International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 1, Pages 47-67
doi:10.1155/S016117128000004X
The continous Legendre transform, its inverse transform, and applications
Lehrstuhl A für Mathematik, Aachen University of Technology, Aachen 51, Germany
Received 15 January 1979
Copyright © 1980 P. L. Butzer et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.