Fc-kl-convolutiontransforms - am58 4 6s

58 (2013) APPLICATIONS OF MATHEMATICS No. 4, 473–486 ON THE FOURIER COSINE—KONTOROVICH-LEBEDEV GENERALIZED CONVOLUTION TRANSFORMS Nguyen Thanh Hong, Trinh Tuan, Nguyen Xuan Thao, Hanoi (Received August 1, 2011) Abstract. We deal with several classes of integral transformations of the form Z 1 −u cosh(x+v) (e + e−u cosh(x−v) )h(u)f (v) du dv, f (x) → D 2 u R+ where D is an operator. In case D is the identity operator, we obtain several operator properties on Lp (R+ ) with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on L2 (R+ ) and define the inversion formula. Further, for an other class of differential operators of finite order, we apply these transformations to solve a class of integro-differential problems of generalized convolution type. Keywords: convolution, Hölder inequality, Young’s theorem, Watson’s theorem, unitary, Fourier cosine, Kontorovich-Lebedev, transform, integro-differential equation MSC 2010 : 33C10, 44A35, 45E10, 45J05, 47A30, 47B15 1. Introduction The Fourier cosine integral transform is of the form (see [10], [11]) r Z ∞ 2 f (x) cos xy dx (1.1) (Fc f )(y) = π 0 for f ∈ L1 (R+ ), and (1.2) r Z N r Z ∞ sin xy 2 2 d dx (Fc f )(y) = lim f (x) cos yx dx = f (x) N →∞ π 0 π dx 0 x This research is funded by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2011.05. 473 for f ∈ L2 (R+ ); here the limit is understood in L2 (R+ ) norm mean. These two definitions are equivalent if f ∈ L1 (R+ ) ∩ L2 (R+ ). The Kontorovich-Lebedev integral transform was first investigated by M. J. Kontorovich and N. N. Lebedev in 1938–1939 and has the form (see [5], [6], [14]) Z ∞ (1.3) K[f ](y) = Kix (y)f (x) dx, 0 which contains as the kernel the Macdonald function Kν (x) (see [1]) of the pure imaginary index ν = iy. The function Kν (z) satisfies the differential equation (1.4) z2 d2 u du +z − (z 2 + ν 2 )u = 0. dz 2 dz The Macdonald function has the asymptotic behaviour (see [6]) (1.5) Kν (z) =  π 1/2 e−z [1 + O(1/z)], 2z z → ∞, and near the origin (1.6) z ν Kν (z) = 2ν−1 Γ(ν) + o(1), (1.7) K0 (z) = − log z + O(1), z → 0, ν 6= 0, z → 0. The following form for the Macdonald function is very useful (see [1], [6], [14]): Z ∞ (1.8) Kiy (x) = e−x cosh u cos yu du, x > 0. 0 The inverse Kontorovich-Lebedev transform (1.3) is of the form (see [5], [6]) Z ∞ 1 2 Kix (y)g(y) dy, (1.9) f (x) = K −1 [g](x) = 2 x sinh(πx) π y 0 here, g(y) = K[f ](y). Throughout this paper, we are interested in the Kontorovich-Lebedev transform (1.3). However, note that there is another version of the Kontorovich-Lebedev integral transform which is of the form (see [1], [6], [16]) Z ∞ e (1.10) g(y) = K[f ](y) = Kiy (x)f (x) dx. 0 A generalized convolution for the Fourier cosine and the Kontorovich-Lebedev integral transforms has been studied in [12]: (1.11) Z ∞Z ∞ γ 1 −u cosh(x+v) 1 [e + e−u cosh(x−v) ]h(u)f (v) du dv, x > 0. (h ∗ f )(x) = 2 π 0 u 0 474 The existence of the generalized convolution (1.11) for two functions in L1 (R+ ) with weight and its application to solving integral equations of generalized convolution type were studied in [12]. Namely, for h ∈ L1 (R+ , 1/x), f ∈ L1 (R+ , 1/ sinh x), the following factorization equality holds (see [12]): (1.12) γ Fc (h ∗ f )(y) = 1 K −1 [h](y)(Fc f )(y), ∀y > 0. y sinh πy In any convolution (h ∗ f ) of two functions h and f , if we fix the function h and let f vary in a certain function space, then one can study convolution transforms of the type f 7→ D(f ∗ h), where D is an operator. The most famous integral transforms constructed in this way are the Watson transforms that are related to the Mellin convolution and the Mellin transform (see [11]) f (x) 7−→ g(x) = Z ∞ k(xy)f (y) dy. 0 Recently, several authors have been interested in the convolution transforms of this type (see [3], [4], [13], [15]). In this paper, we are interested in the transform f 7→ γ γ D(h ∗ f ), where (h ∗ f ) is the generalized convolution (1.11). For the case D is the identity operator, in Section 2 we study several further operator properties in the Lebesgue spaces Lp (R+ ) with weight for the generalized convolution (1.11). In particular, Young’s theorem and Young’s inequality for this generalized convolution are obtained. In Section 3, for a class of differential operators D of infinite order, we obtain the necessary and sufficient condition such that the respective transforms are unitary on L2 (R+ ), and define the inverse transforms. Finally, in Section 4, for an other class of differential operator D of finite order, we obtain the solution in closed form of a class of integro-differential equations. 2. Generalized convolution operator properties In this section, we will prove several norm properties of the generalized convolution (1.11). Throughout the paper, we are interested in the following two-parametric family of Lebesgue spaces. Definition 1 (see [16]). For α ∈ R, 0 < β 6 1, we denote by Lα,β p (R+ ) the space of all functions f (x) defined in R+ such that (2.1) Z ∞ 0 |f (x)|p K0 (βx)xα dx < ∞. 475 The norm of a function in this space is defined by kf kLα,β (R+ ) = p Z ∞ 0 1/p |f (x)|p K0 (βx)xα dx . Using the asymptotics of the Macdonald function (1.5), (1.6), (1.7), formula (2.1) can be expressed in an equivalent form Z 1 0 |f (x)|p | log x|xα dx + Z ∞ 1 |f (x)|p xα−1/2 e−βx dx < ∞. The boundedness of the generalized convolution (1.11) on the spaces L1 (R+ ) is given by the following theorem; here we consider the function h ∈ L−1,β (R+ ). 1 Theorem 2.1. Let h ∈ L−1,β (R+ ) and g ∈ L1 (R+ ), 0 < β 6 1. Then the 1 generalized convolution (1.11) exists for almost all x > 0, belongs to L1 (R+ ), and the following estimation holds: γ (2.2) kh ∗ gkL1 (R+ ) 6 2 khkL−1,β (R+ ) kf kL1 (R+ ) . 