Index transforms with Weber type kernels

Transcription

1 Inde transforms with Weber tpe kernels arxiv: v1 [math.ca] 6 Nov 17 Semon Yakubovich November 7, 17 Department of Mathematics, Facult of Sciences, Universit of Porto, Campo Alegre str., Porto, Portugal Abstract New inde transforms with Weber tpe kernels, consisting of products of Bessel functions of the first and second kind are investigated. Mapping properties and inversion formulas are established for these transforms in Lebesgue spaces. The results are applied to solve a boundar value problem on the wedge for a fourth order partial differential equation. Kewords: Inde Transforms, Bessel functions of the first and second kind, Fourier transform, Mellin transform, Boundar value problem AMS subject classification: 44A15, 33C1, 44A5 1 Introduction and preliminar results Let α R and f(), g(τ), R +, τ R be comple-valued functions. In this paper we will investigate mapping properties of the following inde transforms [1] (F α f)(τ) = (G α g)() = sakubov@fc.up.pt W α (,τ)f()d, (1.1) W α (,τ) g(τ)dτ, (1.) 1

2 S. Yakubovich where W α (,τ) is the Weber tpe kernel defined b the formula W α (,τ) = Im [ J α+iτ ( )Y α iτ ( ) ]. (1.3) Hereiistheimaginarunit,ImistheimaginarpartofacomplenumberandJ ν (z), Y ν (z) are Bessel functions of the first and second kind, respectivel, [], Vol. II, which, in turn, are fundamental solutions of the differential equation z d u dz +zdu dz +(z ν )u =. (1.4) Bessel functions in(1.3) with fied τ form the kernel of the well-known Weber-Orr integral transforms (cf. [3], [4], [5] ) widel used, for instance, in the heat process problems for radial-smmetric regions. Concerning inde transforms of the Weber tpe, we mention integral operators considered in [6], where the variable of integration in the corresponding kernel is involved in the arguments and indices of Bessel functions. Appealing to relation ( ) in [7], Vol. III we find the important Mellin-Barnes integral representation of the kernel (1.3), namel, W α (,τ) = sinh(πτ) 4π 7/ i Γ(s+iτ)Γ(s iτ) Γ(s+α)Γ(1/ s)γ(1 s) s ds, >, (1.5) where Γ(z) is Euler s gamma-function [], Vol. I and γ is chosen from the interval (ma( α, ), 1/). Integral (1.5) converges absolutel for an γ from the open vertical strip ma( α,) < γ < 1/ and uniforml with respect to > b virtue of the Stirling asmptotic formula for the gamma-function [], Vol. I, which ields for a fied τ Γ(s+iτ)Γ(s iτ)γ(s+α)γ(1/ s)γ(1 s) = O ( e π s s γ 3/), s. (1.6) The Mellin-Barnes representation (1.5) can be used to represent W α (,τ) for all >, τ R as the Fourier cosine transform [9]. Precisel, we have Lemma 1. Let >,τ R, α R\{}. Then W α (,τ) has the following representation in terms of the Fourier cosine transform W α (,τ) = sinh(πτ) πsin(πα) where J ν (z) is the Anger function [], Vol. II. [ Jα ( cosh(u/)) J α ( cosh(u/)) ] cos(τu)du, (1.7)

