claim is proved Put ϕ(x) := F −1 DTc − iDTs DTc ,Ts (x) Is is easy to check that f1 (x) := DTc (x) , DTc ,Ts (x) and f2 (x) := satisfy the conditions of the claim Hence, ϕ(x) = F DTs (x) DTc ,Ts (x) DTc +iDTs DTc ,Ts (x) It follows that (F ϕ)(x) = DTc (x)−iDTs (x) , DTc ,Ts (x) −1 −1 DTc (x)+iDTs (x) D c (x) Since Tc = F +F and Ts = F 2i−F , we get (Tc ϕ)(x) = DT T,T , and DTc ,Ts (x) c s (x) DTs (x) DTc ,Ts (x) Therefore, (Tc ϕ)(x), (Ts ϕ)(x) satisfy (3.13) In this system of the equations, multiplying (F −1 ϕ)(x) = (Ts ϕ)(x) = the second equation by the unit imaginary number i and subtracting it from the first, and calculating without difficulty we get n (2π) = (F q)(x) F λϕ(x) + Rn Rn [a1 e− |x+u+v|2 + a2 e− |x+u−v|2 + a3 e− |x−u+v|2 + a4 e− |x−u−v|2 ]k(v)ϕ(u) du dv By the uniqueness theorem of the Fourier transform F , the function ϕ satisfies Equation (3.2) for almost every x ∈ Rn (see [4, Theorem 34]) The theorem is proved It is known that (3.1) is called Fredholm integral equation of first the kind if λ = 0, and that of the second kind if λ = For the second kind, there exists a great class of functions satisfying the assumptions in item (b) of Theorem 3.3 Proposition 3.5 Assume that λ = Then (a) DTc ,Ts (x) = for every x outside a ball with finite radius (b) Assume that DTc ,Ts (x) = for every x ∈ Rn If DTc , DTs ∈ L1 (Rn ), then P r o o f (a) Consider γ1 ∈ S, k ∗ k, k Tc γ1 ∗ Tc ,Ts ,Ts DT c DTc ,Ts D s , DT T,T ∈ L1 (Rn ) c s k ∈ L1 (Rn ) and the Riemann-Lebesgue’s lemma for the Fourier-cosine transform Tc , the function DTc ,Ts (x) is continuous on Rn , and lim|x|→∞ DTc ,Ts (x) = λ (see [24, Theorem 7.5], or [30, Theorem 1]) Thus item (a) follows from λ = and the continuity of DTc ,Ts (x) (b) Due to the continuity of DTc ,Ts (x), and lim|x|→∞ DTc ,Ts (x) = λ = 0, there exist R > 0, ε1 > so that inf |x|>R |DTc ,Ts (x)| > ε1 www.mn-journal.com c 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim 1768 Bui Thi Giang, Nguyen Van Mau, and Nguyen Minh Tuan: Convolutions for Fourier transforms Since DTc ,Ts (x) is continuous, not vanishing in the compact set S(0, R) = {x ∈ Rn : |x| ≤ R}, there exists ε2 > so that inf |x|≤R |DTc ,Ts (x)| > ε2 We then have supx∈Rn |DT ,T1 (x)| ≤ max{ ε11 , ε12 } < ∞ As a c result, the function |DT ,T1 (x)| is continuous and bounded on Rn Therefore, c s provided DTc , DTs ∈ L1 (Rn ) Item (b) is proved 3.2.2 s DT c DT s |DTc ,Ts (x)| , |DTc ,Ts (x)| ∈ L1 (Rn ), Second equation Consider the following integral equation of convolution type λϕ(x) + (2π)n e− |x−u−v| Rn +i h,x−u−v Rn k(v)ϕ(u) du dv = g(x), (3.14) where λ ∈ C, g, k are given functions from L1 (Rn ), ϕ is the function to be determined Theorem 3.6 (a) If the equation (3.14) has a solution, then (Fh g)(x) = whenever λ + γ1 (x)(Fh k)(x) = (b) Assume that λ + γ1 (x)(Fh k)(x) = for every x ∈ Rn , and λ+γF1h(Fg h k) ∈ L1 (Rn ) Then the equation (3.14) has a solution in L1 (Rn ) if and only if Fh−1 of the equation (3.14) is given by ϕ(x) = Fh−1 Fh g λ+γ1 (Fh k) Fh g λ+γ1 (Fh k) ∈ L1 (Rn ) If this is in case, then the solution (x) We need the following lemma Lemma 3.7 (Inversion Theorem) Assume that f, Fh f ∈ L1 (Rn ) Then n (2π) f (x) = ei x,y+h Rn (Fh f )(y)dy := Fh−1 Fh f (y) (x) for almost every x ∈ Rn P r o o f o f t h e l e m m a We have (Fh f )(x) = n (2π) e−i x,y e−i h,y Rn f (y) dy = F e−i h,y f (y) (x) ∈ L1 (Rn ) Since the function e−i h,y is continuous and bounded on Rn , the function e−i h,y f (y) belongs to L1 (Rn ) Using the inversion theorem of the Fourier transform (see [24, Theorem 7.7]), we obtain f (x) = n (2π) ei x,y+h Rn (Fh f )(y)dy for almost every x ∈ Rn The lemma is proved First, we prove item (b) of the theorem Suppose that ϕ ∈ L1 (Rn ) is a solution of (3.14) By (2.2), Equation γ1 (3.14) can be rewritten in the form: λϕ(x) + ϕ ∗ k (x) = g(x) Applying Fh to both sides of this equation Fh and using the factorization identity of convolution (2.2), we get λ + γ1 (x)(Fh k)(x) (Fh ϕ)(x) = (Fh