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. ϱ 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.ގ0 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. ᮊ 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 ϱ 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 ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. 184 MENDEZ´´-PEREZ AND MORALES
An interchange of the roles betweenŽ. 1.1 and Ž. 1.2 gives rise to the discrete Legendre transformation
ϱ 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.ގ0 of the complex valued sequences Ä⌽Ž.n 4 , ގ denoting the set of nonnegative integer ng ގ 0 0 numbers, such that
y␣ n U k ␣,knŽ.⌽ s sup < e ⌳⌽Ž.n< -ϱ, ngގ0 where ␣ is a positive real number, for every k s 0, 1, 2, . . . . Here ⌳* stands for the finite-difference operator
n 1 n U q ⌳*⌽Ž.n s ⌳⌽n Ž.n s ⌽Ž.nq1 q ⌽Ž.Ž.ny1, 1.5 2nq32ny1 and its adjoint is supplied by
Ž.Ž.Ž.n q 1 ⌽ n q 1 q n⌽ n y 1 ⌳⌽Ž.n s ⌳⌽n Ž.n s .1.6Ž. 2nq1 Henceforth ⌽Ž.n must be taken as zero if n is negative. Endowed with the topology generated by the collection of seminorms Ä4 ,lŽ.ގ turns out to be a Frechet space. Its dual space is ␣, kkgގ0 0 ´ represented by by lЈŽ.ގ0 . Since, considered as a function of the discrete variable n,2ŽŽ nq 1..Ž.Ž.r2Pxn glގ0 for any x, y1 F x F 1, the generalized discrete Le- gendre transformation was defined inwx 13 as
2n q 1 ŽlЈFx .Ž.sfx Ž.s Fn Ž., Pxn Ž. , FglЈŽ.Ž.ގ0 ,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 ␥kŽ.s sup R Ž.x - ϱ, 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 Ä4␥ . Thus, L Ž.1, 1 is a Frechet space. L ЈŽ.1, 1 de- kkgގ0 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 ϱ 2n q 1 1 Ý Žlf .Ž.Ž.Ž. n l n s Hfx Ž.Ž.xdx,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.ގ0 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.ގ0 of the multipliers of the space LŽ.ގ00and its dual LЈŽ.ގ 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 MENDEZ´´-PEREZ AND MORALES
Recall, in the end, that the translation operator of the finite calculusw 6, 9x is defined by
E⌽Ž.n s ⌽ Žn q 1 . ,Ž. 1.15 whereas the finite-difference operator is
⌬⌽Ž.n s ⌽ Žn q 1 .y ⌽ Ž.n .Ž. 1.16 In view ofŽ. 1.6 and Ž 1.10 . we obtain
⌳ 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.ގ00and L Ž.ގ and ²: , to denote a 1 functional operating on L Ž.y1, 1 . As usual, l Ž.ގ0 denotes the class of those complex sequences ⌽Ž.n for which
ϱ 2n q 1 55⌽1s ⌽Ž.n Ý 2 ns0
ϱ is finite, while l Ž.ގ0 consists of all complex sequences ⌽Ž.n such that
55⌽ϱssup ⌽Ž.n - ϱ. ngގ0
Remark 1.1. The Legendre equationw 3,Ž. II , p. 79x
2 Ž.1 y x yЉ y 2 xyЈ q nn Ž.q1ys0
2 reduces to the same equationŽ 1 y xy.Љy2xyЈ s 0 for the values n s 0 Ž. Ž. and n sy1. This justifies our assumption Pxy10sPxby convention. Ž. Ž. ގ This argument permits us to put in general Pxyny1 sPxn ,ng 0.
2. THE SPACE LŽ.ގ0 OF THE SEQUENCES OF RAPID DESCENT AND ITS DUAL
We define LŽ.ގ0 as the linear space of all complex-valued sequences Ä⌽Ž.n4 such that, for any pair of nonnegative integers r and s, one has ng ގ 0
rs ⌰⌽r,sŽ.ssup n ⌳⌽Ž.n -ϱ,2.1Ž. ngގ0 DISCRETE LEGENDRE TRANSFORMATION 187
where ⌳ is the finite-difference operatorŽ. 1.6 . Note that each ⌰r, s is a seminorm on LŽ.ގ0 and, in particular, the ⌰r,0 are norms. Then, the topology of LŽ.ގ is that generated by Ä4⌰ . Thus, LŽ.ގ turns out 0 r, sr,sgގ0 0 to be a Frechet´ space. In the following, LЈŽ.ގ00represents the dual space of LŽ.ގ and we assign it the weak topology generated by the collection of seminorms Ä4 , which are defined by ⌽⌽gLŽގ0.
⌽ŽF .s ²² Fn Ž.,⌽ Ž.n:: , FgLЈŽ.ގ0 .
The members of LŽ.ގ0 can be characterized easily.