3 Weber tpe kernels 3 Proof. In fact, appealing to the reciprocal formulae via the Fourier cosine transform (cf. formula (1.14) in [1]) Γ(s+iτ)Γ(s iτ)cos(τ)dτ = π Γ(s) s cosh s, Re s >, (1.8) (/) Γ(s+iτ)Γ(s iτ) = Γ(s) s 1 cos(τ ) cosh s (/) d, (1.9) we replace the gamma-product Γ(s + iτ) Γ(s iτ) in (1.5) b the right-hand side of (1.9). Then, changing the order of integration via the absolute convergence, which can be justified using the Stirling asmptotic formula for the gamma-function, we emplo the reflection and duplication formulae for the gamma-function [], Vol. I to derive W α (,τ) = sinh(πτ) π i cos(τ u) Γ(s+α) ( cosh (u/) ) s dsdu. (1.1) sin(πs) In the meantime, the inner integral with respect to s in (1.1) is calculated in [7], Vol. III, relation ( ), and we have Γ(s+α)Γ(s)Γ(s+1/)Γ(1/ s)γ(1 s) s ds = 4π3 i [ Jα ( ) J α ( ) ], sin(απ) where J ν (z) is the Anger function [], Vol. II. Hence, substituting this result in (1.1), we get (1.7). Further, recallingthemellin-barnesrepresentation(1.5)ofthekernel W α (,τ), wewill derive an ordinar differential equation whose particular solution is W α (,τ). Precisel, it is given b Lemma. Let α < 1/. For each τ R the function W α (,τ) satisfies the following fourth order differential equation with variable coefficients 3d4 W α +6 d3 W α + ( 7+τ α + ) d W α d 4 d 3 d ( + 1+τ α + 5 ) ( ) dwα 1 d + (ατ) W α =, >. (1.11)

4 4 S. Yakubovich Proof. Indeed, the asmptotic behavior at infinit (1.6) of the integrand in (1.5) and the absolute and uniform convergence of the integral and its derivatives with respect to allow us to differentiate under the integral sign. Hence, appling twice the differential operator d to both sides of (1.5) and emploing the reduction formula for gamma-function [], d Vol. I, we obtain ( d ) W α (,τ) = sinh(πτ) d 4π 7/ i Γ(s+α)Γ(1/ s)γ(1 s) s ds s Γ(s+iτ)Γ(s iτ) = sinh(πτ) 4π 7/ i Γ(s+1+iτ)Γ(s+1 iτ) Γ(s+α)Γ(1/ s)γ(1 s) s ds τ W α (,τ) = sinh(πτ) π 7/ γ+1+i γ+1 i Γ(s+iτ)Γ(s iτ) Γ(s+α 1)Γ(3/ s)γ( s) 1 s ds τ W α (,τ). Γ(+α s) After multiplication b α and differentiation of both sides of the obtained equalit, we emplo again the reduction formula for the gamma-function to derive ( d ( α d ) W α (,τ)) = sinh(πτ) γ+1+i Γ(s+iτ)Γ(s iτ) d d 4π 7/ i γ+1 i Γ(s+α 1)Γ(3/ s)γ( s) α s ds τ d d (α W α (,τ)). (1.1) In the meantime, the contour in the integral in the right-hand side of (1.1) can be moved to the left via Slater s theorem [7], Vol. III. Hence, integrating over the vertical line Re s = γ in the comple plane, we take into account assumptions ma( α,) < γ < 1/, α < 1/ to guarantee the condition < γ+α < 1. Then the integral can be treated as follows Γ(s+iτ)Γ(s iτ) Γ(s+α 1)Γ(3/ s)γ( s) α s ds = Γ(s+iτ)Γ(s iτ) (1/ s)(1 s)γ(s+α)γ(1/ s)γ(1 s) α s ds (s+α 1)

5 Weber tpe kernels 5 = α d d Γ(s+iτ)Γ(s iτ) (1/ s)γ(s+α)γ(1/ s)γ(1 s) 1 s ds (s+α 1) = α d d d 3/ d Γ(s+iτ)Γ(s iτ) Γ(s+α)Γ(1/ s)γ(1 s) 1/ s ds (s+α 1) = α d d 3/ d d α 1/ Γ(s+iτ)Γ(s iτ) Γ(s+α)Γ(1/ s)γ(1 s) s α dsd. Substituting this result into (1.1) and using (1.5), we find ( d ( α d ) W α (,τ)) +τ d d d d (α W α (,τ)) = α d d d 3/ d α 1/ α W α (,τ)d. (1.13) Finall, fulfilling the differentiation in (1.13), we end up with equation (1.11), completing the proof of Lemma. In the sequel we will emplo the Mellin transform technique developed in [8] in order to investigate the mapping properties of the inde transforms (1.1), (1.). Precisel, the Mellin transform is defined, for instance, in L ν,p (R + ), 1 p (see details in [9]) b the integral f (s) = f() s 1 d, (1.14) being convergent in mean with respect to the norm in L q (ν i,ν +i ), ν R, q = p/(p 1). Moreover, the Parseval equalit holds for f L ν,p (R + ), g L 1 ν,q (R + ) f()g()d = 1 πi ν i The inverse Mellin transform is given accordingl f() = 1 πi ν i f (s)g (1 s)ds. (1.15) f (s) s ds, (1.16)

