LEBEDEV’S TYPE INDEX TRANSFORMS WITH THE SQUARES OF THE ASSOCIATED LEGENDRE FUNCTIONS S.YAKUBOVICH 7 1 0 2 n ABSTRACT. Theclassical Lebedev index transform (1967), involving squares andproducts oftheLegendre a functions is generalized on the associated Legendre functions ofan arbitrary order. Mapping properties are J investigatedintheLebesguespaces. Inversionformulasareproved. Asaninterestingapplication,asolutionto 3 theboundaryvalueproblemforathirdorderpartialdifferentialequationisobtained. 1 1. INTRODUCTION AND PRELIMINARY RESULTS ] A In 1967N.N. Lebedevproved(cf. [8]) that the antiderivativeof an arbitrary function f definedon the C interval(1,¥ )andsatisfyingintegrabilityconditions f L ((1,a);(x 1) 1/2dx), f L ((a,¥ );logxdx) 1 − 1 . forsomea>1canbeexpandedintermsofthefollowin∈grepeatedinte−gralforx (1,∈¥ ) h ∈ mat x f(y)dy=2(x2 1)1/2 ¥ t tanh(pt ) P 1/2+it(x) 2 Z1 − Z0 − [ ¥ (cid:2) (cid:3) (y2 1)1/2P 1/2+it(y) Q 1/2+it(y)+Q 1/2 it(y) f(y)dydt , (1.1) 1 × 1 − − − − − Z v wherePn (z),Qn (z) are Legendrefunctionsof th(cid:2)e firstand secondkind, res(cid:3)pectively,(see [4], Vol. I, [3]). 6 Thisexpansiongeneratesapairofthedirectandinverseindextransforms[2],namely, 2 6 ¥ 3 F(t )= (y2 1)1/2P 1/2+it(y) Q 1/2+it(y)+Q 1/2 it(y) f(y)dy (1.2) .0 forallt >0, Z1 − − (cid:2) − − − (cid:3) 1 0 d ¥ 17 f(x)=2dxZ0 t tanh(pt ) P−1/2+it(x) 2(x2−1)1/2F(t )dt (1.3) : for almost all x>0. The main goal of the prese(cid:2)nt paper is t(cid:3)o extend Lebedev’s transforms (1.2), (1.3), v consideringLebedev-likekernels, whichinvolveproductsoftheassociatedLegendrefunctionsofthefirst Xi and second kind of an arbitrarycomplexorder m , correspondingly,Pnm (z),Qnm (z) [3]. We will study them r mappingproperties,proveinversiontheoremsandapplytosolvetheboundaryvalueproblemfora higher a orderPDE.Precisely,wewillconsiderthefollowingoperatoroftheindextransform 2 (Fm f)(t )=√p G (cid:18)12+it −m (cid:19)G (cid:18)12−it −m (cid:19)Z0¥ "P−m 1/2+it s1+y y!# f(y)dy, t ∈R, (1.4) Date:January16,2017. 2000MathematicsSubjectClassification. 44A15,33C10,33C45,44A05. Keywordsandphrases. IndexTransforms,AssociatedLegendrefunctions,modifiedBesselfunctions,Fouriertransform,Mellin transform,Mehler-Focktransform,Boundaryvalueproblem. 1 2 S.Yakubovich anditsadjointone 2 (Gm g)(x)=√p Z−¥¥ G (cid:18)12+it −m (cid:19)G (cid:18)21−it −m (cid:19)"P−m 1/2+it r1+x x!# g(t )dt , x∈R+, (1.5) whereG (z)isEuler’sgamma-functionm C,iistheimaginaryunitandtheintegrationin(1.5)isrealized ∈ withrespecttothelowerindexoftheassociatedLegendrefunctionofthefirstkind.Wenotethatthekernel m P (x), x>1 corresponds to the well-known generalized Mehler-Fock transform [2]. Denoting the 1/2+it ke−rnelof(1.4),(1.5)by 2 F t (x)=√p G (cid:18)12+it −m (cid:19)G (cid:18)21−it −m (cid:19)"P−m 1/2+it r1+x x!