PROPOSITION 2.1. The complex sequence Ä⌽Ž.n4 is a member of ng ގ 0 LŽ.ގ0 if and only if ⌽Ž.n is of rapid descent. Proof. Assume that ⌽Ž.n is of rapid descent, that is to say, 1 ⌽Ž.n s o ,asnϱ,2.2Ž. ž/nt ª for any t g ގ0. On the other hand, by a direct computation we get s s ⌳⌽Ž.n s Ýi Ž.n ⌽ Žnqi .,2Ž..3 isys where the iŽ.n stand for certain rational functions, whose denominators rs do not vanish in ގ0. The resultsŽ. 2.2 and Ž. 2.3 prove that n ⌳⌽Ž.n ª0as rs nªϱ. This implies that Some properties of the spaces LŽ.ގ00and LЈ Ž.ގ are remarked below. Ž.i Every complex-valued sequence ÄFnŽ.4 such that ng ގ 0 ϱ ÝFnŽ.-ϱ Ž.2.4 ns0 gives rise to a regular member in LЈŽ.ގ0 through ϱ ²²F, ⌽ :: s ÝFnŽ.⌽ Ž.n, ⌽gL Žގ0 ..2.5 Ž . ns0 Indeed, the linearity being obvious, its continuity is inferred from ϱ ²²F, ⌽ :: F ⌰⌽0,0Ž.ÝFn Ž.sC⌰⌽0,0 Ž.. ns0 188 MENDEZ´´-PEREZ AND MORALES In particular, each member of LŽ.ގ0 fulfils the requirement Ž. 2.4 , since ϱϱ1 ÝÝ⌽Ž.nFÄ4⌰⌽2,0 Ž.q2⌰⌽1,0 Ž.q⌰⌽0,0 Ž. 2 -ϱ. ns0 ns0Ž.nq1 Furthermore, two members of LŽ.ގ0 that generate the same regular member in LЈŽ.ގ0 must necessarily be identical. Thus, we can identify LŽ.ގ000with a subspace of LЈŽ.ގ and the inclusion LŽ.ގ ; LЈ Ž.ގ0is justified. Ž.ii The classical finite-difference operator ⌳ given byŽ. 1.6 is a continuous linear mapping of LŽ.ގ0 into itself. In effect, ⌰⌳⌽ ⌰⌽ ⌽ގ r,srŽ.s,sq10 Ž., gL Ž.. Hence, the generalized finite-difference operator ⌳ defined by means of ²²⌳ F, ⌽ ::s ²F, ⌳⌽ :: Ž.2.6 is also a continuous linear mapping of LЈŽ.ގ0 into itself. Ž.iii L Ž.ގ00is a subspace lŽ.ގ and the topology of LŽ.ގ0is stronger than that induced on it by lŽ.ގ0 . Notice firstly thatŽ. 1.5 can be written ⌳*⌽Ž.n s ␣1,1 Ž.n ⌽ Žn q 1 .q ␣1,0 Ž.nn⌽ Žny1, . where ␣1, 1Ž.n s Žn q 1 .Žr 2n q 3 . and ␣1, 0 Ž.n s 1r Ž2n y 1 . . An induc- tive procedure allows us to obtain s ⌳ 2 s⌽ ␣ ⌽ * Ž.n s Ý2s,2syjŽ.n Ž.n q 2 Žs y j . js0 s ␣ ⌽ qÝ2s,syjŽ.Žnnny1... . Žny2jq1 . Žny2j ., js1 ss0,1,2,...,Ž. 2.7 and 2 s 1 ⌳* y ⌽Ž.n s s ␣ ⌽ Ý2sy1,2syjŽ.n Ž.nq2 Žsyj .q1 js1 s ␣ ⌽ qÝ2sy1, syjŽ.Ž.Ž.Ž.nnny1...ny2jq2 ny2jq1, js1 ss1,2,...,Ž. 2.8 DISCRETE LEGENDRE TRANSFORMATION 189 where the ␣i, jŽ.n ’s denote rational functions whose denominators never are equal to zero in ގ00. Now, for an arbitrary ⌽ g LŽ.ގ , by virtue of Ž. 2.7 andŽ. 2.8 we get y␣ nk ⌳⌽␣,kŽ.ssup e ⌳* ⌽Ž.n F C␣,k⌰⌽0,0 Ž.,2.9 Ž . ngގ0 y␣n since <␣i, jiŽ.ne< FC,jŽ.␣ for any n g ގ0, C␣,kiand C ,jŽ.␣ being certain positive constants. By invokingwx 20, p. 12 andŽ. 2.9 we conclude validity of our assertion. Therefore, the restriction of F g lЈŽ.ގ00to L Ž.ގ is a member of LЈ Ž.ގ0 and the inclusion lЈŽ.ގ00; LЈ Ž.ގ must be understood in this sense. Ž.iv Observe thatŽŽ 2n q 1 .r2 .Pxn Ž., considered as a function of the discrete variable n, does not belong to LŽ.ގ0 , for, 2n 1 r q ⌰r,0Ä4PnnŽ.1 s sup nPŽ.1sϱ,rs0,1,2,..., 2 ngގ0 in view ofŽ. 1.12 . On the contrary,ŽŽ 2n q 1 .r2 .Pxn Ž .is a member of l Žގ0 . wx13 . By this reason the previous inclusion inŽ. iii is strict. PROPOSITION 2.2. The operation ⌽Ž.n ª ŽŽ2n q 1 .r2 .⌽ Ž.n is an iso- morphism of the space LŽ.ގ0 into itself. Proof. By proceeding in a similar way as in the proof of formulasŽ. 2.7 andŽ. 2.8 , we can verify in relation to the operator ⌳, 2n q 1 ⌳⌽s Ž.n 2 wxsr2q ⌽ sÝAn2syjŽ.Ž.nq2 Žsqj .q js wxsr2q ⌽ qÝAnnnsyjŽ.Žy1... . Žny2jq1q .Ž.ny2jq js1 Ž.2.10 190 MENDEZ´´-PEREZ AND MORALES and, analogously, 2 ⌳⌽s Ž.n 2nq1 wxsr2q ⌽ sÝBn2syjŽ.Ž.nq2 Žsyj .q js wxsr2q ⌽ qÝBnnnsyjŽ.Žy1... . Žny2jq1q .