1 π2 Moreover, the factorization property (1.12) holds true. Furthermore, if 0 < β < 1, then the convolution (1.11) belongs to C0 (R+ ), and the Parseval type equality takes place for all x > 0: (2.3) r Z ∞ 1 2 K −1 [h](y)(Fc f )(y) cos xy dy. (h ∗ f )(x) = π 0 y sinh πy γ P r o o f. Using formula (1.8) we obtain 1 2 (2.4) Z ∞ (e−u cosh(x+v) + e−u cosh(x−v) ) dv = K0 (u). 0 Then γ kh ∗f kL1 (R+ ) 6 2 π2 Z ∞Z 0 0 ∞ 2 |h(u)| K0 (u)|f (v)| du dv = 2 khkL−1,β (R+ ,1/x) ·kf kL1 (R+ ) . 1 u π We now prove the Parseval type equality. Using Fubini’s theorem and the formula (2.16.48.19) in [9] Z ∞ π cos byKiy (u) dy = e−u cosh b , 2 0 476 we have γ Z ∞Z ∞ 1 1 −u cosh(x+v) [e + e−u cosh(x−v) ]h(u)f (v) du dv π2 0 0 u Z ∞Z ∞Z ∞ 1 21 = 2 h(u)f (v)Kiy (u)(cos(x + v)y + cos(x − v)y) du dv dy π 0 0 0 πu Z ∞Z ∞Z ∞ 4 1 = 3 h(u)f (v)Kiy (u) cos xy cos vy du dv dy π 0 0 0 u r Z ∞   Z ∞ 1 2 1 2 y sinh πy K (u)h(u) du = iy π 0 y sinh πy π2 u 0 r Z ∞  2 × f (v) cos vy dv cos xy dy π 0 r Z ∞ 2 1 K −1 [h](y)(Fc f )(y) cos xy dy. = π 0 y sinh πy (h ∗ f )(x) = That gives the Parseval identity (2.3), and the proof of the theorem is complete.  The next theorem draws a parallel with a result studied in [16], namely, the boundedness of the generalized convolution (1.11) on spaces Lα,γ r , 1 < r < ∞, α > −1, 0 < γ 6 1 is given. Theorem 2.2. Let 1 < p < ∞ be a real number and q its conjugate exponent, i.e. 1/p + 1/q = 1. Then for any h ∈ L−p,β (R+ ) and f ∈ Lq (R+ ), the generalized p γ convolution (h ∗ f ) (1.11) is well-defined as a bounded continuous function on R+ . γ Moreover, (h ∗ f ) belongs to Lα,γ r (R+ ), 1 6 r < ∞, α > −1, 0 < γ 6 1, and (2.5) γ 1/r kh ∗ f kLα,γ (R+ ) 6 Cα,γ khkL−p,β r (R+ ) kf kLq (R+ ) , p where Cα,γ = (2r+α−1 /π2r γ α+1 )Γ2 ((α + 1)/2). P r o o f. Using the integral representation (2.4) for the function K0 (u), the Hölder inequality, and the fact that e−u cosh(x+v) +e−u cosh(x−v) 6 2e−u for all positive u, x, v, we get Z ∞Z ∞ γ h(u) 1 |f (v)|[e−u cosh(x+v) + e−u cosh(x−v) ] du dv (2.6) |(h ∗ f )(x)| 6 2 π 0 u 0 Z ∞ Z ∞ 1/p 1 h(u) p −u cosh(x+v) 6 2 [e + e−u cosh(x−v) ] du dv π u 0 0 Z ∞Z ∞ 1/q q −u cosh(x+v) −u cosh(x−v) × |f (v)| [e +e ] du dv 2 6 2 π Z 0 0 ∞ 0 h(u) p K0 (u) du u 1/p kf kLq (R+ ) . 477 Therefore, the generalized convolution is well-defined as a bounded operator and the estimation (2.6) holds. Moreover, in view of formula (2.16.2.2) in [9] we get γ kh ∗ f k (R+ ) Lα,γ r Z ∞ 1/r 2 α 6 2 khkL−p,β x K0 (γx) dx (R+ ) kf kLq (R+ ) p π 0 α + 1  γ −α/r 2 khkL−p,β Γ2/r = 2 (2γ)−1/r (R+ ) kf kLq (R+ ) , α > −1. p π 2 2 This yields (2.5)  For the Fourier convolution (see [10]) 1 (h ∗ f )(x) = √ F 2π (2.7) Z ∞ −∞ h(x − y)f (y) dy, Young’s theorem and its corollary, the so-called Young inequality, are fundamental (see [2]). So, it is useful to study similar topics for convolutions and generalized convolutions for other integral transforms. Next, we will prove Young’s type theorem for the generalized