The continous Legendre transform, its inverse transform, and applications
Open Access
- 1 January 1980
- journal article
- Published by Wiley in International Journal of Mathematics and Mathematical Sciences
- Vol. 3 (1) , 47-67
- https://doi.org/10.1155/s016117128000004x
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
Funding Information
- Deutsche Forschungsgemeinschaft (166/27)
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