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The Generalized Discrete Legendre Transformation

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The Generalized Discrete Legendre Transformation JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 218, 183]206Ž. 1998 ARTICLE NO. AY975759 The Generalized Discrete Legendre Transformation J. M. R. Mendez-Perez* and G. Miquel Morales Departamento de Analisis Matematico, Uni¨ersidad de La Laguna, 38271 La Laguna() Tenerife , Canary Islands, Spain View metadata, citation and similar papersSubmitted at core.ac.uk by Richard A. Duke brought to you by CORE provided by Elsevier - Publisher Connector Received March 10, 1997 Ž.Ž.Ž. ` ŽŽ . Ž . Ž . The discrete transformation LF x s fxsÝns0 2nq1r2PxFnn , where the PxnŽ.'s denote the well-known Legendre polynomials, is an isomorphism between the space LŽ.N0 of the complex-valued sequences of rapid descent and the space L Ž.y1, 1 of all infinitely differentiable complex-valued functions defined on y1 - x - 1, of slow growth in the end points y1 and 1. This result is extended to the corresponding dual spaces. A space of multipliers for these spaces is considered and the translation operator and the convolution are investigated on them. The operational calculus generated involves certain finite-difference opera- tors and is applied to find the solution of some ordinary and partial finite- difference equations. Q 1998 Academic Press 1. INTRODUCTION The finite Legendre transformation 1 ŽLf .Ž. n s Fn Ž.sHPxfxdxn Ž.Ž. ,1.1Ž. y1 whose inversion formula is ` 2n 1 y1 q ŽLF .Ž. xsfx Ž.s PxFnnŽ. Ž.,1.2 Ž . Ý 2 ns0 the PxnŽ.'s being the well-known Legendre polynomials, has been studied by different authors, among others, by R. V. Churchillwx 2 , I. N. Sneddon wx17 , and C. J. Tranter wx 19 . * The work of the first author was partially supported by DGICYT Grant PB 94-0591 Ž.Spain . 183 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved. 184 MENDEZÂÂ-PEREZ AND MORALES An interchange of the roles betweenŽ. 1.1 and Ž. 1.2 gives rise to the discrete Legendre transformation ` 2n q 1 ŽlF .Ž. x s fx Ž.s PxFnnŽ. Ž. Ž1.3 . Ý 2 ns0 and its inversion formula 1 y1 Ž.lfnŽ.sFn Ž.sHPxfxdxn Ž.Ž. .1.4Ž. y1 The transformŽ. 1.1 is profusely investigated inwx 5, 20 in a more general context. However, the study of the discrete transformŽ. 1.3 is scarcely found in the literature. Inwx 13 we introduce the space lŽ.N0 of the complex valued sequences ÄFŽ.n 4 , N denoting the set of nonnegative integer ng N 0 0 numbers, such that ya n U k la,knŽ.F s sup < e LFŽ.n< -`, ngN0 where a is a positive real number, for every k s 0, 1, 2, . Here L* stands for the finite-difference operator n 1 n U q L*FŽ.n s LFn Ž.n s FŽ.nq1 q FŽ.Ž.ny1, 1.5 2nq32ny1 and its adjoint is supplied by Ž.Ž.Ž.n q 1 F n q 1 q nF n y 1 LFŽ.n s LFn Ž.n s .1.6Ž. 2nq1 Henceforth FŽ.n must be taken as zero if n is negative. Endowed with the topology generated by the collection of seminorms Ä4l ,lŽ.N turns out to be a Frechet space. Its dual space is a, kkgN0 0  represented by by l9Ž.N0 . Since, considered as a function of the discrete variable n,2ŽŽ nq 1..Ž.Ž.r2Pxn glN0 for any x, y1 F x F 1, the generalized discrete Le- gendre transformation was defined inwx 13 as 2n q 1 Žl9Fx .Ž.sfx Ž.s Fn Ž., Pxn Ž. , Fgl9Ž.Ž.N0 ,1.7 ¦¦2 ;; its inversion theorem and main properties being investigated. DISCRETE LEGENDRE TRANSFORMATION 185 On the other hand, inwx 12 we consider the space L Ž.y1, 1 of all complex-valued infinitely differentiable functions defined on the interval Ž.y1, 1 such that k gwkŽ.s sup R w Ž.x - `, k s 0,1,2,..., y1-x-1 where d 222 RsRxsDxŽ.Ž.y1Ds xy1Dq2xD, D s .1.8Ž. dx We equip L Ž.y1, 1 with the topology generated by the countable multinorm Ä4g . Thus, L Ž.1, 1 is a Frechet space. L 9Ž.1, 1 de- kkgN0 y  y notes its dual space. Now note that, as a function of x, PxnŽ.gL Žy1, 1 . The objective of this paper is to analyse the discrete Legendre transfor- mation in a space of sequences of rapid descent to which, considered as a function of n, the kernelŽŽ 2n q 1 .r2 .Pxn Ž . does not belong. This fact makes it impossible to define the generalized transformation by means of Ž.1.7 and it will be necessary to do it through a quite different method that will be based on the Parseval relation ` 2n q 1 1 Ý Žlf .