convolution (1.11). Theorem 2.3 (Young’s Type Theorem). Let p, q, r be real numbers in (1; ∞) such that 1/p + 1/q + 1/r = 2 and let f ∈ L−p,β (R+ ), 0 < β 6 1, g ∈ Lq (R+ ), p h ∈ Lr (R+ ). Then Z (2.8) 0 P r o o f. means ∞ γ (f ∗ g)(x) · h(x) dx 6 2(p−1)/p kf kL−p,β (R+ ) kgkLq (R+ ) khkLr (R+ ) . p π2 Let p1 , q1 , r1 be the conjugate exponentials of p, q, r, respectively, it 1 1 1 1 1 1 = + = + = 1. + p p1 q q1 r r1 Then it is obvious that 1/p1 + 1/q1 + 1/r1 = 1. Put F (x, u, v) = |g(v)|q/p1 |h(x)|r/p1 [e−u cosh(x+v) + e−u cosh(x−v) ]1/p1 , f (u) p/q1 G(x, u, v) = |h(x)|r/q1 [e−u cosh(x+v) + e−u cosh(x−v) ]1/q1 , u f (u) p/r1 |g(v)|q/r1 [e−u cosh(x+v) + e−u cosh(x−v) ]1/r1 . H(x, u, v) = u We have (2.9) 478 (F · G · H)(x, u, v) = f (u) |g(v)||h(x)|[e−u cosh(x+v) + e−u cosh(x−v) ]. u On the other hand, in the space Lp1 (R3+ ) we have (2.10) Z ∞Z ∞Z ∞ kF kpL1p (R3 ) = |g(v)|q |h(x)|r [e−u cosh(x+v) + e−u cosh(x−v) ] du dv dx + 1 0 0 0 Z ∞Z ∞Z ∞ 62 |g(v)|q |h(x)|r e−u du dv dx 0 0 0 = 2kgkqLq (R+ ) khkrLr (R+ ) . Further, the fact that K0 (u) 6 K0 (βu) for 0 < β 6 1 (see [16]) yields (2.11) Z ∞Z ∞Z ∞ f (u) p kGkpL1q (R3 ) = |h(x)|r [e−u cosh(x+v) + e−u cosh(x−v) ] du dv dx + 1 u Z0 ∞Z0 ∞Z0 ∞ f (u) p 6 K0 (βu)|h(x)|r du dx u 0 0 0 khkrLr (R+ ) , = kf kp −p,β Lp and similarly, (2.12) kHkrL1r 3 1 (R+ ) (R+ ) Z ∞Z ∞Z ∞ f (u) p |g(v)|q [e−u cosh(x+v) + e−u cosh(x−v) ] du dv dx u Z0 ∞Z0 ∞Z0 ∞ f (u) p 6 K0 (βu)|g(v)|r du dv u 0 0 0 = kf kpL−p,β (R ) kgkqLq (R+ ) . = p + From (2.10), (2.11) and (2.12) we have (2.13) kF kLp1 (R3+ ) kGkLq1 (R3+ ) kHkLr1 (R3+ ) 6 2(p−1)/p kf kL−p,β (R+ ) kgkLq (R+ ) khkLr (R+ ) . p From (2.9) and (2.13), by three-function form of the Hölder inequality [2] we have Z ∞ γ (f ∗ g)(x) · h(x) dx 0 Z ∞Z ∞Z ∞ f (u) 1 |g(v)||h(x)|[e−u cosh(x+v) + e−u cosh(x−v) ] du dv dx 6 2 π 0 0 0 u Z ∞Z ∞Z ∞ 1 = 2 F (x, u, v)G(x, u, v)H(x, u, v) du dv dx π 0 0 0 1 6 2 kF kLp1 (R3+ ) kGkLq1 (R3+ ) kHkLr1 (R3+ ) π 2(p−1)/p 6 kf kL−p,β (R+ ) kgkLq (R+ ) khkLr (R+ ) . p π2 The proof is complete.  479 The following Young’s type inequality is the direct corollary of the above theorem Corollary 2.1 (A Young’s Type Inequality). Let 1 < p < ∞, 1 < q < ∞, 1 < r < ∞ be such that 1/p + 1/q = 1 + 1/r and let f ∈ L−p,β (R+ ), 0 < β 6 1, p g ∈ Lq (R+ ). Then the generalized convolution (1.11) is well-defined in Lr (R+ ), moreover, the following inequality holds: γ (2.14) kf ∗ gkLr (R+ ) 6 2(p−1)/p kf kL−p,β (R+ ) kgkLq (R+ ) . p π2 3. A Watson type theorem An important part of the integral transforms theory is to study unitary transforms. In this section, for a class of differential operators of infinite order, we give a condition on the kernel h such that the convolution transformation (3.3) defines a unitary operator in L2 (R+ ), and calculate the inverse transformation. By an argument similar to that in the proof of Theorem 2.1, one can easily prove the following