# , (1.6) wewillfindforfurtheruseitsrepresentationintermsofFouriercosinetransform[7]anddeduceanordinary differentialequationwithpolynomialcoefficients,whosesolutionisF t (x),employingtheso-calledmethod of the Mellin-Barnes integrals, which is already being successfully applied by the author for other index transforms. In fact, appealingto relation (8.4.41.47)in [5], Vol. III, we find the followingMellin-Barnes integralrepresentationforthekernel(1.6),namely 1 g+i¥ G (s+1/2+it )G (s+1/2 it )G (1/2+s)G ( m s) F t (x)= 2p i g i¥ G (1+s)G (−1+s m ) − − x−sds, x>0, (1.7) Z − − where g is taken from the interval( 1/2, Rem ). The absoluteconvergenceof the integral(1.7)follows immediatelyfromtheStirlingasymp−toticfo−rmulaforthegamma-function[4],Vol. I,becauseforallt R ∈ G (s+1/2+it )GG ((s1++1s/)G2(−1i+t )sG (1m/)2+s)G (−m −s) =O e−p |s||s|−Rem −3/2 , |s|→¥ . (1.8) − (cid:16) (cid:17) Moreover,itcanbedifferentiatedundertheintegralsignanynumberoftimesduetotheabsoluteanduniform convergencebyx x >0. Wehave 0 Lemma1. Let≥x 1,t R, Rem <1/2.Thenthekernel(1.6)hasthefollowingrepresentationinterms ≥ ∈ ofFouriercosinetransformofthesecondkindLegendrefunction,namely x ¥ F t (x)=2 p cos(t u)Q 1/2 m 2xcosh2(u/2)+1 du. (1.9) r Z0 − − (cid:0) (cid:1) Proof. Infact,appealingtothereciprocalformulaeviatheFouriercosinetransform(cf. formula(1.104)in [2]) ¥ p G (2s+1) 1 G (s+1/2+it )G (s+1/2 it )cos(t y)dt = , Res> , (1.10) 0 − 22s+1cosh2s+1(y/2) −2 Z G (2s+1) ¥ cos(t y) G (s+1/2+it )G (s+1/2 it )= dy, (1.11) − 22s 0 cosh2s+1(y/2) Z wereplacethegamma-productG (s+1/2+it )G (s+1/2 it )intheintegral(1.9)byitsintegralrepresen- − tation (1.11) and change the order of integration via Fubini’s theorem. Then, employing the duplication formulaforthegamma-function[4],Vol. I,wederive F t (x)= 2p √1p i 0¥ cocoshs((tu/u2)) gg+i¥i¥ [G (s+G 1(/12+)]s2G (m−)m −s) xcosh2(u/2) −sdsdu Z Z − − (cid:0) (cid:1) IndextransformswiththesquaresandproductsoftheassociatedLegendrefunctions 3 x ¥ =2 p cos(t u)Q 1/2 m 2xcosh2(u/2)+1 du, r Z0 − − wherethe innerintegralwith respectto s iscalculatedv(cid:0)ia Slater’s theorem(cid:1)asa sumof residuesat simple right-handsidepolesofthegamma-functionG ( m s)andrelation(7.3.1.71)in[5],Vol. III.Lemma1is − − proved. (cid:3) Lemma 2. For eacht R the functionF t (x) given by formula (1.6)is a fundamentalsolutionof the ∈ followingthirdorderdifferentialequationwithpolynomialcoefficients 2x3(1+x)d3F t +3x(1+2x)d2F t +x 2x(1 m 2)+2t 2+1 dF t 1+t 2 F t =0, x>0. (1.12) dx3 dx2 − 2 dx − 4 (cid:18) (cid:19) (cid:18) (cid:19) Proof. Asitwasmentionedabove,theasymptoticbehavior(1.8)oftheintegrandin(1.7)permitsadifferen- tiationundertheintegralsignanynumberoftimes. Henceemployingthereductionformulaforthegamma- function[4],Vol. I,wederive d 2 1 g+i¥ s2G (s+1/2+it )G (s+1/2 it )G (1/2+s)G ( m s) xdx F t = 2p i g i¥ G (1+s)G (1−+s m ) − − x−sds (cid:18) (cid:19) Z − − = 21p i gg+i¥i¥ G (s+3/2+it )GG ((s1++3s/)G2(−1i+t )sG (1m/)2+s)G (−m −s)x−sds− 14+t 2 F t +xddF xt . (1.13) Z − − (cid:18) (cid:19) Meanwhile,withasimplechangeofvariable 1 g+i¥ G (s+3/2+it )G (s+3/2 it )G (1/2+s)G ( m s) 2p i g i¥ G (1+s)G (−1+s m ) − − x−sds Z − − 1 1+g+i¥ G (s+1/2+it )G (s+1/2 it )G (s 1/2)G (1 m s) = 2p iZ1+g−i¥ G (s)G −(s−m ) − − − x1−sds 1 1+g+i¥ s(s m )( m s)G (s+1/2+it )G (s+1/2 it )G (s+1/2)G ( m s) = 2p iZ1+g−i¥ − − − (s−1/2)G (1+s)G (1+−s−m ) − − x1−sds 1 1+g+i¥ (s2 m 2)G (s+1/2+it )G (s+1/2 it )G (s+1/2)G ( m s) =−2p iZ1+g−i¥ − G (1+s)G (1+s−−m ) − − x1−sds 1 1+g+i¥ (s2 m 2)G (s+1/2+it )G (s+1/2 it )G (s+1/2)G ( m s) −4p iZ1+g−i¥ − (s−1/2)G (1+s)G (−1+s−m ) − − x1−sds. Therefore, moving the contour of two latter integrals to the left via Cauchy’s theorem, dividing by √x, differentiatingagainandusing(1.13),weobtain d 2 x d 2F t + 1+t 2 F t xdF t +x x d 2F t xm 2F t dx"√x"(cid:18) dx(cid:19) (cid:18)4 (cid:19) − dx (cid:18) dx(cid:19) − ## 1 d 2 = x F t m 2F t . √x"(cid:18) dx(cid:19) − # Thisisequivalenttothefollowingoperatorequation 4 S.Yakubovich d 3 d 2 3 d 1 2(1+x) x F t 3 x F t + +2(t 2 xm 2) x F t +t 2 F t =0, x>0, dx − dx 2 − dx − 4 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) whichdrivesto(1.12),fulfillingsimpledifferentiation.Lemma2isproved. (cid:3) 2. BOUNDEDNESS AND INVERSION PROPERTIES FOR THEINDEX TRANSFORM(1.4) In this section we will investigate the mapping properties of the index transform (1.4), involving the Mellin transform technique developed in [6]. Precisely, the Mellin transform is defined, for instance, in Ln ,p(R+), 1 p 2(seedetailsin[7])bytheintegral ≤ ≤ ¥ f (s)= f(x)xs 1dx, (2.1) ∗ − Z0 beingconvergentinmeanwithrespecttothenorminL (n i¥ ,n +i¥ ), n R, q=p/(p 1). Moreover, q theParsevalequalityholdsfor f Ln ,p(R+), g L1 n ,q(R−+) ∈ − ∈ ∈ − ¥ 1 n +i¥ f(x)g(x)dx= f (s)g (1 s)ds. (2.2) 0 2p i n i¥ ∗ ∗ − Z Z − TheinverseMellintransformisgivenaccordingly 1 n +i¥ f(x)= f (s)x sds, (2.3) 2p i n i¥ ∗ − Z − wheretheintegralconvergesinmeanwithrespecttothenorminLn ,p(R+) ¥ 1/p f n ,p= f(x) pxn p−1dx . (2.4) || || 0 | | (cid:18)Z (cid:19) In particular, letting n =1/p we get the usual space L (R ). Further, denoting byC (R ) the space of 1 + b + boundedcontinuousfunctions,weestablish Theorem 1. Let Rem <1/2. The index transform (1.4) is well-defined as a bounded operator Fm : L1 n ,1(R+) Cb(R), n ( 1/2, Rem )andthefollowingnorminequalitytakesplace − → ∈ − − ||Fm f||Cb(R)≡tsuRp|(Fm f)(t )|≤Cm ,n ||f||1−n ,1, (2.5) ∈ where 22n 1 1 1 n +i¥ [G (s+1/2)]2G ( m s) Cm ,n = p √−p B n +2, n +2 n i¥ (cid:12) G (1+s m−) − ds(cid:12) (2.6) (cid:18) (cid:19)Z − (cid:12) − (cid:12) afndLB1(an,b,p),i1st<hepEul2erabnedtan-fun(cti1o/n([24q]),,Vomli.n1(.0M, oRreeomv)e)r,,(cid:12)(cid:12)(cid:12)(qFm=f)p(/t()p→10),,|tth|e→n ¥ .Bes(cid:12)(cid:12)(cid:12)ides,if,inaddition, ∈ − ≤ ∈ − − − √p ¥ 1 1 1 dx (Fm f)(t )= cosh(pt ) 0 Kit √x Iit √x +I−it √x j (x) x , (2.7) Z (cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) where 1 n +1/2+i¥ G (1/2 m s)G (s)G (1 s) j (x)= 2p iZn +1/2−i¥ G (s+1/2)G−(1/−2−s)G (1/2−+s−m )f∗(3/2−s)x−sds (2.8) andintegrals(2.7), (2.8)convergeabsolutely. IndextransformswiththesquaresandproductsoftheassociatedLegendrefunctions 5 Proof. Indeed,followingideaselaboratedintheproofofLemma1andcalculatinganelementaryintegral withthehyperbolicfunction(seerelation(2.4.4.4)in[5],Vol. I),wehave ¥ 1 ¥ du (Fm f)(t ) ≤ 0 |F t (y)||f(y)|dy≤ 2p √p 0 [cosh(u/2)]2g+1 Z Z (cid:12) (cid:12) (cid:12)n +i¥ [G ((cid:12)s+1/2)]2G ( m s) ¥ n × n i¥ (cid:12) G (1+s m−) − ds(cid:12) 0 |f(y)|y− dy=Cm ,n ||f||1−n ,1. (2.9) Z − (cid:12) − (cid:12)Z This proves(2.5). Furtherm(cid:12) ore,Fubini’s theoremand(cid:12)definition of the Mellin transform(2.1) allow us to (cid:12) (cid:12) write the composition repr(cid:12)esentation for index transfo(cid:12)rm (1.4) in terms of the Fourier cosine transform, namely, (Fm f)(t )= 2p √1p i 0¥ cocoshs((tu/u2)) nn +i¥i¥ [G (s+G 1(/12+)]s2G (m−)m −s)f∗(1−s) cosh2(u/2) −sdsdu, Z Z − − (cid:0) (cid:1) where Y (u)= nn +i¥i¥ [G (s+G 1(/12+)]s2G (m−)m −s)f∗(1−s) cosh2(u/2) −s−1ds∈L1(R+; du). Z − − Henceittendstozero,when t ¥ viatheRiemann-Lebes(cid:0)guelemma.(cid:1)Ontheotherhand,returningtothe | |→ Mellin-Barnesrepresentation(1.7),appealingtotheParsevalequality(2.2)andmakingasimplesubstitution, wededucetheformula 1 n +1/2+i¥ G (s+it )G (s it )G (s)G (1/2 m s) (Fm f)(t )= 2p iZn +1/2−i¥ G (1/2+−s)G (1/2+s−−m ) − f∗(3/2−s)ds. (2.10) Meanwhile,takingtheMellin-BarnesrepresentationfortheproductofmodifiedBesselfunctionsofthe thirdkind(cf. relation(8.4.23.24)in[5],Vol. III) √p 1 1 1 xcosh(pt )Kit √x Iit √x +I−it √x (cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) 1 g+i¥ G (s 1/2) = 2p i g i¥ G−(s) G (1+it −s)G (1−it −s)x−sds, x>0, Z − where 1/2<g <1 and using again the Parseval equality (2.2), we find that (2.10) becomesthe Lebedev indextransformwith the productofthe modifiedBessel functions[8] givenbyformula(2.7),where j (x) is defined by integral(2.8). Its convergencefor each x>0 is absolute due to the estimate with the aid of Ho¨lder’sinequality Znn++1/12/−2+i¥i¥ (cid:12)G (s+G (11//22)G−(1m/−2−s)Gs)(Gs)(G1/(12−+ss)−m )f∗(3/2−s)x−sds(cid:12)≤x−n −1/2(cid:18)Znn−+i¥i¥ |f∗(1−s)|q|ds|(cid:19)1/q (cid:12) (cid:12) (cid:12)(cid:12) n +i¥ G ( m s)G (s+1/2)G (1/2 s) p (cid:12)(cid:12) 1/p p − − − ds <¥ , q= . ×(cid:18)Zn −i¥ (cid:12) G (s+1)G (−s)G (1+s−m ) (cid:12) | |(cid:19) p−1 The convergence of the latt(cid:12)er integral by s is justified, recal(cid:12)ling the Stirling formula for the