6 6 S. Yakubovich where the integral converges in mean with respect to the norm in L ν,p (R + ) ( 1/p f ν,p = f() p d) νp 1. (1.17) In particular, letting ν = 1/p we get the usual space L 1 (R + ; d). The so-called Lebedev epansion theorem [1], which is related to the representation of the antiderivative in terms of the repeated integral with the product of the modified Bessel functions of the first and second kind [], Vol. II will pla an important role to prove our main results. Precisel, we have Theorem 1. Let f L 1/,1 (,1) L 3/,1 (1, ). Then for all > f()d = 4 τ sinh(πτ)k π iτ ()dτ f()[i iτ ()+I iτ ()]K iτ ()d, (1.18) where I µ (z),k µ (z) are modified Bessel functions of the first and second kind, respectivel, and the inner integral b converges absolutel and the eterior integral b τ is understood as the Riemann improper integral. Boundedness and inversion properties of the inde transform (1.1) Let C b (R;[sinh(πτ)] 1 ) be the space of functions f(τ) such that f(τ)/sinh(πτ) is bounded continuous on R. Theorem. Let ma( α,) < γ < 1/. The inde transform (1.1) is well-defined as a bounded operator F α : L 1 γ,1 (R + ) C b (R;[sinh(πτ)] 1 ) and the following norm inequalit takes place where F α f Cb sup τ R (F α f)(τ) sinh(πτ) C γ f L1 γ,1 (R + ), (.1) C γ = (γ 1) Γ(s+α) B(γ,γ) ds (.) π sin(πs) and B(a, b) is the beta-function [], Vol. I. Moreover, it admits the integral representation (F α f)(τ) = sinh(πτ) π K iτ ( )[ Iiτ ( ) +I iτ ( )] ϕα ()d, τ R, (.3)

7 Weber tpe kernels 7 where ϕ α () = 1 1 γ+i πi 1 and integrals (.3), (.4) converge absolutel. Proof. In fact, from (1.1) we have the estimate Γ(1 s+α)γ (s) f (s) s ds, > (.4) Γ(s+α) W α (,τ) γ sinh(πτ) π du γ+i cosh γ (u) where the constant C γ is defined b (.). Hence, F α f Cb = sup (F α f)(τ) τ R sinh(πτ) C γ Γ(s+α) sin(πs) ds = C γ γ sinh(πτ), (.5) γ f() d, which ields (.1) via (1.17). Net, recalling Parseval equalit (1.15), the Mellin-Barnes representation (1.5) of the kernel W α (,τ) and formula (1.16) of the Mellin transform, we have (F α f)(τ) = sinh(πτ) 4π 7/ i Γ(s+iτ)Γ(s iτ) Γ(s+α)Γ(1/ s)γ(1 s) f (1 s)ds. Therefore, appealing to relation ( ) in [7], Vol. III, it implies the representation π cosh(πτ) K ( )[ ( ) ( )] 1 iτ Iiτ +I iτ = πi Γ(1/ s) Γ(1 s) s ds, >, < γ < 1. Γ(s+iτ)Γ(s iτ) Hence emploing again Parseval equalit (1.15), it can be rewritten in the form (.3), where integrals (.3), (.4) converge absolutel owing to Stirling asmptotic formula for the gamma-function, the asmptotic behavior of the kernel in (.3) (see [], Vol. II) K iτ ( )[ Iiτ ( ) +I iτ ( )] = O(log),,