lemma. Lemma 3.1. Let h ∈ L−2,β (R+ ), 0 < β 6 1, and f ∈ L2 (R+ ). Then the gener2 alized convolution (1.11) satisfies the factorization equality (1.12). Furthermore, the following generalized Parseval identity holds: (3.1) γ (h ∗ f )(x) = r Z ∞ 1 2 K −1 [h](y)(Fc f )(y) cos xy dy, π 0 y sinh πy where the integral is understood in the L2 (R+ ) norm, if necessary. Theorem 3.1. Let h ∈ L−2,β (R+ ), 0 < β 6 1. Then the condition 2 |K −1 [h](τ )| = (3.2) 1 cosh(πτ ) is necessary and sufficient for the transformation f 7→ g given by formula (3.3) g(x) = Z Z ∞ 4 d2  ∞ ∞ 1 −u cosh(x+v) d2 Y  (e 1 − π2 dx2 k 2 dx2 0 0 u k=0 + e−u cosh(x−v) )h(u)f (v) du dv, 480 to be unitary on L2 (R+ ). Moreover, the inverse transformation can be written in the symmetric form (3.4) Z Z N d2 Y  4 d2  ∞ ∞ 1 −u cosh(x+v) f (x) = lim 2 2 (e 1− 2 2 N →∞ π dx k dx u 0 0 k=0 + e−u cosh(x−v) )h̄(u)f (v) du dv. Here, the limit is understood in the L2 (R+ ) norm. P r o o f. Sufficiency. Suppose that the function h satisfies condition (3.2). Applying Lemma 3.1, it is easy to see that the generalized convolution transform (3.3) can be written in the form (3.5) r Z N d2 Y  4 d2  ∞ 1 2 lim (Kiy [h])(Fc f )(y) cos xy dy, 1 − g(x) = 2 2 2 π N →∞ dx k dx y sinh πy 0 k=0 or equivalently, g(x) = lim gN (x), where N →∞ gN (x) = N  d2 Y  1 4 d2   F (K [h])(F f )(y) (x). 1 − c iy c dx2 k 2 dx2 y sinh πy k=0 It is well-known that h(y), yh(y), y 2 h(y) ∈ L2 (R+ ) if and only if (F h)(x), (d(F h)(x)/dx), (d2 (F h)(x)/dx2 ) ∈ L2 (R+ ) (Theorem 68, page 92, [11]). Therefore, h(y), yh(y), y 2 h(y), . . ., y n h(y) ∈ L2 (R+ ) if and only if (F h)(x), (d(F h)(x)/dx), (d2 (F h)(x)/dx2 ), . . . , (dn (F h)(x)/dxn ) ∈ L2 (R+ ). Moreover, for each positive integer n we have d2n (F h)(x) = (−1)n F (y 2n h(y))(x). dx2n Therefore, if y 2 N Q k=0 (1 + 4y 2 /k 2 )h(y) ∈ L2 (R+ ) then the following formula holds:  Y  N  N 4y 2  d2 Y  4 d2  2 1 + 2 h(y) (x). 1 − 2 2 (Fc h)(x) = −Fc y dx2 k dx k (3.6) k=0 k=0 From condition (3.2) and the infinite product form of sinh z (see formula (4.5.68) in [1]) we have y2 N Y k=0 (1 + (4y 2 /k 2 ))(1/y sinh πy)K −1 [h](y) = 1  Y ∞ (1 + (4y 2 /k 2 )) < 1, k=N +1 481 and hence it is bounded. Therefore N  Y 4y 2  1 2 y 1+ 2 K −1 [h](y)(Fc f )(y) ∈ L2 (R+ ), k y sinh πy k=0 and formula (3.6) yields   Y N  1 4y 2  −1 2 K [h](y)(Fc f )(y) (x) ∈ L2 (R+ ). gN (x) = Fc y 1+ 2 k y sinh πy k=0 This shows that gN belongs to L2 (R+ ). Applying the Fourier cosine transform to both sides of the above relation, we have (Fc gN )(y) = y 2 N  Y 1 4y 2  K −1 [h](y)(Fc f )(y). 