The continous Legendre transform, its inverse transform, and applications

Abstract

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 Pλ(x) with λ real. Another approach to T.M. MacRobert's inversion formula is found; for this purpose an inverse Legendre transform, mapping L1(ℝ+) into L2(−1,1), 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 about their behaviour near the point x=−1.

Keywords

• continuous Legendre transform
• inverse Legendre transform
• inversion formula
• Shannon sampling theorem
• truncation error estimates
• behaviour of Legendre functions near singular points.