asymptotic (cid:12) (cid:12) behaviorofthegamma-func(cid:12)tion,whichgives (cid:12) 6 S.Yakubovich G (−G (ms+−1s))GG ((s+s)1G /(21)+G (s1/2m−)s)=O |s|−2n −1 , |s|→¥ − − andn ischosenfromtheinterval( 1/(2q), min(0, Rem )).M(cid:0)oreover,w(cid:1)efindthatj (x)=O(x n 1/2), x> − − − − 0.Thisasymptoticbehaviorisusedtoestablishtheabsoluteconvergenceoftheintegral(2.7). Infact,em- ployingtheasymptoticformulaeforthemodifiedBesselfunctions[4],Vol. 2forfixedt R ∈ Kit √x Iit √x +I it √x =O(logx), x 0+, − → (cid:0) (cid:1)(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) Kit √x Iit √x +I it √x =O x−1/2 , x ¥ , − → wederiveviaelementarysubs(cid:0)tituti(cid:1)o(cid:2)n (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) (cid:16) (cid:17) ¥ 1 1 1 dx 1 Kit Iit +I it j (x) = O xn −1/2logx dx Z0 (cid:12)(cid:12) (cid:18)√x(cid:19)¥ (cid:20) (cid:18)√x(cid:19) − (cid:18)√x(cid:19)(cid:21) (cid:12)(cid:12) x Z0 (cid:16) (cid:17) (cid:12)(cid:12) + O xn −1 dx<¥ , n ( 1/(2q(cid:12)(cid:12)), min(0, Rem )). 1 ∈ − − Z Theorem1isproved. (cid:0) (cid:1) (cid:3) Writing(2.7)intheform 2√p ¥ dx (Fm f)(t )= cosh(pt ) 0 Kit (x)[Iit (x)+I−it (x)]j (1/x2) x , (2.11) Z wewillappealtotheLebedevexpansiontheoremin[8],whichimpliesthefollowingrepresentationofthe antiderivative ¥ 1 dy 1 ¥ x j y2 y = p 2√p 0 t sinh(2pt )Ki2t (x)(Fm f)(t )dt , x>0, (2.12) Z (cid:18) (cid:19) Z which holds under condition j (1/x2)x 1 L (0,1); x 1/2dx L (1,¥ ); x1/2dx . By straightforward − 1 − 1 substitutionswe see thatthisconditionis∈equivalentto (cf. (2.4)∩)j L (1,¥ ) L (0,1). On the (cid:0) (cid:1) ∈(cid:0) 1/4,1 ∩(cid:1) −1/4,1 otherhand,lettingin(2.12)1/√xinsteadofxandchangingthevariableinitsleft-handside,weobtain x dy 2 ¥ 1 0 j (y) y = p 2√p 0 t sinh(2pt )Ki2t √x (Fm f)(t )dt . Z Z (cid:18) (cid:19) Then,appealingtorelation(8.4.23.28)in[5],Vol. IIIanddifferentiatingtwosidesofthelatterequalitywith respecttox,wefind x d ¥ g+i¥ G ( s)G (it s)G ( it s) j (x)= 2p 3idx 0 t sinh(2pt )(Fm f)(t ) g i¥ − G (1−/2 s)− − x−sdsdt , g <0. (2.13) Z Z − − Ourgoalnowistomotivatethedifferentiationunderintegralsignintheright-handsideof(2.13).Todothis weuse(1.11)togettheuniformestimateoftheproductofgamma-functions ¥ y G (it s)G ( it s) 2Res+1 G ( 2s) cosh2Res dy, Res<0. | − − − |≤ | − | 0 2 Z Consequently,assuming the integrabilityconditionFm L1(R; t e2p |t|(cid:16)dt )(cid:17), the repeatedintegralin (2.13) ∈ | | canbeestimatedasfollows IndextransformswiththesquaresandproductsoftheassociatedLegendrefunctions 7 ¥ g+i¥ G ( s)G (it s)G ( it s) 0 t sinh(2pt ) (Fm f)(t ) g i¥ − G (1−/2 s)− − x−sds dt Z Z − (cid:12) − (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ≤Cx−g||F(cid:12)m ||L1(R+;|t(cid:12)|e2pt dt)Z(cid:12)(cid:12)g−g+i¥i¥ (cid:12)G (G −(12/s2)G+(−s)s)ds(cid:12)<¥ , (cid:12)(cid:12) whereC>0 is an