Ž.ny2jq , js1 Ž.2.11 where wxs represents as usual the integer part of s g ޒ; s 0 when s is even and s 1 when s is odd, and AniiŽ.and Bn Ž.denote rational coefficients with non-zero denominators. Our assertion is now an immediate consequence of expressionsŽ. 2.10 andŽ. 2.11 . COROLLARY 2.1. The operator ⌽ ¬ ⌳*⌽ is a continuous linear mapping of LŽ.ގ0 into itself. Hence, the operator F ¬ ⌳*F defined by ²²⌳*F, ⌽ ::s ²F, ⌳*⌽ ::, F g LЈŽ.ގ00, ⌽ g L Ž.Žގ 2.12 . is also a continuous linear mapping of LЈŽ.ގ0 into itself. Proof. This result is deduced by combining propertyŽ. ii , Proposition 2.2, and the relation 2n q 12 ⌳*⌽Ž.ns ⌳⌽Ž.n 22nq1 between both the finite-difference operatorsŽ. 1.5 and Ž. 1.6 . 3. THE DISCRETE LEGENDRE TRANSFORMATION IN THE SPACES LŽ.ގ00AND LЈ Ž.ގ In order to show the main result of this paper, we need to establish first the next assertion. s PROPOSITION 3.1.Ž. a The operation Žx .¬ x Ž.x , s s 0, 1, 2, . . . , is a continuous linear mapping of L Ž.y1, 1 into itself. s Ž.b The operation f Ž x .¬ xfŽ. x,defined through s s ² xfŽ. x , Ž.x:²s fx Ž.,x Ž.x:, is also a continuous linear mapping of L ЈŽ.y1, 1 into itself. DISCRETE LEGENDRE TRANSFORMATION 191 Proof. Some straightforward manipulations lead to obtain the formula ss s s2 RxŽ.Ž.x sxRqssŽ.Ž.q1xysy1xy 2 s 1 q2sxŽ.y1xy D. The general case k ks Žk. kyi RxŽ.Ž.xsÝpxRs,i Ž. Ž.x is0 ky1 2 Žk. kyiy1 qŽ.xy1ÝqxDRsy1, i Ž.Ž.Ž.x 3.1 is0 Žk.Ž. Žk. Ž. can be proved by induction on k, where pxs, isand qxy1, i stand for Žk. s polynomials of sth and Ž.s y 1 th degrees, respectively, with pxs,0Ž.sx for every k s 0, 1, 2, . . . . On the other hand, g L Ž.y1, 1 implies that Ž.x and its derivatives are bounded as x ª 1 y 0 and x ª y1 q 0wx 12, Proposition 2.1 . Taking into account the last result, a simple integration yields x 2 Ž.Ž.Ž.x y 1 D x s HR tdt Ž.3.2 y1 in view ofŽ. 1.8 . By substitutingŽ. 3.2 into the second addend of the right-hand side inŽ. 3.1 we obtain k ks Žk.␥ RxŽ.Ž.x FÝCs,ikyiŽ.. is0 This inequality completes the proof of partŽ. a . Part Ž. b of our assertion is inferred bearing in mindŽ. a andwx 20, Theorem 1.10-1 . Now we can state the following. THEOREM 3.1. The discrete Legendre transformation gi¨en by Ž.1.3 is an isomorphism of the space LŽ.ގ0 into L Žy1, 1 . , its in¨erse being defined by Ž.1.4 . Proof. It is clear that l is a linear mapping. Next, let ⌽ be an arbitrary member of LŽ.ގ0 . Notice that ϱ 2n 1 k q ␥kkÄ4Ž.Ž.l⌽ x s ␥ Ž.s sup R ⌽Ž.Ž.Ž.nPn x .3.3 Ý2 y1-x-1ns0 192 MENDEZ´´-PEREZ AND MORALES It can be easily seen thatwx 4, p. 110 2k kj RŽ.xsÝpxDj Ž. Ž.x, js1 where pxjŽ.are polynomials of jth degree. On the other hand, the series Ž.1.3 and those obtained after any number of term-by-term differentiation ofŽ. 1.3 converge uniformly on y1 F x F 1. Indeed, according to Ž 1.14 . we have ϱ 2n q 1 ⌽Ž.nDjÄ4 P Ž. x Ý 2 n ns0 ϱ 2n 1 q 2j F n ⌽Ž.n Ý 2 ns0 ϱ 1 ⌰⌽⌰⌽ ϱ F82jq3,0Ž.q 2 j,0 Ž.Ý 2 -. ns0Ž.nq1 Thus, it is permissible to interchange the differential operator R with the summation sign to get, in view ofŽ. 1.13 and Ž. 1.11 , ϱ 2n q 1 k ⌽Ž.nnn Žq1 .Pxn Ž. Ý 2 ns0 ϱ 1 kq3⌰⌽⌰⌽ F2 2kq3,0Ž.qk,0 Ž.Ý 2 . ns0Ž.nq1 Finally, bearing in mind this result inŽ. 3.3 , we arrive at ␥ ⌽ ⌰⌽ ⌰⌽ kkkŽl .F C ,0 Ž.q 2kq3,0 Ž., for a certain positive constant Ck . This inequality proves that s Ž.Ž.l⌽ x gLŽ.y1, 1 and that the linear mapping l is also continuous from LŽ.ގ0 into L Ž.y1, 1 . Conversely, assume that g L Ž.y1, 1 and evaluate 1 s y1 rs n⌳Ž.lŽ.nsn⌳HPxnŽ.Ž.xdx y1 rs1 snxPxH nŽ.