g)(x) (3.15) (Fh g)(x) λ+γ1 (x)(Fh k)(x) Using Lemma 3.7, we function ϕ = Fh−1 λ+γF1h(Fg h k) ∈ L1 (Rn ) From the assumption λ + γ1 (x)(Fh k)(x) = it follows that (Fh ϕ)(x) = obtain ϕ = Fh−1 Fh g λ+γ1 (Fh k) By Lemma 3.7, (Fh ϕ)(x) = ∈ L1 (Rn ) Conversely, consider the (Fh g)(x) λ+γ1 (x)(Fh k)(x) This implies that λ + γ1 (x)(Fh k)(x) (Fh ϕ)(x) = (Fh g)(x) By γ1 formula (2.2) Fh λϕ + ϕ ∗ k (x) = (Fh g)(x) Again by Lemma 3.7 ϕ(x) satisfies (3.14) for almost every Fh x ∈ Rn Item (b) is proved Item (a) now is just a consequence of (3.15) and the fact that the functions on both sides of (3.15) are continuous on Rn Theorem 3.6 is proved Analogously to Proposition 3.5 we can prove the following proposition c 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim www.mn-journal.com Math Nachr 283, No 12 (2010) / www.mn-journal.com 1769 Proposition 3.8 Let λ = Then λ + γ1 (x)(Fh k)(x) = for every x outside a ball with finite radius Fh g ∈ L1 (Rn ) Moreover, if Fh g ∈ L1 (Rn ) and if λ + γ1 (x)(Fh k)(x) = for every x ∈ Rn , then λ+γ Fh k Remark 3.9 If λ ∈ C \ (−∞, 0] and k(x) is Gaussian, then λ + γ1 (x)(Fh k)(x) = for every x ∈ Rn 3.2.3 Third equation Let α = (α1 , , αn ) ∈ Rn+ (αi > 0, i = 1, , n) be given The notation α · x has been defined in Sub-subsection 2.2.2 We consider the equation n j=1 λϕ(x) + αi |α| (2π)n k(v)e− Rn |α·(x−u−v)|2 Rn ϕ(u) du dv = g(x), (3.16) where λ ∈ C, g, k are given functions in L1 (Rn ), ϕ is to be determined Theorem 3.10 (a) If Equation (3.16) has a solution, then (Fα g)(x) = whenever λ + γ1 (x)(Fα k)(x) = (Fα g) (b) Assume that λ + γ1 (x)(Fα k)(x) = for every x ∈ Rn , and λ+γ ∈ L1 (Rn ) Then the equation (Fα k) (3.16) has a solution in L1 (Rn ) if and only if Fα−1 of Equation (3.16) is given by ϕ(x) = Fα−1 Fα g λ+γ1 (Fα k) Fα g λ+γ1 (Fα k) ∈ L1 (Rn ) If this is in case, then the solution (x) As Lemma 3.7 we can prove the following lemma Lemma 3.11 (Inversion Theorem) Assume that f, Fα f ∈ L1 (Rn ) Then n f (x) = j=1 αj n |α|(2π) ei Rn α·x,y (Fα f )(y)dy := Fα−1 (Fα f ) (x) for almost every x ∈ Rn We first prove item (b) By (2.4) Equation (3.16) can be rewritten in the form of the following convolution γ1 equation: λϕ(x) + f ∗ k (x) = g(x) Applying Fα to both sides of this equation and using (2.4), we get Fα λ + γ1 (x)(Fα k)(x) (Fα ϕ)(x) = (Fα g)(x) (3.17) α g)(x) Using Lemma 3.11, we obtain ϕ(x) = Fα−1 λ+γF1α(Fg α k) (x) Hence (Fα ϕ)(x) = λ+γ(F (x)(Fα k)(x) The conversion is proved similarly as that of Theorem 3.6 Thus item (b) is proved Item (a) is just a consequence of (3.17) and the fact that the functions in this equation are continuous on Rn Theorem 3.10 is proved The proof of Proposition 3.12 below is similar to that of Proposition 3.5 Proposition 3.12 Let λ = Then λ + γ1 (x)(Fα k)(x) = for every x outside a ball with finite radius Fα g ∈ L1 (Rn ) Moreover, if Fα g ∈ L1 (Rn ), and if λ + γ1 (x)(Fα k)(x) = for every x ∈ Rn , then λ+γ Fα k Remark 3.13 If λ ∈ C \ (−∞, 0] and g(x) is Gaussian, then λ + γ1 (x)(Fα k)(x) = for every x ∈ Rn In the general theory of integral equations, each one of the requirements DTc ,Ts (x) = 0, λ + γ1 (x)(Fh k)(x) = 0, and λ + γ1 (x)(Fα k)(x) = for every x ∈ Rn as in Theorems 3.3, 3.6, 3.10 is the normally solvable condition of the corresponding equation (see [2, 16, 30]) Comparison In constructing convolutions for 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Convolutions for the Fourier transforms with geometric variables Convolutions for the Fourier transform with shift Let h ∈ Rn be fixed Let F denote the Fourier and inverse transforms as (F f... commutative algebra, constructs three convolutions for the Fourier transforms with geometric variables and four generalized convolutions for the Fourier- cosine, Fourier- sine transforms With respect... convolutions of the Fourier- cosine and Fourier- sine transforms As usual, there exist different convolutions for the same transform, and a transform may be a convolution for different transforms Section