The generalized convolutions related to the fourier sine and Kontorovich - Lebedev integral transformations

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In this paper, we establish several properties in a two parametric family of Lebesgue spaces for certain convolution related to the Fourier sine and Kontorovich - Lebedev integral transformations. Norm estimations in the weighted Lp - spaces and applications to solve a class of the integral equations and systems of integral equations are obtained.

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Nội dung Text: The generalized convolutions related to the fourier sine and Kontorovich - Lebedev integral transformations

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2011, Vol. 56, No. 7, pp. 3-13 THE GENERALIZED CONVOLUTIONS RELATED TO THE FOURIER SINE AND KONTOROVICH - LEBEDEV INTEGRAL TRANSFORMATIONS Nguyen Xuan Thao and Nguyen Tien Trung(∗) Hanoi University of Science and Technology (∗) E-mail: tientrungbk@gmail.com Abstract. In this paper, we establish several properties in a two parametric family of Lebesgue spaces for certain convolution related to the Fourier sine and Kontorovich - Lebedev integral transformations. Norm estimations in the weighted Lp - spaces and applications to solve a class of the integral equations and systems of integral equations are obtained. Keywords: Convolution, weight function, Kontorovich - Lebedev, Fourier transform. 1. Introduction The convolutions for the integral transform have been studied since the be- ginning of the 20th century, such as convolutions for the Fourier, Laplace, Mellin, Hankel, etc. Theory of convolutions have many interesting applications in solving problems of mathematical physics, inverse problems, integral equations, differential equations and it attracted the attention of many mathematicians. In 1998, Kachichev V.A. and Thao N.X. proposed a method for constructing convolutions with a weight function for three integral transformations. In their recent article in a prestigious journal [5], Yakubovich S.B. and Britvinar L.E. presented their research on gener- alized convolutions whose factorization identity contained two integral transforma- tions: Kontorovich - Lebedev and Fourier cosine. In this paper, we construct and study two generalized convolutions whose factorization equation contains two in- tegral transformations Kontorovich - Lebedev and Fourier sine. Our study results, together with those demonstrated in [5] will give an overview of the classes of con- volution with Fourier and Kontorovich - Lebedev transformations. 3
Nguyen Xuan Thao and Nguyen Tien Trung 2. Well-known result 2.1. Fourier sine tranformation The Fourier sine integral transformation of function f (x) was studied in ref- erences [3, 5, 7] r Z ∞ 2 (Fs f )(y) = sin yx f (x)dx. (2.1) π 0 The Fourier sine transform (2.1) is well-defined on the space L1 (R+ ; dx). More- over, if g(y) = (Fs f )(y) ∈ L1 (R+ ; dy), we have the reciprocal inversion formula f (x) = (Fs g)(x). In case of L2 (R+ ; dx) we should define the Fourier sine transfor- mation in the mean-square convergence sense, namely r Z N 2 (Fs f )(y) = lim sin yx f (x)dx (2.2) N →∞ π N1 and Plancherel’s theorem [3, 5] said that Fs : L2 (R+ ; dx) → L2 (R+ ; dx) defines an isometric isomorphism with the reciprocal inverse formula r Z N 2 f (x) = lim (Fs f ) sin yxdy, (2.3) N →∞ π N1 and satisfies the Plancherel’s equality ||Fs f ||L2(R+ ) = ||f ||L2(R+ ;dx) . (2.4) 2.2. Kontorovich - Lebedev transformation The Kontorovich - Lebedev transformation was studied in [5] Z ∞ Kiy [f ] = Kiy (x)f (x)dx, (2.5) 0 where the modified Bessel function Kiy (x) can be represented by the Fourier integral [3, 8] Z ∞ Kiy (x) = e−x cosh τ cos yτ dτ, x>0 (2.6) 0 based on the following reciprocities of the product of the Bessel functions (2.16.52.13 in [2]) Z ∞ π p Kiy (x)Kiy (v) cos yudy = K0 ( x2 + y 2 + 2xv cosh u) (2.7) 0 2 Z ∞ √ Kiy (x)Kiy (v) = K0 ( x2 + v 2 + 2xv cosh u) cos yudu (2.8) 0 4
The generalized convolutions related to the Fourier sine and Kontorovich - Lebedev... On the space L2 (R+ ; xdx), the Kontorovich - Lebedev transformation is of the form Z N Kiy [f ] = lim Kiy (x)f (x)dx, (2.9) N →∞ 1 N where the limit is taken in the mean-square sense with respect to the norm of the space L2 (R+ , y sinh πydy). It has been proved that Kiy (x) : L2 (R+ ; xdx) → L2 (R+ ; y sinh πydy) forms an isometric isomorphism operator, with the Parseval’s identity [5] Z ∞ Z ∞ 2 2 2 y sinh πy|Kiy [f ]| dy = |f (x)|2xdx, (2.10) π 0 0 The inverse operator is defined by the formula: Z N 2 Kiy (x) f (x) = lim 2 y sinh πy Kiy [f ]dy, (2.11) N →∞ π 0 x where the convergence is in mean-square with respect to the norm of L2 (R+ ; xdx). 2.3. Two parametric family of Lebesgue spaces Definition 2.1. Two parametric family of Lebesgue spaces Lα,β p (R+ ) are respectively defined by: ( Z ! p1 ) ∞ Lα,β p (R+ ) = f: p α |f (x)| x K0 (βx)dx 0 (3.2) 2πx 0 0 and Z ∞ Z ∞ 1 (f ∗g)2 (x) = f (u)g(v)[κ(u, v, x−1)−κ(u, v, x+1)]dudv, x > 0 (3.3) 2 0 0 5
Nguyen Xuan Thao and Nguyen Tien Trung Corollary 3.1. Under certain conditions we will establish the following factorization equalities r π sin y Kiy [(f ∗ g)1 ] = (Fs f )(y)Kiy [g] (3.4) 2 y sinh πy r 2 (Fs (f ∗ g)2 )(y) = sin y Kiy [f ]Kiy [g], y > 0. (3.5) π It shows that convolution (3.2) is noncommutative and convolution (3.3) is commu- tative. 3.1. The non-commutative convolution (equ. 3.2) Theorem 3.1. Let f (u) ∈ L1 (R+ ; du), g ∈ L0,β 1 (R+ ), 0 < β ≤ 1, then convolution (3.2) exists for all x > 0 and belongs to Lα,β p (R+ ) with α > p − 1, 1 ≤ p < ∞. Moreover, the following estimation holds ||(f ∗ g)1 (x)||Lα,β p (R+ ) ≤ Aα,p,β ||f ||L1(R+ ;du) ||g||L0,β (R+ ) (3.6) 1 where (see (2.16.6.3) in [2]) 1 1 −1 Γ2 (α − p − 1) p 1− α+1 Aα,p,β = π 2p(2p) p . Γ(α, p, β) α−p+1 α−p 3 β2 2 F1 , + 1, α − p + ; 1 − 2 , (3.7) 2 2 2 p here Γ(x) is Gamma Euller function , 2 F1 (x, y, z) is the Gauss hypergeometric func- tion [1]. Besides, the generalized Parseval type equality holds √ Z ∞ 2 (f ∗ g)1 (x) = √ sin y(Fs f )(y)Kiy [g]Kiy (x)dy, x > 0 (3.8) π πx 0 and finally, with Kiy [g] ∈ L2 ((0, 1); dy) then the factorization equality (3.4) takes place. Proof. Using (2.6), (3.1) we have Z ∞ Z ∞ −(x+v) cosh τ −x κ(x, v, u) ≤ e dτ ≤ e e−v cosh τ dτ ≤ e−x K0 (βv). (3.9) 0 0 Therefore Z ∞Z ∞ 1 |(f ∗ g)1 (x)| ≤ |f (u)||g(v)||(κ(x, v, u − 1) + κ(x, v, u + 1))|dudv 2πx 0 0 Z Z e−x ∞ ∞ ≤ |f (u)||g(v)|K0(βv)dudv. πx 0 0 6