Ž.xdx,3.4Ž. y1 where we have made use ofŽ. 1.17 . DISCRETE LEGENDRE TRANSFORMATION 193 2 By integrating by parts 2r times and setting R s DxŽ y1.D, the right-hand side inŽ. 3.4 adopts the form 11 11sr rs rrHHxŽ.xRPxdxÄ4nn Ž. s PxRxŽ. Ž.xdx Ž.nq1y1 Ž.nq1y1 Ž.3.5 j since the limits terms are equal to zero because of D Ž.x s O Ž.1as xª"1, for any j g ގ0. By virtue of Proposition 3.1 and combiningŽ. 3.4 andŽ. 3.5 we are led to r rs⌳ y1 Žr.␥ n Ž.l Ž.nF2ÝCs,iryiŽ., is0 in other words, for all g L Ž.y1, 1 the inequality r ⌰ y1 Žr.␥ r,ssŽ.l F ÝC ,iryiŽ. is0 Žr. holds, where Cs, i represent certain positive constants. This completes the proof of the second part of our assertion. y1 y1 Moreover, llŽ.⌽s⌽and ll Ž.Žs for any ⌽ g L ގ0 .and for each g L Ž.y1, 1 , respectively. The generalized discrete Legendre transformation is defined by lЈ: LЈŽ.ގ0¬ L Ј Žy1,1 . F¬lЈF, where 2n q 1 ²:ŽlЈFx .Ž.Ž,l⌽ .Ž.xs Fn Ž., ⌽ Ž.n , ⌽gLŽ.ގ0 . ¦¦2 ;; Ž.3.6 This definition is suggested by the classical Parseval relationŽ. 1.9 . We remark that the right-hand side inŽ. 3.6 has a sense in view of Proposi- tion 2.2. From Theorem 3.1 andwx 20, Theorem 1.10-2 we immediately obtain. THEOREM 3.2. The generalized discrete Legendre transformation lЈ is an isomorphism between the spaces LЈŽ.ގ0 and L Ј Žy1, 1 . . 194 MENDEZ´´-PEREZ AND MORALES By setting l⌽ s and lЈF s f the expressionŽ. 3.6 can be rewritten 2n 1 1 q 1 Ž.lЈy fnŽ., Ž.ly Ž.n s²:f,,3.7Ž. ¦¦ 2 ;; for any g L Ž.y1, 1 and f g L Ј Ž.y1, 1 . We can consider that Eq.Ž. 3.7 defines the inverse of lЈ. Next, we establish the most interesting operational formulas of the discrete Legendre transform: PROPOSITION 3.2. Let k g ގ0. Ž.a If ⌽ g L Žގ0 ., then k k lŽ.⌳ ⌽ s x Ž.l⌽ . Ž.b If F g LЈ Žގ0 ., then k k lЈ⌳Ž.Ž.*F sx lЈF. Proof. To deriveŽ. a , assume firstly that k s 1. By making use of the well-known finite difference operatorsŽ. 1.15 and Ž. 1.16 we find that ϱ 2n q 1 lŽ.⌳⌽ s PxnŽ.Ž.⌳⌽ n Ý2 ns0 12ϱϱnq1 sPxnnŽ.⌬n⌬⌽ Žn y 1 .q PxŽ.⌽ Ž.n.3.8 Ž . 22ÝÝ ns0ns0 Now, having resorted twice to the rule of summation by parts of the finite calculuswx 9, p. 29 , the first term of the right-hand side inŽ. 3.8 may be expressed as 1 ϱ Ä4EPŽ.n1Ž. x и⌬ n⌬⌽ Žn y 1 . 2Ýy ns0 1 nªϱ sPxnn 1Ž.⌬⌽ Žn y 1 . 2 y ns0 1 nªϱ y Ž.ny1Ä4⌬Pxn2 Ž.Ž.и⌽ ny1 2 y ns0 1ϱ q ⌬n⌬Pxn 1Ž.и⌽ Ž.n 2Ý y ns0 1ϱ s ⌬n⌬Pxn 1Ž.⌽ Ž.n, 2Ý y ns0 DISCRETE LEGENDRE TRANSFORMATION 195 since the terms within the braces vanish in view of Remark 1.1 and the fact that ⌽ is of rapid descent. By taking into account this result and the relation ⌬ n ⌬⌽Žn y 1 .s Ž2n q 1 .Ä4⌳⌽ Ž.n y ⌽ Ž.n Ž3.9 . between the classical finite-difference operator ⌬ and the operator ⌳,we deduce fromŽ. 3.8 that ϱ 2n q 1 lŽ.⌳⌽ s Ž.⌳ PxnŽ.Ž.Ž.Ž.⌽nsxl⌽ x, Ý2 ns0 according to the operational ruleŽ. 1.17 . The general case follows by induction on k. To verifyŽ. b , we infer from Ž 3.6 .Ž , 2.12 . , and formula Ž. a that ²:lЈ⌳Ž.Ž.*F, l⌽ 2nq12nq1 s⌳*FnŽ., ⌽ Ž.n s Fn Ž.,⌳* ⌽Ž.n ¦¦22 ;; ¦¦ ;; 2nq1 sFnŽ., ⌳⌽ Ž.n s²: ŽlЈF .Ž, l⌳⌽ . ¦¦2 ;; s²:²:ŽlЈF .Ž.,xl⌽ s xl ŽЈF .Ž., l⌽ , by virtue of Proposition 3.1 and the relation 2n q 12nq1 ⌳*⌽Ž.ns⌳⌽Ž.n 22 between both the finite-difference operatorsŽ. 1.5 and Ž. 1.6 . COROLLARY 3.1. If F is an arbitrary member of LŽ.ގ0 , then the classical transformation lF, gi¨en by Ž.1.3 , is a particular case of the generalized discrete transform lЈF, defined by Ž.3.6 . Proof. Note that lЈF has a sense because LŽ.ގ00; LЈ Ž.ގ . On the other hand, for any ⌽ g LŽ.ގ0 , it follows fromŽ. 3.6 that 2n q 1 ²:Ž.Ž.lЈF,l⌽sFn Ž., ⌽ Ž.n ¦¦2 ;; ϱ 2 n q 1 s FnŽ.⌽ Ž.n, Ž 3.10 . Ý 2 ns0 196 MENDEZ´´-PEREZ AND MORALES by invoking Proposition 2.2 and propertyŽ. i of Section 2. Next in accor- dance with Theorem 3.1, one has ²:²:lF, l⌽ s f , , Ž.3.11 where f s lF and s l⌽ belong to L Ž.y1, 1 . But each member of LŽ.y1, 1 generates a regular member of L Ј Ž.y1, 1 . By this reason the right-hand side inŽ. 3.11 may be rewritten ϱ 1 2n q 1 ²:f,sHfxŽ.