THE CONTINUOUS LEGENDRE TRANSFORM , ITS INVERSE TRANSFORM , AND APPLICATIONS

This paper is concerned with the continuous Legendre transform, derived from the classical discrete Legendre transform by replacing the Legendre polynomial Pk(X) by the function P%(x) with % real. Another approach to T.M. MacRobert's inversion formula is found; for this purpose an inverse Legendre transform, mapping LI(+) into L2(-I,I), is defined. Its inversion in turn is naturally achieved by the continuous Legendre transform. One application is devoted to the Shannon sampling theorem in the Legendre frame together with a new type of error estimate. The other deals with a new representation of Legendre functions giving information


INTRODUCTION
In the early fifties C.J. Tranter (15) and R.V. Churchill (7) examined in some detail the (discrete) Legendre transform, which is defined in terms of the Legendre polynomials Pk(X), kEP={0,1,2,...}, and discussed its applicability mainly to the solution of partial differential equations.The present authors dealt with applications of this transform to a variety of problems in approximation theory in (14), ( 4), (5).
The first aim of this paper is to generalize this transform by replacing k by an arbitrary real , thus to study the transform (x)dx f^(1) := I f(x)Pl -I (1.1) {Pl(x); I R} being the system of Legendre functions.It will be referred to as the first continuous Legendre transform of f.Of basic importance in such a study is the existence of a further transform leading to an inversion formula, enabling one to reconstruct the original func- tion f from the values f^(1) of f^.T.M. MacRobert found such a formula in (I0), and gave sufficient conditions for its validity in (I I).The latter result was recalled with another proof by L. Robin (12, pp. 131).The transform in question is defined by AF(x) := 4 I F(1)Pl_i/2(-x)% sin A dl O.
Given the transform (1.2), there also arises the question as to its inversion.
One naturally expects that the transform leading to it is (I.I).It will be shown that this is indeed so for a special class of functions, with [F](-I12) F() (kER+).(1.4)This result, which does not seem to have been considered previously, is dealt with in Theorem 2.
Two applications will be considered.The first is concerned with a version of the Shannon sampling theorem in the Legendre setting which is more constructive than that of L.L. Campbell (6).Legendre transform methods will also enable us to present a new type of truncation error estimate for the corresponding sampling sum.The second application deals with a representation of the Legendre function Pk(x) as a sum of an infinite series of Legendre polynomials and an extra term describing the singularity of Pk(x) at the point x=-1.

PRELIMINARIES
The Legendre functions will be defined by means of the hypergeometric series.
Let a,b,c be real numbers, c #0,-I ,-2,... (2.11 k=O is absolutely and uniformly convergent on each compact subinterval of (-I, ).The series also converges for x =-I if c-a-b >-I, and for x provided that c-a-b > O.
In the latter case lim F(a,b;c;x) F(a,b;c;1) F c-a)r(c-b provided the right-hand side is meaningful.(For these properties see e.g.W.N.
At this stage some elementary properties of the Legendre polynomials are nee- ded (cf.( 14)).In the particular case that k=nP the series in (2.3) is finite, so that P (x) reduces to a polynomial of degree n, the classical Legendre polyno- n mial of ath degree in view of (2.4) and (2.6).These polynomials form an orthogo- nal set on [-1,1], i.e., for m,nq, If fqX, X denoting one of the spaces C[-1,1] or LP(-I,I), Igp<, the (discrete Legendre transform is defined by it is a bounded, linear mapping from X into the space c of all null sequences.o There holds the uniqueness theorem f(k) 0 (kP) * f(x) 0 (a.e.), (2.13)   and to each fX one may associate its Legendre series This series, which can be interpreted as the discrete counterpart of the inverse transform (1.2), does generally not represent the function f, but for fL2(-1,1) valid for f,g6L2(-1,1), and the fact that IIf-Snfll2 En(f)_ (2.16) Here S f denotes the nth partial sum of the series in (2.1) and En(f) is the n best approximation to f by algebraic polynomials Pn of degree n in L2(-1,1) -space, i.e., E n(f) inf llf-pnll2 (2.17)

Pn
Let us now return to the Legendre functions.Noting that Pk(X)= (-I)kPk(-X) for I =k6P, it follows from (2.10) and (2.11) that P1(k) for kP.Since PI L2(-1'I)' one has ba taking f=g=P1 in (2.15) that for -/ I sinai 1 2 (2k+l) (I-(I+k+I (2.19) This formula enables us to deduce LEMMA 2. a) For eac compact interval [a,b] (-1,1) there holds there exists M>O such that lIP I (x) P(x)ll 2 M 11- PROOF.For a) see (8, formulae 3.9.1 (2) and 1.18 (5) This gives (2.20) if l,v are not integers.If one (or both) of the reals l,v be- long to , the same proof applies with obvious modifications.Part b) follows by similar arguments.