Ž.Ž.Ž. n lw n s Hfx Ž.Ž.wxdx,1.9 Ž . 2 1 ns0 y in a similar way as the Fourier transform is extended to a space of tempered distributionswx 16 . In order to attain this goal, we first establish that the discrete Legendre transformation is an isomorphism of the space LŽ.y1, 1 into certain space L Ž.N0 of sequences of rapid descent and later the generalized transformation is defined between their duals by adjoint- ness. Next, inspired by the works of L. Schwartzwx 16 about the Fourier transform and J. J. Betancor and I. Marrerowx 1, 11 with regard to the Hankel transformation, we introduce the discrete space OŽ.N0 of the multipliers of the space LŽ.N00and its dual L9Ž.N and investigate the properties of the translation operator and the convolution on these spaces. Finally, the results obtained are applied in solving a certain kind of ordinary and partial difference equations. Along this work we shall frequently use the following formulas and properties of the Legendre polynomialswx 2, 3, 10, 18 : Žn q 1 .Pxnq1 Ž.qnPny1 Ž. x s Ž2n q 1 .xPn Ž. x Ž1.10 . PxnŽ.F1, 1 F x F 1, n s 0,1,2,...Ž. 1.11 n PnnŽ.1s1, P Ž.Ž.y1 sy1Ž. 1.12 RPxxnÄ4Ž.snn Žq1 .Pxn Ž. Ž.1.13 k 2k DPnŽ. x Fn , y1FxF1, k s 0,1,2,... .Ž. 1.14 186 MENDEZÂÂ-PEREZ AND MORALES Recall, in the end, that the translation operator of the finite calculusw 6, 9x is defined by EFŽ.n s F Žn q 1 . ,Ž. 1.15 whereas the finite-difference operator is DFŽ.n s F Žn q 1 .y F Ž.n .Ž. 1.16 In view ofŽ. 1.6 and Ž 1.10 . we obtain L PxnnŽ.sxP Ž. x , Ž.1.17 which will play the same role in our finite calculus as that of the differential relationŽ. 1.13 inwx 12 . As for the notation, the symbol²² , :: will be used to represent a functional acting on the spaces lŽ.N00and L Ž.N and ²: , to denote a 1 functional operating on L Ž.y1, 1 . As usual, l Ž.N0 denotes the class of those complex sequences FŽ.n for which ` 2n q 1 55F1s FŽ.n Ý 2 ns0 ` is finite, while l Ž.N0 consists of all complex sequences FŽ.n such that 55F`ssup FŽ.n - `. ngN0 Remark 1.1. The Legendre equationw 3,Ž. II , p. 79x 2 Ž.1 y x y0 y 2 xy9 q nn Ž.q1ys0 2 reduces to the same equationŽ 1 y xy.0y2xy9 s 0 for the values n s 0 Ž. Ž. and n sy1. This justifies our assumption Pxy10sPxby convention. Ž. Ž. N This argument permits us to put in general Pxyny1 sPxn ,ng 0. 2. THE SPACE LŽ.N0 OF THE SEQUENCES OF RAPID DESCENT AND ITS DUAL We define LŽ.N0 as the linear space of all complex-valued sequences ÄFŽ.n4 such that, for any pair of nonnegative integers r and s, one has ng N 0 rs QFr,sŽ.ssup n LFŽ.n -`,2.1Ž. ngN0 DISCRETE LEGENDRE TRANSFORMATION 187 where L is the finite-difference operatorŽ. 1.6 . Note that each Qr, s is a seminorm on LŽ.N0 and, in particular, the Qr,0 are norms. Then, the topology of LŽ.N is that generated by Ä4Q . Thus, LŽ.N turns out 0 r, sr,sgN0 0 to be a Frechet space. In the following, L9Ž.N00represents the dual space of LŽ.N and we assign it the weak topology generated by the collection of seminorms Ä4j , which are defined by FFgLŽN0. j FŽF .s ²² Fn Ž.,F Ž.n:: , FgL9Ž.N0 . The members of LŽ.N0 can be characterized easily. PROPOSITION 2.1. The complex sequence ÄFŽ.n4 is a member of ng N 0 LŽ.N0 if and only if FŽ.n is of rapid descent. Proof. Assume that FŽ.n is of rapid descent, that is to say, 1 FŽ.n s o ,asn`,2.2Ž. ž/nt ª for any t g N0. On the other hand, by a direct computation we get s s LFŽ.n s Ýri Ž.n F Žnqi .,2Ž..3 isys where the riŽ.n stand for certain rational functions, whose denominators rs do not vanish in N0. The resultsŽ. 2.2 and Ž. 2.3 prove that n LFŽ.n ª0as rs nª`. This implies that <n LFŽ.n<is bounded for any n g N0 and, therefore, F g LŽ.N0 . Conversely, if F g LŽ.N0 we deduce immediately fromŽ. 2.1 with s s 0 that F is of rapid descent.
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