Ž.xdxsÝ FnŽ.⌽ Ž.n, Ž 3.12 . 1 2 y ns0 after we have made use of the classical Parseval equalityŽ. 1.9 . Finally, by combiningŽ.Ž.Ž. 3.10 , 3.11 , and 3.12 we arrive at lЈF s lF, which is what we wished to prove. Let F be an arbitrary member of lЈŽ.ގ0 . In view of the inclusion lЈŽ.ގ00;LЈ Ž.Žގ remember propertyŽ. iii of Section 2. , F admits two discrete Legendre transforms, namely: f defined byŽ. 1.7 and lЈF given by Ž.3.6 . In order to guarantee that the developed theory is consistent, we have to prove that both definitions agree. This is shown in the following assertion: COROLLARY 3.2. For e¨ery F g lЈŽ.ގ0 , f s lЈF in the sense of the equality in the space L ЈŽ.y1, 1 . Proof. We only outline the proof. Assume that g L Ž.y1, 1 and ϱ recall that f g C Žwxy1, 1.wx 13 . This implies that f, defined byŽ. 1.7 , generates a regular member in L ЈŽ.y1, 1 . Moreover, the technique of Riemann sumswx 20, p. 148 allows us in the sequel to interchange the functional symbol with the integral sign. All these considerations justify the following steps, 112n q 1 ²:f, fxŽ.Ž.xdx FnŽ., Px Ž. Ž.xdx sHHs ¦¦n ;; y 1 y 1 2 2 n q 1 1 s FnŽ., PxnŽ.Ž.xdx ¦¦ 2 H ;; y 1 2 n q 1 s FnŽ., ⌽ Ž.n ,Ž. 3.13 ¦¦2 ;; y 1 where ⌽ s l g LŽ.ގ0 in view of Theorem 3.1. On the other hand, since F g LЈŽ.ގ0 the last term inŽ. 3.13 converts into ²:²:lЈF,l⌽slЈF,. DISCRETE LEGENDRE TRANSFORMATION 197 From this we see that ²:²f, s lЈF, :for any g L Žy1, 1 . . Thus the proof is ended. 4. THE TRANSLATION OPERATOR AND THE CONVOLUTION The classical convolution corresponding to the discrete transformŽ. 1.3 is investigated by I. I. Hirschman inwx 7 . Let m g ގ0 be arbitrarily fixed. We 1 define the translation tm of the sequence ⌽ g l Ž.ގ0 by means ofw 13, Ž.4.3 x ϱ 2n q 1 tm⌽Ž.l s ⌸Ž.Ž.l, m, n ⌽ n ,4.1Ž. Ý 2 ns0 where 1 HPxPxPxdxlmnŽ. Ž. Ž.s⌸ Žl,m,n .,4.2 Ž . y1 for any l, m, n g ގ0. The formula ltŽmm⌽ .sPxl Ž.Ž.Ž.⌽x Ž.4.3 holds. Notice that ⌸Ž.l, m, n vanishes for n ) l q m wx7, p. 339 . We now list some properties involving the kernel ⌸Ž.l, m, n wx7, 8 : ⌸Ž.l, m, n G 04Ž..4 ϱ2nq1 PxPxlmŽ. Ž.s ⌸Žl,m,nP .n Ž. x Ž4.5 . Ý 2 ns0 ϱ2nq1 ⌸Ž.l,m,ns1.Ž. 4.6 Ý2 ns0 Next the convolution is introduced through ϱ 2l q 1 Ž.Ž.⌽ ⌿ n s ⌽ Ž.Ž.ltn⌿l ( Ý2 ns0 ϱϱ2lq12mq1 s ⌸Ž.Ž.Ž.Ž.l,m,n⌽ l ⌿ m,4.7 ÝÝ 22 ls0ms0 1 1 for every ⌽, ⌿ g l Ž.ގ00. It is easily deduced that ⌽(⌿ g l Ž.ގ , 55⌽(⌿ 1 F5555⌽⌿11, and lŽ⌽(⌿ .Ž.xs Žl⌽ .Ž.Žxиl⌿ .Ž.x.4.8Ž. 198 MENDEZ´´-PEREZ AND MORALES In the following assertions we will reproduce these properties in the spaces lŽ.Ž.ގ00, L ގ , and their duals. PROPOSITION 4.1. Let m g ގ0 arbitrarily fixed. Then, Ž.i The operation ⌽ ¬ tm⌽ is a continuous linear mapping from LŽ.ގ0 into itself. Ž.ii If ⌽ g L Žގ00 .and k g ގ , one has kk tmm⌳⌽s⌳t⌽. Ž.iii The mapping Ž.⌽, ⌿ ¬ ⌽(⌿ is continuous from LŽ.ގ00= L Ž.ގ into LŽ.ގ0 . Ž.iv For e¨ery ⌽, ⌿ g LŽ.ގ0 it is fulfilled that ⌳⌽Ž.Ž.(⌿ s ⌳⌽ (⌿ q ⌽( Ž.⌳⌿ . Proof. Note first that ϱϱ2lq12nq1 ⌸Ž.Ž.l,m,n⌽n ÝÝ22 ls0 ns0 ϱϱ2nq12lq1 F ⌽Ž.n ⌸Ž.l,m,n ÝÝ22 ns0 ls0 ϱ2nq1 s⌽Ž.n-ϱ, Ý 2 ns0 by invokingŽ. 4.4 and Ž. 4.6 , and taking into account that ⌽ g LŽ.ގ0 ; 1 lŽ.ގ0 . Now it is as deduced fromŽ. 4.3 that y1 tmm⌽ s lPxlŽ.Ž.Ž.⌽x.4.9Ž. For brevity, we set l⌽ s . Since PxnŽ.Ž.xgL Žy1, 1 . for all g LŽ.y1, 1 , the statementŽ. i follows immediately fromŽ. 4.9 and Theo- rem 3.1. To verifyŽ. ii , observe that we can write k y1 k tmm⌳ ⌽ s l Ž.PxlŽ.Ž⌳⌽ .Ž.x y1k y1 k slŽ.Ž.PxxmmŽ. Žl⌽ .