1+ 2 k y sinh πy k=0 Besides, from the Parseval equality for the Fourier cosine transform kFc f kL2 (R+ ) = kf kL2 (R+ ) , it follows that kFc gN − Fc gkL2 (R+ ) = kgN − gkL2 (R+ ) → 0, N → ∞. Therefore, using formula (4.5.68) in [1] we conclude that (Fc g)(y) = y 2 ∞  Y 1+ k=0 1 4y 2  K −1 [h](y)(Fc f )(y) k 2 y sinh πy 1 K −1 [h](y)(Fc f )(y) 2y sinh πy = cosh πyK −1 [h](y)(Fc f )(y). = y sinh 2πy From condition (3.2), it is easy to see that |(Fc g)(y)| ≡ |(Fc f )(y)|, then kf kL2 (R+ ) = kgkL2 (R+ ) , which implies that the transform (3.3) is unitary. Again from condition (3.2) we obtain cosh πyK −1 [h̄](y)(Fc g)(y) = (Fc f )(y). Thus, in the same manner as above it corresponds to (3.4) and the inversion formula of the transform (3.3) follows. Necessity. Suppose that transform (3.3) is unitary on L2 (R+ ) and the inversion formula is defined by (3.4). Then using the Parseval type identity (3.1), the Parseval identity for the Fourier cosine transform, and formula (4.5.68) in [1] we obtain kgkL2(R+ ) = k cosh πyK −1 [h](y)(Fc f )(y)kL2 (R+ ) = kFc f kL2 (R+ ) = kf kL2 (R+ ) . The middle equality holds for all f ∈ L2 (R+ ) if and only if h satisfies the condition (3.2). This completes the proof of the theorem.  482 4. A class of integro-differential problems In spite of having many useful applications (see [7]), not many integro-differential equations can be solved in closed form. No application of convolution type transforms of solving integro-differential was presented in recent investigations [3], [4], [13], [15]. In this section, we apply a general class of Fourier cosine and Kontorovich-Lebedev generalized convolution transforms to solve a class of integro-differential problems, which seems to be difficult to solve in closed form by using other techniques. Namely,
n−1
Q γ d2 d2 2 in case D = dx − dx , the transform f 7→ Kh (f ) := D(h ∗ f ) is of the 2 2 + k form (4.1) (Kh f )(x) = k=1 n−1  Z ∞Z ∞ 1 1 d2 Y  d2 2 + k [e−u cosh(x+v) − π2 dx2 dx2 u 0 0 k=1 + e−u cosh(x−v) ]h(u)f (v) du dv. We consider the integro-differential problem (4.2) f (x) + (Kh f )(x) = g(x), d2k−1 f (0) = 0, k = 1, n, dx2k−1 lim f (k) (x) = 0, k = 0, 2n − 1. x→∞ Here, h, g are given functions in L1 (R+ ), and f is the unknown function. In order to give a solution of the above problem, note that, for h ∈ L1 (R+ ) such that h(0) = 0, lim h′ (x) = 0, the Fourier sine and Fourier cosine transforms of h, h′ x→∞ exist. Furthermore, (4.3) Z ∞ 1 h′ (x) sin xy dx (Fs h′ )(y) = √ 2π 0   Z ∞ ∞ 1 h(x) sin xy − y =√ h(x) cos xy dx 0 2π 0 = − y(Fc h)(y), and (4.4) 1 (Fc h )(y) = √ 2π ′ Z ∞ h′ (x) cos xy dx = y(Fs h)(y). 0 483 Theorem 4.1. Suppose the following condition holds:   γ (2n − 1)! 1 1− √ (y) 6= 0, Fc h ∗ cosh2n τ /2 2π · 22n−1 (4.5) ∀y > 0. Then problem (4.2) has a unique solution in L1 (R+ ) whose closed form is (4.6) f (x) = g(x) + (g ∗ l)(x), Fc where l ∈ L1 (R+ ) is defined by √ γ ((2n − 1)!/ 2π · 22n−1 )Fc (h ∗ cosh−2n τ /2)(y) . (Fc l)(y) = √ γ 1 − ((2n − 1)!