The generalized convolutions related to the Fourier sine and Kontorovich - Lebedev... or equivalent e−x |(f ∗ g)1(x)| ≤ ||f ||L1(R+ ;du) ||g||L0,β (R+ ) < ∞. (3.10) πx 1 Hence convolution (3.2) exists, for all x > 0. We now prove the estimation (3.2), using (3.10) we have Z ∞ p p1 e−x ||(f ∗ g)1 (x)||Lα,β p (R+ ) ≤ ||f ||L1(R+ ;du) ||g||L0,β (R+ ) K0 (βx)xα dx 0 πx 1 Z ∞ p1 1 −px α−p = e K0 (βx)x dx ||f ||L1(R+ ;du) ||g||L0,β (R+ ) π 0 1 = Aα,p,β ||f ||L1(R+ ;du) ||g||L0,β (R+ ) < ∞, 1 where Aα,p,β is given by (3.7). Using (2.7) and appealing to Fubini’s theorem we write convolution (3.2) in the form Z ∞Z ∞Z ∞ 1 (f ∗ g)1(x) = f (u)g(v)Kiy (v)Kiy (x)[cos y(u − 1) π2x 0 0 0 − cos y(u + 1)]dudvdy Z ∞ Z ∞ Z ∞ 2 = f (u) sin yudu g(v)Kiy (v)dv Kiy (x) sin ydy π2x 0 0 0 √ Z ∞ 2 = √ sin y(Fs f )(y)Kiy [g]Kiy (x)dy, π πx 0 which proves the generalized Paseval equality (3.8). In order to prove (3.4), we assume first that Kiy [g] ∈ L2 ((0, 1); dy). Hence we have the right hand side of (3.4) belonging to L2 (R+ ; y sinh πydy) [5]. Therefore, by Par- seval equality (3.8), reciprocities (2.11) we get (3.4). Theorem (3.1) is proved. To demonstrate as in theorem (3.1), using (3.9) and Schwartz inequality we obtain the following theorem. Theorem 3.2. Let f (u) ∈ L1 (R+ ; du), g ∈ L0,12 (R+ ) then convolution (3.2) ex- ists for all x > 0 as a bounded continuous function and belongs to L2 (R+ ; xα dx). Moreover, we have ||(f ∗ g)1||L2 (R+ ;xα dx) ≤ Cα¸||f ||L1(R+ ;du) .||g||L0,1(R+ ) , (3.11) 2 where Γ(α − 1) Cα = α−1 3 1 , α > 1. (3.12) 2 2 π 4 Γ 2 (α − 12 ) Besides, the Parseval type equality (3.8) holds true. 7
Nguyen Xuan Thao and Nguyen Tien Trung Theorem 3.3. Let f (u) ∈ L2 (R+ ; du), g ∈ L0,11 (R+ ) and Kiy [g] = O(1), y → 0. Then convolution (3.2) exists, continuous, bounded for all x > 0, belongs to L2 (R+ ; xα dx). Furthermore, we have ||(f ∗ g)1||L2 (R+ ;xαdx) ≤ Cα ||f ||L2(R+ ;du) .||g||L01,1(R+ ) (3.13) where Cα is given by formula (3.12). Besides, the Parseval type equality (3.8) and factorization property (3.4) hold true. Corollary 3.2. Under conditions of theorem (3.1) or theorem (3.3), the convolution (3.2) belongs to the space L2 (R+ ; xdx) and the following identity holds: Z ∞ Z 2 1 ∞ dy |(f ∗ g)1(x)| xdx = | sin y(Fs f )(y)Kiy [g]|2 (3.14) 0 π 0 y sinh πy 3.2. The commutative convolution (equ. 3.3) Similar to demonstrate as in [5], we obtain following results: Theorem 3.4. Let f, g ∈ L0,β 1 (R+ ), 0 < β ≤ 1 then convolution (3.3) exists for all x > 0, belonging to L1 (R+ ; dx) and ||(f ∗ g)2||L1 (R+ ;dx) ≤ ||f ||L0,β (R+ ) ||g||L0,β (R+ ) (3.15) 1 1 Moreover, it satisfies the factorization identity (3.5). Further, if β ∈ (0, 1) then (f ∗ g)2 (x) vanish at infinity and for all x > 0 the Parseval type equality holds: Z 2 ∞ (f ∗ g)2 (x) = sin yKiy [f ]Kiy [g] sin yxdy. (3.16) π 0 Theorem 3.5. Let f ∈ L0,β 0,β p (R+ ), g ∈ Lq (R+ ), 1 < p < ∞, p −1 + q −1 = 1, 0 < β ≤ 1. Then the convolution (3.3) exists for all x > 0 as a bounded continuous function and belonging to Lα,θ r (R+ ), 1 < r < ∞, 0 < γ < 1 and − αr −1 θ 2 α+1 ||(f ∗ g)2 ||Lα,θ (R+ ) ≤ (2θ) r Γr ||f ||L0,β p (R+ ) ||g||L0,β q (R+ ) (3.17) r 2 2 Besides, it satisfies the factorization identity (3.5). Furthermore, if β ∈ (0, 1) then (f ∗ g)2(x) vanish at infinity and for all x > 0, the Parseval type equality (3.16) holds. Corollary 3.3. Under conditions of theorem 3.5, the convolution (3.3) exists for all x > 0 and belongs to Lp (R+ ; dx). Besides, it satisfies the following inequality π p1 ||(f ∗ g)2 (x)||Lp (R+ ;dx) ≤ ||f ||L0,β p (R+ ) ||g||L0,β q (R+ ) (3.18) 2β Specially, in case p = 2, the following Fourier type Parseval identity takes place Z ∞ Z 2 2 ∞ |(f ∗ g)2 (x)| dx = | sin yKiy [f ]Kiy [g]|2dy (3.19) 0 π 0 8