Ž.xslŽ.⌳PxŽ.Ž l⌽ .Ž.x ky1k s⌳lŽ.PxlmmŽ.Ž.Ž.⌽xs⌳t⌽, by resorting toŽ. 4.9 , part Ž. a in Proposition 3.2, andŽ. 1.17 . DISCRETE LEGENDRE TRANSFORMATION 199 Starting fromŽ. 4.8 we obtain 1 ⌽(⌿ s ly Ž.Ž.Ž.l⌽ l⌿ .Ž. 4.10 Inasmuch as и g L Ž.y1, 1 whenever , g L Ž.y1, 1w 12, Proposition 2.1x , the assertionŽ. iii is a simple consequence ofŽ. 4.10 and Theorem 3.1. UsingŽ. 4.10 and the operational formulaŽ. a in Proposition 3.2 we arrive at 1 1 Ž⌳⌽ .(⌿ s ly Ž.Ž. Žl⌳⌽ .Ž. и l⌿ s ly xl Ž.Ž.⌽иl⌿ , whose left-hand side has sense in view of Proposition 3.1. Similarly, 1 ⌽(Ž.⌳⌿ s ly Ž.xl Ž.Ž.⌽иl⌿ . On the other hand, Proposition 3.1 andŽ. 4.8 allow us to write 1 1 ⌳⌽Ž.(⌿ slly ⌳⌽ Ž.(⌿ slxly Ž. Ž.Ž.⌽иl⌿ . Finally, by combining these results we inferŽ. iv . Next we define the convolution of F g LЈŽ.ގ00and ⌽ g L Ž.ގ by 2lq1 F⌽Ž.nsFl Ž., tn⌽ Ž.l .Ž. 4.11 (¦¦2 ;; In particular, if F g LŽ.ގ0 , then F generates a regular member in LЈŽ.ގ0 according to Ž. 2.5 . Thus, Ž 4.11 . adopts the form ϱ 2l q 1 F ⌽Ž.n s FltŽ.n⌽ Ž.l, ( Ý 2 ls0 in other words, in this case the generalized convolutionŽ. 4.11 reduces to the classical definitionŽ. 4.7 . Henceforth OŽ.ގ0 will represent the space of all sequences Mn Ž.such k that <⌳ MnŽ. PROPOSITION 4.2. OŽ.ގ00is a multiplier space for LŽ.ގ and LЈ Ž.ގ0. Proof. Let M be any member of OŽ.ގ00. We have that M и⌽gLŽ.ގ , for whatever ⌽ g LŽ.ގ0 . Indeed, by virtue ofŽ. 2.3 and the fact that ⌽Ž.n 200 MENDEZ´´-PEREZ AND MORALES is taken as zero for negative values of n, we get ⌰r,sŽ.M⌽ rs ssup n ⌳ MnŽ.⌽ Ž.n ngގ0 s FÝiŽ.0 ⌽ Ž.i is0 s ry␣␣ii qÝsup niŽ.Ž.Ž.Ž.Ž.nnqiMnqinqi ⌽nqi isys ngގ0 nqi)0 s C ⌰⌽-ϱ F Ý␣i rŽi.,0Ž. isys Ž.r,sގ, for suitable positive constants C and riŽ.ގ, s i s. g0 ␣i g 0 yF F We have really established that the operation Ž.M, ⌽ ¬ M и⌽ is a continuous linear mapping from OŽ.ގ00= L Ž.ގ into L Ž.ގ 0. If F g LЈŽ.ގ0 we define as usual ²²MF, ⌽ ::s ²²F, M и⌽ ::, ⌽gLŽ.ގ0. Ž 4.12 . The above considerations imply that the operation Ž.M, F ª M и F, given byŽ. 4.12 , is also a continuous linear mapping from O Ž.ގ00= LЈ Ž.ގ into LЈŽ.ގ0. Proposition 4.2 justifies that the space OŽ.ގ0 be endowed with the topology generated by the family of seminorms ⌸ r,s;BrŽ.M s sup ⌰ ,s ŽM и⌽ ., ⌽gL Ž.ގ0, ⌽gB where B runs all bounded subsets of LŽ.ގ00.SoO Ž.ގ becomes a Hausdorff, nonmetrizable, complete, topological vector space. Note that LŽ.ގ000is a proper subspace of O Ž.ގ . Since each element of O Ž.ގ provides a regular member in LЈŽ.ގ00through Ž. 2.5 , the inclusion OŽ.ގ ; LЈŽ.ގ0 is justified. PROPOSITION 4.3. Assume that F g lЈŽ.ގ00. For any ⌽ g LŽ.ގ , the exchange formula lFŽ(⌽ .Ž.xs ŽlЈFx .Ž.Žиl⌽ .Ž.x, xgy Ž1, 1 . , Ž 4.13 . holds. The mapping ⌽ ¬ F(⌽ is continuous from LŽ.ގ0 into itself. More- o¨er, the sequence ŽŽ2n q 1 .r2 .F (⌽ gi¨es rise to a regular member in DISCRETE LEGENDRE TRANSFORMATION 201 LЈŽ.ގ0 satisfying 2n q 12nq1 F⌽,⌽sF,⌽⌿,⌿gLŽ.ގ0. ¦¦22( ;; ¦¦( ;; Proof. Since lЈŽ.ގ00; LЈ Ž.ގ , then F (⌽ is well-defined in accordance withŽ. 4.11 . Hence, by definition, N 2n q 12lq1 lFŽ(⌽ .Ž.xslim Ý FlŽ., tnn⌽ Ž.lPx Ž. Nϱ22¦¦ ;; ªns0 2lq12Nnq1 slim FlŽ., Ý tnn⌽Ž.lP Ž x .. Ž 4.14 . Nªϱ¦¦ 22 ;; ns0 By invokingw 13,Ž. 2.8x and Proposition 4.1Ž. ii , as well as making use of Ž.4.9 and the operational rulew 12,Ž. 3.22x we are led to 2l q 12ϱ nq1 ey␣lk⌳* t⌽Ž.lP Ž x . 22Ý nn nsNq1 2l12ϱn1 y␣lkqq setnn⌳⌽Ž.lPx Ž . 22Ý nsNq1 2l12ϱn1 y␣lqqy1 k selPtllnŽ.Ž⌳⌽ .Ž.ŽtnPx . Ž . 22Ý nsNq1 2l12ϱn1 y␣lqqy1 k selPttllnŽ. Ž⌽ .Ž.ŽtnPx . Ž . 22Ý nsNq1 2l12ϱn1 y␣lq qy12 k selRPtttlÄ4Ž. Ž.tnPx Ž .n Ž ., 2 Ý2 nsNq12nnŽ.q1 Ž.4.15 where s lŽ.