/ 2π · 22n−1 )Fc (h ∗ cosh−2n τ /2)(y) P r o o f. The equation (4.2) can be rewritten in the form (4.7) f (x) + n−1  γ d2 Y  d2 2 {(h ∗ f )(x)} = g(x). + k − 2 2 dx dx k=1 Applying the Fourier cosine transform to both sides of (4.7), by original conditions (4.2) and by virtue of the factorization equality (1.12) and formulas (4.3), (4.4) we obtain (4.8) (Fc f )(y) − y 2 n−1 Y k=1 (y 2 + k 2 ) · 1 K −1 [h](y)(Fc f )(y) = (Fc g)(y). y sinh πy Using formula (see relation (1.9.3) in [5]) Fc  √ n−1 2π · 22n−1 y Y 2 (y + k 2 ), (y) = (2n − 1)! sinh πy cosh2n τ /2 k=1 1  we have   (2n − 1)! 1 (Fc f )(y) − √ (y)K −1 [h](y)(Fc f )(y) = (Fc g)(y), Fc cosh2n τ /2 2π · 22n−1 or equivalently,  i  h γ 1 (2n − 1)! (y) = (Fc g)(y). Fc h(τ ) ∗ (Fc f )(y) 1 − √ 2n 2π · 22n−1 cosh τ /2 484 From condition (4.5) we get (4.9) √   γ ((2n − 1)!/ 2π · 22n−1 )Fc (h ∗ cosh−2n τ /2)(y) (Fc f )(y) = 1 + (Fc g)(y). √ γ 1 − ((2n − 1)!/ 2π · 22n−1 )Fc (h ∗ cosh−2n τ /2)(y) Recall that the Wiener-Levy theorem [8] states that if f is the Fourier transform of an L1 (R) function, and ϕ is analytic in a neighborhood of the origin that contains the domain {f (y), ∀y ∈ R}, and ϕ(0) = 0, then ϕ(f ) is also the Fourier transform of an L1 (R) function. For the Fourier cosine transform it means that if f is the Fourier cosine transform of an L1 (R+ ) function, and ϕ is analytic in a neighborhood of the origin that contains the domain {f (y), ∀y ∈ R+ }, and ϕ(0) = 0, then ϕ(f ) is also the Fourier cosine transform of an L1 (R+ ) function. By the given condition (4.5) the function ϕ(z) = z/(1 + z) satisfies the conditions of the Wiener-Levy theorem, and therefore, there exists a unique function l ∈ L1 (R+ ) such that √ γ ((2n − 1)!/ 2π · 22n−1 )Fc (h ∗ cosh−2n τ /2)(y) . (Fc l)(y) = √ γ 1 − ((2n − 1)!/ 2π · 22n−1 )Fc (h ∗ cosh−2n τ /2)(y) Therefore the equation (4.9) becomes (Fc f )(y) = (1 + (Fc l)(y))(Fc g) = Fc (g + g ∗ l)(y), Fc which implies f (x) = g(x) + (g ∗ l)(x). The proof is complete. Fc  References [1] M. Abramowitz, I. A. Stegun: Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables. U.S. Department of Commerce, Washington, 1964. [2] R. A. Adams, J. J. F. Fournier: Sobolev Spaces, 2nd ed. Pure and Applied Mathematics 140. Academic Press, New York, 2003. [3] F. Al-Musallam, V. K. Tuan: Integral transforms related to a generalized convolution. Result. Math. 38 (2000), 197–208. [4] L. E. Britvina: A class of integral transforms related to the Fourier cosine convolution. 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Britvina: Convolutions related to the Fourier and Kontorovich-Lebedev transforms revisited. Integral Transforms Spec. Funct. 21 (2010), 259–276. Authors’ addresses: Nguyen Thanh Hong, High School for Gifted Students, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam, e-mail: hongdhsp1@ yahoo.com; Trinh Tuan, Department of Mathematics, Electric Power University, 235 Hoang Quoc Viet, Hanoi, Vietnam, e-mail: [email protected]; Nguyen Xuan Thao, Faculty of Applied Mathematics and Informatics, Hanoi University of Technology, No. 1, Dai Co Viet, Hanoi, Vietnam, e-mail: [email protected]. 486