8 8 S. Yakubovich K iτ ( )[ Iiτ ( ) +I iτ ( )] = O ( 1 ),, and the condition ma( α,) < γ < 1/. Theorem is proved. The inversion formula for the transform (1.1) is given b Theorem 3. Let 1/ < α <,γ in (.4) satisf the condition ma( α,1/4) < γ < min(1/,1 a) for some a < b, b > 5/4. Let also f be locall integrable on R +, i.e. f() L loc (R + ) and behave as f() = O( a ),, f() = O ( b),. Then if its Mellin transform f (s) L 1 (ν i,ν+i ) for an ν from the vertical strip a < Res < b and τ 1 α e πτ (F α f)(τ) L 1 (R + ), the following inversion formula for the inde transform (1.1) holds for almost all > ( f() = π α d ) d α and integral (.6) converges absolutel. [ Im [ J1+α+iτ ( )Y α iτ ( ) J α+iτ ( )Y 1+α iτ ( ) ] 4τRe [ J α+iτ ( )Y α iτ ( ) ]] (F α f)(τ)dτ (.6) Proof. In fact, making a simple change of variable in (.3), it becomes (F α f)(τ) = sinh(πτ) π K iτ ()[I iτ ()+I iτ ()]ϕ α ( ) d. (.7) Our goal now is to verif conditions of Theorem 1 for function ϕ α ( ). To do this, we emplo (1.17) and (.4) to derive the estimate 1 ϕα ( ) 1/ d 1 π f L 1 γ,1 (R + ) (4γ 1)π 1 1 γ+i 1 1 γ+i γ 3/ d 1 Γ(1 s+α)γ (s) Γ(s+α) Γ(1 s+α)γ (s) ds Γ(s+α) <, γ > 1 4, f (s)ds Onthe other hand, since f() L loc (R + ) andbehaves asf() = O( a ),, f() = O ( b),, a < b, it is eas to show via the absolute convergence of the integral (1.14) that f (s) is analtic and bounded in the strip a < Res < b. Therefore, shifting the contour in (.4) to the right, we integrate over a vertical line (µ i,µ+i ) in the strip of the analticit of the integrand. Moreover,

9 Weber tpe kernels 9 1 ϕα ( ) 3/ d 1 π f L µ,1 (R + ) (4µ 5)π 1 µ+i µ i 3/ µ d µ+i µ i Γ(1 s+α)γ (s) f (s)ds Γ(s+α) Γ(1 s+α)γ (s) ds Γ(s+α) <, µ > 5 4, and this is true, because 1 γ and µ belong to the interval (a,b). Thus conditions of Theorem 1 are satisfied for the function ϕ α ( ). Hence we emplo (1.18) to invert (.6), and after simple changes of variables we come up with the equalit for all > ϕ α ()d = 4 τk iτ ( )(F α f)(τ)dτ. (.8) Meanwhile, according to relation ( ) in [7], Vol. III the function K iτ( ) has the following Mellin-Barnes integral representation Kiτ ( ) = 1 4π 1/ i ν i Γ(s+iτ)Γ(s iτ)γ(s) s ds, ν >, >. (.9) Γ(1/+s) Substitutingthisvaluein(.8),weobservethatunderconditionsofthetheoremτ(F α f)(τ) L 1 (R + ) the differentiation under the integral sign with respect to is guaranteed via the absolute and uniform convergence for >. Consequentl, after differentiation of both sides in (.8), we obtain ϕ α () = 1 Γ(s+iτ)Γ(s iτ)γ(1+s) π 1/ i ν i Γ(1/+s) τ(f α f)(τ) s dsdτ, >. (.1) Moreover, one can take the Mellin transform (1.14) of both sides in (.1). Hence taking in mind (.4), we find = π Γ(s+iτ)Γ(s iτ)γ(1+α+s) (F α f)(τ) τdτ, (.11) Γ(1+s)Γ(1/+s)Γ(α s) where Res = ν and ν >, a < 1 + ν < b. Then, dividing both sides of (.11) b (α+s)(α s) and inverting the Mellin transform, we obtain 1 ν+i πi ν i (α+s)(α s) s ds = 1 π 1/ i ν i Γ(s+iτ)Γ(s iτ) Γ(1+s)Γ(1/+s) Γ(α+s) (F αf)(τ) s τdτds. (.1)