⌽ g L Žy1, 1 . . Ž j. 2j On the other hand, if we resort to the inequality < PxlŽ. ŽŽ . . ϱ ŽŽ . . In short, we have checked that 2l q 1 r2 ÝnsNq1 2n q 1 r2 tnn⌽Ž.lP Ž x .converges to zero as N ª qϱ in the topology of the space lŽ.ގ0 . Therefore, it is deduced fromŽ. 4.14 that lFŽ.Ž.(⌽ x 2lq12ϱnq1 sFlŽ., Ý tnn⌽Ž.lP Ž x . ¦¦ 22 ;; ns0 2lq12lq1 sFlŽ., Žltll⌽ .Ž.x s Fl Ž., Px Ž.Ž.Ž. l⌽ x ¦¦ 22;; ¦¦ ;; 2lq1 sŽ.Ž.l⌽xFl Ž., Pxl Ž.s Ž.Ž.Ž.Ž.l⌽ xиlЈFx, ¦¦2 ;; by virtue ofŽ. 1.7 and Ž. 4.3 . This completes the proof ofŽ. 4.13 . ϱ r Next, since Ž.Ž.lЈFxgCŽ.y1, 1 and DlŽ.Ž.ЈFx is bounded Žr s 0, 1, 2, . . ..Ž. , for every F g lЈ ގ0 and y1 F x F 1w 13, Proposition 3.1 and Proof of Proposition 3.2x , we get that Ž.Ž.l⌽иlЈFgL Žy1, 1 . because of wx12, Proposition 2.1 . This implies that 1 F(⌽ s ly Ž.Ž.Ž.lЈF и l⌽ . So, we conclude that F (⌽ is continuous in accordance with Theorem 3.1. Finally, inasmuch as F (⌽ g LŽ.ގ00then ŽŽ 2n q 1 ..r2 F(⌽ g L Ž.ގ in line with Proposition 2.2. Consequently,ŽŽ 2n q 1 .r2 .F(⌽ gives rise to a regular member in LЈŽ.ގ0 through 2n q 12ϱ nq1 F⌽,⌿sŽ.Ž.Ž.F⌽n⌿n ¦¦22( ;; Ý ( ns0 ϱ2nq12lq1 s FlŽ., tn⌽ Ž.l ⌿ Žn . Ý 22¦¦ ;; ns0 2lq12ϱnq1 sFlŽ., Ý tl⌽Ž.Ž.n⌿n ¦¦ 22 ;; ns0 2lq1 sFlŽ., ⌽ ⌿ ¦¦2 ( ;; for any ⌿. Remark 4.1. If we replace in Proposition 4.3 the hypothesis F g lЈŽ.ގ0 for the widest one F g LЈŽ.ގ0 the conclusion is false in general. Indeed, DISCRETE LEGENDRE TRANSFORMATION 203 FnŽ.s1 is a regular distributional sequence in LЈ Žގ0 .and it fulfils ϱ 2l q 1 F ⌽Ž.n s tn⌽Ž.l ( Ý 2 ls0 ϱϱ2mq12lq1 s ⌽Ž.m ⌸ Žm,n,l . ÝÝ22 ms0 ls0 ϱ2mq1 s⌽Ž.m, Ý 2 ms0 in other words, F(⌽Ž.n is a constant depending on ⌽ g LŽ.ގ0 . Hence ⌰r,0Ž.F(⌽sqϱ, for each r s 1, 2, 3, . . . . This implies that F(⌽ f LŽ.ގ0 . However, in this particular case, it is obvious that F(⌽ g OŽ.ގ0 . Inwx 14 we have established that the last result holds in general, that is, F(⌽gOŽ.ގ000for every F g LЈ Ž.ގ and ⌽ g L Ž.ގ . 5. APPLICATIONS By way of illustration the applications of the discrete Legendre transfor- mation, we will tackle the following problems: Ž.a First we propose to find a generalized sequence um Ž,n .,mand n g ގ0 , satisfying the partial difference equation with variable coefficients Ž.Ž.m q 1 m q 2 umŽ.q2, n Ž.Ž.2mq32mq5 2 2 mmŽ.q1 qqumŽ.,n Ž.Ž.Ž.Ž.2my12mq12mq12mq3 mmŽ.y1 q umŽ.Ž.Ž.y2, n q um,nq2s05.1 Ž.Ž.2my12my3 and the initial conditions umŽ.,0 sAm Ž.Ž.gLЈގ0 Ž.5.2 umŽ.,1 sBm Ž.Ž.gLЈގ0.5.3Ž. By employing the finite-difference operatorŽ. 1.5 , Eq. Ž. 5.1 becomes U 2 ⌳ mumŽ.Ž,n qum,nq2 .s0.Ž. 5.4 204 MENDEZ´´-PEREZ AND MORALES If we set UxŽ.,nsÄlЈumŽ.,nx4Ž.and apply the discrete Legendre trans- formation, bearing in mind the operational ruleŽ. b in Proposition 3.2, Eq.Ž. 5.4 converts into the ordinary finite-difference equation 2 x UxŽ.,nqUx Ž,nq2 .s0 whose general solution is n n n UxŽ.,nsxC12cos q C sin .Ž. 5.5 ž/22 y1 The values of the constants C12s axŽ.s ŽlЈAx .Ž.and C s xbxŽ.s xly1Ž.Ž.ЈBxare suggested by the conditionsŽ. 5.2 and Ž. 5.3 . Now, by invertingŽ. 5.5 in agreement with Theorem 3.2, we obtain the required solution 1 umŽ.,n sÄ4lЈyUx Ž.Ž.,nm, defined by means of the functional 2m q 1 umŽ.,n, ⌽ Ž.m ¦¦ 2 ;; s²:Ä4Ä4lЈumŽ.Ž.Ž.Ž.,nx,l⌽mx n n n 1 s²:UxŽ,n ., Ž.x s xax Ž.cos q xbxyŽ.sin , Ž.x ¦;½522 n n nn1 scos ²:xaŽ. x, Ž.x qsin ²:xbxyŽ., Ž.x 22 n 1n scos Ä4lЈyŽ.xaŽ. x Ž m . ¦¦ 2 n 2 m 1 1 n 1 q q sin Ä4lЈy Ž.xbxmy Ž.Ž., ⌽ Ž.m , 22;; for all ⌽ g LŽ.ގ0 . Hence, n n 1 n 1 n 1 umŽ.,n scos Ä4lЈy Ž.xaŽ. x Ž m .qsin Ä4lЈy Ž.xbxmy Ž. Ž .. 