The generalized convolutions related to the Fourier sine and Kontorovich - Lebedev... 4. Applications 4.1. Convolution integral equations of the first kind We will consider three integral equations of the first kind (f ∗ µ)1 (x) = g(x), x ∈ R+ ; (4.1) (µ ∗ f )1 (x) = g(x), x ∈ R+ ; (4.2) (µ ∗ f )2 (x) = g(x), x ∈ R+ , (4.3) where g, µ are given and f is unknown. We will establish conditions, which will guarantee the existence and uniqueness of solutions of these equations. Similar to demonstrate as in [5], we obtain following results: Theorem 4.1. Let g ∈ L2 (R+ ; xdx), µ ∈ L0,1 1 (R+ )and Kiy [µ] = O(1), y → 0. Then there exists the unique solution in L2 (R+ ; du) of equation (4.1) if and only y sinh πyKiy [g] if ∈ L2 (R+ ; dy). Moreover, f (u) is given by the formula sin yKiy [µ] Z N 2 y sinh πyKiy [g] f (u) = lim sin yudy, (4.4) π N →∞ N1 sin yKiy [µ] where the convergence is with respect to the norm in L2 (R+ ; du) Definition 4.1. Denoted by KL1iy ≡ h ∈ L2 (R+ ; y sinh πydy), h = Kiy [f ], f ∈ 0,1 Lα,β 1 p (R+ ). We will consider a restriction of map to Kix : L1 ∩ L2 (R+ ; udu) → KLiy and we have KL1iy ∈ C0 (R+ ) [see 5]. Theorem 4.2. Let g ∈ L2 (R+ ; xdx), µ ∈ L1 (R+ ; dv), and y −1(Fs µ)(y) = O(1), y → 0. Suppose that y(sinh πy)2Kiy [g] ∈ L1 (R+ ; dy) (4.5) sin y(Fs µ)(y) Z ∞
Z ∞
y(sinh πy) 2 K iy [g]
sin ytdy
tdt < ∞, (4.6)
sin y(Fs µ)(y)
0 0 then the solvability of equation (4.2) in the L0,11 ∩ L2 (R+ ; udu)is guaranteed if and only if r 2 y sinh πyKiy [g] ∈ KL1iy , (4.7) π sin y(Fs µ)(y) and the explicit solution is unique and given by the formula √ Z ∞ 2 2 [y sinh πy]2Kiy [g] f (u) = 2 √ Kiy (u)dy (4.8) π πu 0 sin y(Fs µ)(y) where the latter integral exists as a Lebesgue sense. 9
Nguyen Xuan Thao and Nguyen Tien Trung Proof. Necessity. We assume that g, µ, f belong to the corresponding L-classes and equation (4.2) is satisfied then via Corollary 3.2 we have the equality r 2 y sinh πy Kiy [g] Kiy [f ] = π sin y(Fs µ)(y) r 2 y sinh πy Kiy [g] But Kiy [f ] ∈ KL1iy hence ∈ KL1iy and solution in L2 (R+ ; udu) π sin y(Fs µ)(y) is given reciprocally by the formula √ Z N 2 2 [y sinh πy]2 Kiy [g] f (u) = lim 2 √ Kiy (u)dy (4.9) N →∞ π πu 0 sin y(Fs µ)(y) But the latter integral is absolutely convergent due to the (4.5) and the boundedness for all u > 0 (see Theorem 4.2 in [5]). Hence, we obtain the solution in form (4.8). We will show that this solution belongs to L0,1 1 (R+ ) as well. R ∞ (−u cosh t) Indeed, using formula yKiy (u) = u 0 e sinh t sin ytdt (see 4.16 in [5]) and substituting it in (4.8) we change the order of integration by Fubinis theorem to obtain: √ Z ∞ Z N 2 2 (−u cosh t) y(sinh πy)2 Kiy [g] f (u) = 2 √ e sinh t sin ytdydt π πu 0 0 sin y(Fs µ)(y) Hence with relation (2.16.6.2) in [2] and (4.6) we derive √ Z ∞ Z ∞
Z
2 2
N y(sinh πy)2 K [g]
(−u cosh t)