10 1 S. Yakubovich In order to change the order of integration in the right hand-side of (.1), we observe that integral representation (1.9) of the product of gamma-functions gives the uniform estimate b τ. Precisel, we derive Γ(s+iτ)Γ(s iτ) C ν Γ(s), C ν = (1 ν) d cosh ν (). Then as in (1.6) we appl the Stirling asmptotic formula and the duplication formula for the gamma-function to get Γ(s)Γ(s + α) Γ(1/+s)Γ(1+s) = s 1 Γ(s+α) s π = O( s (ν 1)), Im s. Hence the corresponding iterated integral in (.1) converges absolutel under conditions α < ν < 1/ and τ(f α f)(τ) L 1 (R + ). Therefore b Fubini s theorem the interchange of the order of integration is allowed, and we obtain from (.1) 1 ν+i πi ν i (α+s)(α s) s ds = 1 π 1/ i ν i Γ(s+iτ)Γ(s iτ)γ(α+s) Γ(1+s)Γ(1/+s) τ(f α f)(τ) s dsdτ. (.13) In the meantime, relation(8.4..1) in[7], Vol. III gives the Mellin-Barnes representation for the following Weber tpe kernel Re [ J α+iτ ( )Y α iτ ( ) ] = 1 π Γ(s+iτ)Γ(s iτ)γ(α+s) πi ν i Γ(s)Γ(1/+s) s ds, (.14) where the integral converges for each > and τ R under the condition α < ν < 3/4. Hence the inner integral in (.13) b s can be calculated with the use of the addition formula for the gamma-function, asmptotic behavior of Bessel functions at infinit [], Vol. II and Parseval equalit (1.15) for functions Re [ J α+iτ ( )Y α iτ ( ) ] L ν, (, ) and 1 L 1 ν, (, ), >, < ν < 1/. Namel, appealing to (.14), we find the formula 1 π Γ(s+iτ)Γ(s iτ)γ(α+s) πi ν i Γ(1+s)Γ(1/+s) s ds

11 Weber tpe kernels 11 = Re[J α+iτ ( )Y α iτ ( )] d. (.15) Moreover, the integral in the right-hand side of (.15) is calculated via relation ( ) in [7], Vol. II and we have = Re[J α+iτ ( )Y α iτ ( )] d τα Im[ J 1+α+iτ ( )Y α iτ ( ) J α+iτ ( )Y 1+α iτ ( ) ] 1 α Re[ J α+iτ ( )Y α iτ ( ) ], α, τ. (.16) On the other hand, the left-hand side of (.13) can be treated as follows 1 ν+i πi ν i (α+s)(α s) s ds = 1 [ 4παi ν i α+s s ds ] + ν i α s s ds, α. Hence combining with (.15), equalit (.13) can be written in the form [ 1 ν+i 4παi ν i α+s s ds+ = π ν i ] α s s ds τ Re[J α+iτ ( )Y α iτ ( )](F α f)(τ) dτd. (.17) The interchange of the order of integration in the right-hand side of (.17) can be motivated b Fubini theorem, emploing an estimate for the product of Bessel functions, which, in turn, is a direct consequence of the Poisson integral [], Vol. II. In fact, we have Re[J α+iτ ( )Y α iτ ( )] C α Γ(α+1/+iτ), (.18) where >, α > 1/,C > and does not depend on,τ. Therefore since due to Stirling s formula Γ(α+1/+iτ) = O(τ α e πτ ), τ + we derive owing toconditions of the theorem the following estimate τ Re[J α+iτ ( )Y α iτ ( )](F α f)(τ) dτd

12 1 S. Yakubovich α 1 d τ 1 α e πτ (F α f)(τ) dτ <, 1 < α <, which guarantees the change of the order of integration in the right-hand side of (.17). Moreover, under assumption f (1+s) L 1 (1+ν i,1+ν+i ) one can differentiate b in (.17) and after simple manipulations we derive [ 1 ν+i 4πi ν i α+s s ds = π Adding (.17) and (.19), we find ν i ] α s s ds τre [ J α+iτ ( )Y α iτ ( ) ] (F α f)(τ)dτ. (.19) 1 ν+i πi ν i α+s s ds = πα τre[j α+iτ ( )Y α iτ ( )](F α f)(τ) ddτ +π τre [ J α+iτ ( )Y α iτ ( ) ] (F α f)(τ)dτ. (.) Meanwhile, the left-hand side of (.) can be represented with the use of (1.16) and integration under the integral sign. Thus we get 1 ν+i πi ν i α+s s ds = α α f()d. Finall, substituting the right-hand side of the latter equalit in(.), taking into account (.16) and differentiating both sides with respect to, we establish for almost all > the inversion formula (.6). Theorem 3 is proved. Remark 1. If the differentiation with respect to is allowed under the integral sign in (.6), this formula can be written as follows τ Re[J α+iτ ( )Y α iτ ( )](F α f)(τ) dτd f() = πα π τ d [ [ Re Jα+iτ ( )Y α iτ ( ) ]] (F α f)(τ)dτ. (.1) d Letting in (1.5) α = and appealing to relation ( ) in [7], Vol. III, operator (1.1) turns to be an adjoint operator of the inde transform with Nicholson s function as the kernel (cf. [11])

