The continous Legendre transform, its inverse transform, and applications

Abstract

ABSTRACT. This

The continuous Jacobi transform

“…Then by Corollary 3.1, we have Gk(X)eC(-l,l)nLw2(-l,l).. The idea now is to evaluate the discrete Jacobi transform of Gk(X) and show that it is equal to k(e'8)(j) whence by (2.5) we obtain Gk(X)=Pk [5]. m The case of ==-where m is a non-negative integer requires a separate analysis and should lead to the continuous version of the associated Legendre transform [9].…”

“…We note that if =8=0, then (c,l)()t) reduces to the continuous Legendre transform of Butzer, Stens and Wehrens [5]. Further, if X--neP, then (,8)() reduces to the discrete Jacobi transform of Debnath [3].…”

“…Y. DEEBA AND E. L. KOH transform generalizes, on the one hand, the continuous Legendre transform studied by Butzer, Stens and Wehrens [5] and on the other, the discrete Jacobi transform studied by Debnath [3]. The study of such transforms is interesting in its own right as well as in their applications to boundary value problems and in sampling theory.…”

Operational calculs for the continuous Legendre transform with applications

“…The discrete analog of the transform ill (1) has been studied by Churchill [2] and Churchill and Dolph [3]. The object of this paper is to develop an operational calculus for the transforin which is useful in solving paxtial differential equations whose underlying differential forln is given by D=xx (1-x)xx (2) Ill section 2 we present the background material needed in the sequel.…”

“…PRELIMINARIES. We recall basic properties of the transform (Tf)(A) (see [1] (2). Fher, i satisfies tim relations P(1) P (1) lira.__ , (1+…”