absolute constant. A similar estimate h(cid:12)olds for the repe(cid:12)ated integral for the derivative (cid:12) (cid:12) withrespecttox,andthedifferentiationundertheintegral(cid:12)signoftheinneri(cid:12)ntegralbysispossibleviathe Stirlingasymptoticformulaforthegamma-function. Thusthedifferentiationin(2.13)ispermittedaswell asthechangeoftheorderofintegration,owingtotheabsoluteanduniformconvergencebyx x >0,and 0 ≥ withtheuseofthereductionformulaforthegamma-functionweobtain 1 g+i¥ G (1 s) ¥ j (x)= 2p 3i g i¥ G (1/2− s)x−s 0 t sinh(2pt )(Fm f)(t )G (it −s)G (−it −s)dt ds. (2.14) Z − − Z Returningtointegralrepresentation(2.8)forj (x), wefindthatifthefunction f (3/2 s)/sisanalyticin ∗ theopenstripg <Res<n +1/2and(1+ s)1 2Resf (3/2 s)/sisabsolutelyintegrab−leoveranyvertical − ∗ | | − lineoftheclosureofthisstrip,thenviaCauchy’stheoremwecanmovethecontourtotheleft,integrating in(2.8)over(g i¥ ,g +i¥ ), g <0. ThereforeunderconditionsofTheorem1andviatheaboveestimates − weobservethatbothsidesof(2.14)areinverseMellintransformsofabsolutelyintegrablefunctions.Hence theuniquenesstheoremcanbeappliedfortheMellintransforminL [7],andwederive 1 1 G (s+1/2)G (1/2+s m ) f∗(3/2−s)= p 2 G (1/2 m s)G (s−) − − ¥ 1 1 t sinh(2pt )(Fm f)(t )G (it s)G ( it s)dt , max , Rem <g <0. (2.15) × 0 − − − −2 −2 Z (cid:18) (cid:19) Afterthesimplechangeofvariablethelatterequalitybecomes 1 G (2 s)G (2 s m ) ¥ f∗(s)= p 2G (s −1 m )G −(3/−2 s) 0 t sinh(2pt )(Fm f)(t )G (s−3/2+it )G (s−3/2−it )dt , − − − Z whereRes=g0=3/2 g .Hence,undercondition f L1/2 g,1(R+)wetaketheinverseMellintransform ofbothsidesofthelatte−requalityandchangetheorde∈rofint−egrationintheright-handsideoftheobtained equality. Thuswededuce,finally,theinversionformulafortheindextransform(1.4),whichcanbewritten (withtheuseofthereductionformulaforthegamma-functionanddifferentiationundertheintegralsign) intheform x 1/2 d ¥ f(x)= −p 2 dx x3/2 t sinh(2pt )Sm (x,t )(Fm f)(t )dt , x>0, (2.16) 0 Z where Sm (x,t )= 21p iZg0g−0+i¥i¥ G G(s(2−−1s−)Gm()2G −(5s/−2−m )s)G (s−3/2+it )G (s−3/2−it )x−sds, x>0. (2.17) Thiskernelcanbecalculatedintermsofthederivativeoftheproductofthefirstandsecondkindassociated Legendrefunctions. Wewilldoitwiththeaidofrelation(8.4.42.34)in[5],Vol. III.Infact,employingthe reflectionformulaforthegamma-function[4],Vol. 1,weobtain 8 S.Yakubovich 2√p e−imp G (1/2−m +it )Pm 1+x Qm 1+x +Qm 1+x x2 G (1/2+m +it ) 1/2+it x 1/2+it x 1/2 it x − r !" − r ! − − r !# = 21p iZg0g−0+i¥i¥ G (2−s)G G(2(s−−mm−−s)1G)(s−3/2)(cid:20)GG ((5s/−23+/2it+−its))+GG ((5s/−23−/2it−−its))(cid:21)x−sds = coshp(ipt )Zg0g−0+i¥i¥ G (2−s)G (2−Gm(s−−s)mG −(s1−)G3/(52/+2−it )sG)(s−3/2−it )x−sds, x>0. Henceowingtotheidentity[4],Vol. 1 e−imp GG ((11//22−+mm ++iitt ))Qm 1/2+it 1+x x =eimp Q−1m/2+it 1+x x , − r ! − r ! weendupwiththefollowingvalueofthekernel(2.17) Sm (x,t )= x2√cposehi(mppt )Pm 1/2+it 1+x x − r ! m 1+x m 1+x ×"Q−−1/2+it r x !