13 Weber tpe kernels 13 (F f)(τ) sinh(πτ) F(τ) = [ ( ) ( )] J iτ +Y iτ f()d. (.) Corollar 1. Let conditions of Theorem 3 hold and α =. Then for almost all > the following inversion formula for the inde transform (.) takes place f() = π d d where the integral converges absolutel. τim [ J iτ ( ) ] F(τ)dτ, (.3) Proof. Indeed, in the limit case α =, we find directl from (.13), (.14) and relation ( ) in [7], Vol. III 1 πi ν i Re [ J iτ ( )Y iτ ( ) ] = Im[J iτ ( )], sinh(πτ) s s ds = 1 πi = π ν i τim [ J iτ ( ) ] F(τ) ddτ. s 1dds s Meanwhile, the latter integral b τ converges absolutel via the estimate (see (.15)) τim [ Jiτ ( ) ] d sinh(πτ) (F f)(τ) dτ A ν Γ(s) sγ(1 s) ds ν i τ (F f)(τ) dτ <, < ν < 1, where A > is an absolute constant. Hence, for almost all > we have 1 πi ν i sds s = π d d τim [ J iτ ( ) ] F(τ) ddτ. (.4) However, similarl as in (.18) it follows from the Poisson integral for Bessel functions that Im [ J iτ ( ) ] cosh(πτ),. Therefore the differentiation with respect to under the integral sign is allowed b the absolute and uniform convergence b > since

14 14 S. Yakubovich Hence we get τ [ Im J iτ ( ) ] F(τ) dτ τ cotanh(πτ) (F f)(τ) dτ <. 1 πi ν i sds s = π τim [ J iτ ( ) ] F(τ)dτ. Repeating, analogousl, the same process and differentiating with respect to again, we derive the inversion formula (.3), completing the proof of Corollar 1. 3 Inde transform (1.) In this section we will stud the boundedness properties and prove an inversion formula for the inde transform (1.). We begin with Theorem 4. Let ma( α,) < γ < 1/ and g(τ) L 1 (R;e π τ dτ). Then γ (G α g)() is bounded continuous on R + and it holds sup γ (G α g)() C γ g L1 (R; e dτ), (3.1) π τ > where C γ is defined in (.). Besides, if (G α g)() L γ,1 (R + ) and its Mellin transform s 1/ γ e π s / (G α g) (s) L 1 (γ i, γ +i ), then for all > 1 γ+i πi Γ(α+s)Γ(1 s) (G αg) (s) s ds = π e / K iτ ) sinh(πτ)g(τ)dτ. (3.) Proof. In fact, the norm estimate (3.1) follows immediatel from (.5) and conditions of the theorem. Then taking the Mellin transform (1.14) of both sides in (1.) under the condition (G α g)() L γ,1 (R + ), we emplo (1.5) and change the order of integration via Fubini theorem in the right-hand side of the obtained equalit. Hence we obtain = Γ(1/ s) π 5/ Γ(α+s)Γ(1 s) (G αg) (s) Meanwhile, relation ( ) in [7], Vol. III sas Γ(s+iτ)Γ(s iτ)sinh(πτ)g(τ)dτ. (3.3)