22 Otherwise, by resorting to Proposition 3.2, we can show that y1 n Un y1 ny1 UŽny1. ÄlЈ Ž xaxŽ..4Ž. m s⌳mmAmŽ.and ÄlЈ ŽxbxmŽ..4Ž.s⌳ BmŽ. which implies that n n U n UŽ ny1. umŽ.,n s cos ⌳ mmAmŽ.q sin ⌳ BmŽ..5.6 Ž . ž/22 ž/ DISCRETE LEGENDRE TRANSFORMATION 205 It is not difficult to demonstrate thatŽ. 5.6 is truly the solution of Ž. 5.1 that satisfies the conditionsŽ. 5.2 and Ž. 5.3 . Ž.b Finally we will try to seek a generalized sequence U g LЈ Žގ0 . satisfying the ordinary difference equation n q 1 n 2UnŽ.y Un Ž.q1y Un Ž.Ž.y1sFn, 2nq32ny1 where F is a given member of LЈŽ.ގ0 . Notice that this equation can be rewritten 2UnŽ.y⌳*Un Ž.sFn Ž..5Ž..7 If we set Ž.Ž.Ž.lЈUxsux and apply the operational ruleŽ. b in Proposition 3.2, Eq.Ž. 5.7 becomes Ž.Ž.Ž.2 y xux sfx, whose solution is fxŽ. uxŽ.s .5.8Ž. 2yx y1 y1 By virtue ofw 15, p. 422Ž. 13x one has that lЈ ŽŽ2 y xn . .Ž .s2Qn Ž2. . Now, in view of the asymptotic representation of the Legendre function of the second kindw 10,Ž. 7.11.11x , we can prove that QnŽ.2 g L Žގ0 .. There- fore, applyingŽ. 4.13 we can invert Ž. 5.8 to get the tentative solution UnŽ.s2Fn Ž.(Qn Ž.2. Ž.5.9 ThatŽ. 5.9 is truly a solution ofŽ. 5.7 can be easily shown, in view of Propositions 2.2 and 3.2, the relationŽ. 2.8 inwx 13 , and definitionŽ. 3.6 . REFERENCES 1. J. J. Betancor and I. Marrero, The Hankel convolution and the Zemanian spaces  and X ,Math. Nachr. 160 Ž.1993 , 277᎐298. 2. R. V. Churchill, ‘‘Operational Mathematics,’’ McGraw᎐Hill, New York, 1972. 3. A. Erdelyi,´ ‘‘Higher Transcendental Functions,’’ Vols. I, II, and III, McGraw᎐Hill, New York, 1953; reprinted by Krieger, Melbourne, FL, 1981. 4. H. J. Glaeske and A. Hess, On the convolution theorem of Mehler᎐Fock transform for a class of generalized functions, Math. Nachr. 131 Ž.1987 , 107᎐117. 5. H. J. Glaeske and T. Runst, The discrete Jacobi transform of generalized functions, Math. Nachr. 132 Ž.1987 , 239᎐251. 6. F. B. Hildebrand, ‘‘Finite-Difference Equations and Simulations,’’ Prentice Hall, Engle- wood Cliffs, NJ, 1968. 206 MENDEZ´´-PEREZ AND MORALES 7. I. I. Hirschman, Variation diminishing transformations and ultraspherical polynomials, J.Analyse Math. 8 Ž.1960᎐1961 , 337᎐360. 8. I. I. Hirschman, Variation diminishing transformations and orthogonal polynomials, J.Analyse Math. 9 Ž.1961 , 177᎐193. 9. W. G. Kelley and A. C. Peterson, ‘‘Difference Equations,’’ Academic Press, New York, 1991. 10. N. N. Lebedev, ‘‘Special Functions and Their Applications,’’ Dover, New York, 1972. 11. I. Marrero and J. J. Betancor, Hankel convolution of generalized functions, Rend. Mat. 15 Ž.1995 , 351᎐380. 12. J. M. R. Mendez-Perez´´ and G. Miquel Morales, The finite Legendre transformation of generalized functions, Rocky Mountain J. Math., in press. 13. J. M. R. Mendez-Perez´´ and G. Miquel Morales, The discrete Legendre transform of sequences of exponential growth, J. Math. Anal. Appl. 201 Ž.1996 , 57᎐74. 14. J. M. R. Mendez-Perez´´ and G. Miquel Morales, The discrete Legendre convolution of generalized sequences and its applications, preprint. 15. A. P. Prudnikov, Yu. A. Brychokov, and O. I. Marichev, ‘‘Integrals and Series. Vol. 2. Special Functions,’’ Gordon & Breach, New York, 1986. 16. L. Schwartz, ‘‘Theorie´ de distributions,’’ Hermann, Paris, 1978. 17. I. N. Sneddon, ‘‘The Use of Integral Transforms,’’ McGraw᎐Hill, New York, 1972. 18. G. Szego,¨ ‘‘Orthogonal Polynomials,’’ Amer. Mat. Soc. Colloq. Publ., Vol. 23, Amer. Math. Soc., Providence, 1975. 19. C. J. Tranter, Legendre transforms, Quart. J. Math. Oxford 1 Ž.1950 , 1᎐8. 20. A. H. Zemanian, ‘‘Generalized Integral Transformations,’’ Interscience, New York, 1968, reprinted by Dover, New York, 1987.