THE FIRST CONTINUOUS LEGENDRE TRANSFORM
The aim of this section is to study the continuous analogue of the discrete Legendre transform (2.12) and its properties, including an inversion formula.We restrict the matter to functions f 6 L2(-I ,I).For the inversion formula two further results will be needed.
PROOF.Denoting the left-hand side of (3.6) by J(x), it follows from La. h that J6C(-1,1) NL2(-1,1).For the discrete Legendre transform of J one has for kEP f J(X)Pk(X)dX f f^(-I/2)I /I Px-I/2(-X)Pk(X)dx I sin d o the interchange of the order of integration being justified by a double use of Fubini's theorem.Now the inner integral is equal to Pk(-y)= (-I)kPk(y) by Prop.
I. This implies that J^(k)=f^(k), kP, so that the proof of part a) follows by (2.13).Part b) follows since in this case both sides of (3.6) are continuous.reproduces F under certain assumptions upon F.
PROOF.One has by Fubini's theorem and La. that f l ^F(x)PI_I/2(x)dx 4 f F(o) [ /I PO-I/2(-x)Pk-I/2(x)dx] o sinwo do Applying Fubini' s theorem once more shows that H 6 LI (+) and To calculate the Fourier-cosine transform of h we proceed in a manner similar to the proof of Prop.I.One has for v>O + yields Replacing the latter integral by a contour integral along R+ C R since I Jl (z)dz vanishes for R/.If v>w, then the last integral in (4.6) also vanishes for R/, so that Fc[hG](v)_ =O for wv<.On the other hand, if 0 v < , then limR__ C j2 (z)dz O, so that F [he](v) lim j2(z)dz F sin(-v)e c 2 R C R (O<v<; >0), the latter equality holding again by the residue theorem.So it follows by 6+ by the uniqueness theorem for the Fourier-cosine transform.If F6 C(+), then (4.3) holds for all 6+ since the left-hand side of (4.3) defines a con- + tinuous function on in view of La. 4 and La. 3.This completes the proof.
The results of Theorems and 2 show that the first and second Legendre trans- form are inverse to another.Indeed, if f6L2(-1 ,I) satisfies the assumptions of for almost all x (-1,1).On the other hand, if FL tions of Thm. 2, then, for almost all ER+, I(R+) satisfies the assump- [^F]^(-I12) F(k) 5. APPLICATIONS 5. THE SAMPLING THEOREM L.L. Campbell (6) stated a sampling theorem for functions F which are repre- sentable as the first Legendre transform of f E L2(-I ,I).We add a simple proof of this result (Thm. 3) based on Parseval's formula (2.15) (for the analogue in the classical setting see e.g. ( 2), (13)).Combining this result with our Thm. 2 yields that a function F, band-limited in the sense of the Fourier-cosine trans- form, satisfies the hypotheses of this theorem.This will be shown in Cor.I.

+
Thin. 3 and Cor show that a continuous function F on satisfying certain B+ conditions can be represented by its sampling sum on Denoting the nth partial sum of the series in (5.2) by (Sw,nF)(1)' one may ask for the error which occurs if one approximates F by S W F. This leads to the following truncation error ,n estimate.
PROOF Denoting the ath partial sum of the Fourier-Legendre series (2.h) of f by Sn f' one can rewrite the S w,nF in view of (2.Note that Lemmas 5 and 6 may also be formulated for functions F,F* satis- fying the assumptions of Cor.I. Then f and f* in (5.3) and (5.5) have to be replaced by AF(./W) and ^F*(-/W), respectively.
Let us conclude this subsection by showing that Cor.
on F imply those for the sampling theorem, so that the latter yields, when extending F to the whole real axis E as an even function,

REPRESENTATION FORMULA FOR LEGENDRE FUNCTIONS
Let us give another application of the theorems to a new representation of the Legendre functions Pl(x) describing in particular the behaviour of Pl(x) for x/ (-I)+.A first result of this type, basic for this paper, was the estimate (2.7).As will be seen, the log (2/(I+x)) term there is sharp.

O 4 f
F(o)h (k)o sin wo do H(k) o where h (l) := cos o cos I .(t2-o 2 [H](v) =O for <v<.Since F [F](v) =O for <v< by assumption, c c one finally has F [H](v)=F [F](v) for all vR+.This implies (4.3) for almost c c all

THEOREM 3 .
If F E C(R+) has the representation