15 Weber tpe kernels 15 e / K iτ ( ) = cosh(πτ) π 3/ i Γ(s+iτ)Γ(s iτ)γ(1/ s) s ds, < γ < 1. Hence an application of the inverse Mellin transform (1.16) to both sides of (3.3) under conditions of the theorem drives us to (3.). The convergence of the integral with respect to s in (3.) is absolute under the condition s 1/ γ e π s / (G α g) (s) L 1 (, γ+i ), and it can be easil verified, recalling the Stirling asmptotic formula for the gammafunction. Theorem 4 is proved. The inversion formula for the inde transform (1.) is given b Theorem 5. Let 1/ < α <, g(z/i) be an odd analtic function in the strip D = {z C : Rez 1/}, g() = and e π Imz / g(z/i) be absolutel integrable over an vertical line in D. If under conditions of Theorem 4 (G α g) (s) is analtic in D and s 1/ ν e π s / (G α g) (s) L 1 (ν i, ν+i ) over an vertical line Res = ν in the strip, then for all R the following inversion formula holds for the inde transform (1.) g() = π (α ( t ( t )] Re [J α+i )Y α i (G α g)() dtd t d + d [Re[J α+i( )Y α i ( ) )]](G α g)()d. (3.4) Proof. Indeed, since the integrand in the left-hand side of (3.) is analtic in the strip Res 1/ and absolutel integrable there over an vertical line, we shift the contour to the left, integrating over (ν i,ν + i ),ν <. Then multipling both sides b e / K i (/) 1 and integrating with respect to over (, ), we change the order of integration in the left-hand side of the obtained equalit due to the absolute convergence of the iterated integral. Moreover, appealing to relation ( ) in [7], Vol. III, we calculate the inner integral to find the equalit 1 ν+i πi ν i = π π Γ( s+i)γ( s i) (G α g) (s)ds Γ(1/ s)γ(α+s)γ(1 s) K i ) K iτ ) sinh(πτ)g(τ) dτd. (3.5) In the meantime the right-hand side of (3.5) can be written, emploing the representation of the Macdonald function in terms of the modified Bessel function of the first kind I z () [], Vol. II

16 16 S. Yakubovich i ) ) ) π sinh(πτ)k iτ = I iτ I iτ, the substitution z = iτ and the propert g( z/i) = g(z/i). Hence it becomes π π = π π K i ) K i iτ K ) i i I z( ) sinh(πτ)g(τ) dτd ) ( z ) dzd g i. (3.6) On the other hand, according to our assumption g(z/i) is analtic in the vertical strip Rez µ, µ >, and e π Imz / g(z/i) is absolutel integrable in the strip. Hence, appealing to the inequalit for the modified Bessel function of the first kind (see [8], p. 93) I z () I Rez () e π Imz /, < Rez µ, one can move the contour to the right in the latter integral in (3.6). Then π π = π π K i ) i i I z( ) µ+i K i I z µ i ) ( z ) dzd g i ) ( z ) dzd g i. (3.7) Hence one can interchange the order of integration in the right-hand side of (3.7) due to the absolute and uniform convergence. Then using the value of the integral (see relation ( ) in [7], Vol. II) we find π π = µ+i π π µ i K i ()I z () d = 1 +z, ) µ+i K i I z g(z/i) +z dz = 1 π π µ i ( µ i µ+i ) ( z ) dzd g i + µ+i µ i ) g(z/i) dz z(z i). (3.8) ThuswearereadtoappltheCauchformulaintheright-handsideofthelatterequalit in (3.8) under conditions of the theorem. Hence π π ) µ+i ) ( z ) dzd K i I z g i µ i

17 Weber tpe kernels 17 Combining with (3.5), we establish the equalit 1 ν+i πi ν i = g(), R\{}. (3.9) π Γ( s+i)γ( s i) (G α g) (s)ds Γ(1/ s)γ(α+s)γ(1 s) = g(), R\{}. (3.1) π The left-hand side of (3.1) can be treated, in turn, appealing to the Parseval equalit (1.15), conditions of the theorem and integral representation (.15). Therefore we derive g() = π where (see (1.16)) h() ( t ) ( t )] dtd Re [J α+i Y α i, R, (3.11) t h() = 1 (α s )(G α g) (s) s ds πi ν i ( d = α (G α g)() d d ) (G α g)(). (3.1) d Substituting the latter epression in (3.11), we integrate twice b parts, eliminating the integrated terms in order to arrive at the inversion formula (3.4). Theorem 4 is proved. 4 Boundar value problem In this section we will emplo the inde transform (1.) to investigate the solvabilit of a boundar value problem for the following fourth order partial differential equation, involving the Laplacian [ ( r + ) ] ( +1 1 α u+ + ) 3 u ( +r + ) u+ 3 ( r + ) u+ ur =, (,) R, (4.1)