+Q−−1/2−it r x !#, x>0. Wesummarizetheresultsofthissectionasthefollowingintegraltheorem. Theorem2. LetconditionsofTheorem1holdandj L 1/4,1(0,1) L1/4,1(1,¥ ), Fm L1(R; t e2p t dt ).Letbesides,theMellintransform f (3/2 s∈)be−suchthat f∩(3/2 s)/sbean∈alyticin|th|eopen | | ∗ v×erticalstripg <Res<n +1/2forsomeg <0, f (3/−2 s)/s L ((Res∗ i¥ ,R−es+i¥ );(1+ s)1 2Res ds) 1 − overanyverticallineoftheclosureofthisstripa∗nd f −L1/2 ∈g,1(R+). T−henforx>0thean|ti|derivativ|eo|f thefunctionx1/2f(x)canbeexpandedintermofthere∈peated−integral ¥ y1/2f(y)dy= 2eimp ¥ t sinh(pt )G 1+it m G 1 it m Pm 1+x Zx − p √x Z0 (cid:18)2 − (cid:19) (cid:18)2− − (cid:19) −1/2+it r x ! 2 ×"Q−−1m/2+it r1+x x!+Q−−1m/2−it r1+x x!#Z0¥ "P−m 1/2+it s1+y y!# f(y)dydt , (2.18) generatingthepair(1.4),(2.16)ofthedirectandinverseindextransforms,respectively. Remark1. TheclassicalLebedevexpansion(1.1)foradjointkernelscanbeobtained,lettingm =0in (2.18).Then,substituting (1+x)/x, (1+y)/ybynewvariables,wederive x p p¥ h(y)dy=2(x2 1)1/2 t tanh(pt )P 1/2+it(x) Q 1/2+it(x)+Q 1/2 it(x) 1 − 0 − − − − Z Z ¥ (cid:2) (cid:3) (y2 1)1/2 P 1/2+it(y) 2h(y)dydt , × 1 − − Z whereh(y)=2yf (y2 1) 1 (y2 1) 5/2. (cid:2) (cid:3) − − − − Remark2. BythesametechniqueonecanestablishthegeneralizedLebedevexpansion(1.1)intheform (cid:0) (cid:1) IndextransformswiththesquaresandproductsoftheassociatedLegendrefunctions 9 ¥ y1/2f(y)dy= 2eimp ¥ t sinh(pt )G 1+it m G 1 it m Pm 1+x 2 Zx − p √x Z0 (cid:18)2 − (cid:19) (cid:18)2− − (cid:19)" −1/2+it r x !# ×Z0¥ P−m 1/2+it s1+y y!"Q−−1m/2+it s1+y y!+Q−−1m/2−it s1+y y!#f(y)dydt , x>0, leadingtothereciprocalformulas(1.2),(1.3)whenm =0. Weleavedetailstothereader. 3. INDEX TRANSFORM(1.5) Theboundednessofthe adjointoperator(1.5)andinversionformulaforthistransformationwill bees- tablishedbelow.Webeginwith Theorem 3. Let Rem <1/2. The index transform (1.5) is well-defined as a bounded operator Gm : L1(R) Ln ,¥ (R+), n ( 1/2, Rem )andthefollowingnorminequalitytakesplace → ∈ − − n ||Gm g||n ,¥ ≡esssupx>0|x (Gm g)(x)|≤Cm ,n ||g||L1(R), (3.1) whereCm ,n isdefinedby(2.6). Moreover,if(Gm g)(x) Ln ,1(R+),thenforally>0 ∈ 1 n +i¥ G (1+s)G (1+s m )G ( s) ¥ y g(t ) 2p i n i¥ G (1/2+s)G (−m s)− (G∗m g)(s)y−sds =√p y ¥ ey/2 Kit 2 cosh(pt )dt . (3.2) Z − − − Z− (cid:16) (cid:17) Proof. The norm inequality (3.1) is a direct consequence of the estimates (2.9). Indeed, since the kernel (1.6)hasabound F t (x) Cm ,n x−n , x>0, | |≤ whereCm ,n isdefinedby(2.6),wehave |(Gm g)(x)|≤Cm ,n x−n R|g(t )|dt =Cm ,n x−n ||g||L1(R), Z and (3.1) follows. Now, taking the Mellin transform (2.1) of both sides of (1.5), we change the order of integrationbyFubini’stheoremandtakeintoaccount(1.7)toobtaintheequality G (1+s)G (1+s m ) ¥ 1 1 G (1/2+s)G ( m− s)(G∗m g)(s)= ¥ G s+2+it G s+2−it g(t )dt , (3.3) − − Z− (cid:18) (cid:19) (cid:18) (cid:19) where 1/2<Res< Rem . Hence theinverseMellin transform(2.3)andrelation(8.4.23.5)in [5], Vol. IIIwill−leadusto(3.2)−,completingtheproofofTheorem3. (cid:3) Theinversionformulafortheindextransform(1.5)isgivenby Theorem4.LetRem <1/2,g(z/i)beanevenanalyticfunctioninthestripD= z C: Rez <a <1/2 , { ∈ | | } suchthatg(0)=g′(0)=0 and g(z/i) be absolutely integrableover any vertical line in D. If (Gm g)(y) L1 g,1(R+), 3/2<g <min(2, 2 Rem ), then for all x R the inversion formula holds for the inde∈x tra−nsform(1.5) − ∈ g(x)= eimp xsinh(p x) ¥ y 1/2 d y 1/2 Pm 1+y p √p Z0 (cid:18) − dy − (cid:19)" −1/2+ix s y ! 10 S.Yakubovich m 1+y m 1+y Q− +Q− (Gm g)(y)dy. (3.4) ×" −1/2+ix s y ! −1/2−ix s y !## Proof. Indeed,recalling(3.2),wemultiplyitsbothsidesbye y/2K (y/2)ye 3/2forsomepositivee (0,1) − ix − andintegratewithrespecttoyover(0,¥ ). Hencechangingtheorderofintegrationintheleft-hand∈sideof theobtainedequalityduetotheabsoluteconvergenceoftheiteratedintegral,weappealtorelation(8.4.23.3) in[5],Vol. IIItofind 1 n +i¥ G (1+s)G (1+s m )G (e s 1/2+ix)G (e s 1/2 ix)G ( s) 2p i n i¥ −G (1/2+−s)−G (e s)G ( m −s)− − − (Gm g)∗(s)ds Z − − − − = 0¥ Kix 2y ye−1 ¥¥ Kit 2y cosgh((tpt) )dt dy. (3.5) Z (cid:16) (cid:17) Z− (cid:16) (cid:17) Inthemeantime,theright-handsideof(3.5)canbetreated,usingtheevennessofgandtherepresentation ofthemodifiedBesselfunctionK (y)intermsofthemodifiedBesselfunctionofthefirstkindI (y)[4],Vol. z z II.Hencewithasimplesubstitutionwefind 0¥ Kix 2y ye−1 ¥¥ Kit 2y cosgh((tpt) )dt dy Z (cid:16) (cid:17) Z− (cid:16) (cid:17) =2p i ¥ K y ye 1 i¥ I y g(z/i) dzdy. (3.6) 0 ix 2 − i¥ z 2 sin(2p z) Z (cid:16) (cid:17) Z− (cid:16) (cid:17) Ontheotherhand,accordingtoourassumptiong(z/i)isanalyticintheverticalstrip0 Rez<a <1/2, ≤ g(0)=g(0)=0andintegrableintheclosureofthestrip.Hence,appealingtotheinequalityforthemodified ′ Besselfunctionofthefirstkind(see[6],p. 93) Iz(y) IRez(y)ep |Imz|/2, 0<Rez<a , | |≤ onecanmovethecontourtotherightinthelatterintegralin(3.6).Then 2p i ¥ K y ye 1 i¥ I y g(z/i) dzdy 0 ix 2 − i¥ z 2 sin(2p z) Z (cid:16) (cid:17) Z− (cid:16) (cid:17) =2p i ¥ K y ye 1 a +i¥ I y g(z/i) dzdy. 0 ix 2 − a i¥ z 2 sin(2p z) NowRez>0, andit ispossibletZo passt(cid:16)ot(cid:17)he limiZtu−ndert(cid:16)he i(cid:17)ntegralsignwhene 0 andtochangethe → orderofintegrationdueto the absoluteanduniformconvergence. Thereforethe valueof the integral(see relation(2.16.28.3)in[5],Vol. II) ¥ dy 1 K (y)I (y) = 0 ix z y x2+z2 Z leadsustotheequalities lim2p i ¥ K y ye 1 i¥ I y g(z/i) dzdy e 0 0 ix 2 − i¥ z 2 sin(2p z) → Z (cid:16) (cid:17) Z− (cid:16) (cid:17) a +i¥ g(z/i) a i¥ a +i¥ g(z/i)dz =2p i dz=p i − − + . (3.7) Za −i¥ (x2+z2)sin(2p z) (cid:18)Z−a +i¥ Za −i¥ (cid:19)(z−ix)zsin(2p z)