18 18 S. Yakubovich where r = +, = + is the Laplacian in R. In fact, writing (4.1) in polar coordinates (r,θ), we end up with the equation r 4 u r + 4 u u 4 r θ +6r 3 r u 3 r r θ +( 7 α +r ) u r ( α u 1 α r θ ) u r r + u r =. (4.) Lemma 3. Let g(τ) L 1 ( R;e (β+π) τ dτ ), β (,π). Then the function u(r,θ) = W α (r,τ) e θτ g(τ)dτ, (4.3) satisfies the partial differential equation (4.) on the wedge (r,θ) : r >, θ β < β, vanishing at infinit. Proof. The proof is straightforward b substitution (4.3) into (4.) and the use of (1.11). The necessar differentiation with respect to r and θ under the integral sign is allowed via the absolute and uniform convergence, which can be verified using inequalit (3.1) and the integrabilit condition g L 1 ( R;e (β+π) τ dτ ), β (,π) of the lemma. Finall, the condition u(r,θ), r is due to inequalit (.5). Finall we will formulate the boundar value problem for equation (4.) and give its solution. Theorem 6. Let g() be given b formula (3.4) and its transform (G α g)(t) G α (t) satisfies conditions of Theorem 5. Then u(r,θ), r >, θ < β b formula (4.3) will be a solution of the initial value problem for the partial differential equation (4.) subject to the initial condition u(r,) = G α (r). Acknowledgments The work was partiall supported b CMUP(UID/MAT/144/13), which is funded b FCT(Portugal) with national(mec), European structural funds through the programs FEDER under the partnership agreement PT, and Project STRIDE - NORTE FEDER- 33, funded b ERDF - NORTE.

19 Weber tpe kernels 19 References [1] Yakubovich S. Inde transforms. Singapore: World Scientific Publishing Compan; [] Erdéli A, Magnus W, Oberhettinger F, Tricomi FG. Higher transcendental functions. Vols. I, II. New York: McGraw-Hill; [3] Zhang X, Tong D. A generalized Weber transform and its inverse formula. Appl. Math. Comp. 7; 193: [4] Malits P. On a certain class of integral equations associated with Hankel transforms. Acta Appl. Math. 7; 98: [5] Nasim C. Dual integral equations involving Weber-Orr transforms. Internat. J. Math. and Math. Sci. 1991; 14; 1: [6] Al-Musallam F, Vu KT. A modified and a finite inde Weber transforms. J. Anal. Appl. ; 1; : [7] Prudnikov AP, Brchkov YuA, Marichev OI. Integrals and series: Vol. I: Elementar functions. New York: Gordon and Breach; 1986; Vol. II: Special functions. New York: Gordon and Breach; 1986; Vol. III: More special functions. New York: Gordon and Breach; 199. [8] Yakubovich S, Luchko Yu. The hpergeometric approach to integral transforms and convolutions, Mathematics and its applications. Vol. 87. Dordrecht: Kluwer Academic Publishers Group; [9] Titchmarsh EC. An introduction to the theor of Fourier integrals. New York: Chelsea; [1] Lebedev NN. On an integral representation of an arbitrar function in terms of squares of Macdonald functions with imaginar inde. Sibirsk. Mat. Zh. 196; 3: 13- (in Russian). [11] Yakubovich S. On the inde integral transformation with Nicholson s function as the kernel. J. Math. Anal. Appl. ; 69: [1] Brchkov YuA, Glaeske H-Yu, Marichev OI. Factorization of integral transformations of convolution tpe. Mathematical analsis. Moscow: Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesouz. Inst. Nauchn. i Tekhn. Inform. 1983; 1: (in Russian).