P. L. Butzer, R. L. Stens, M. Wehrens, "The continous Legendre transform, its inverse transform, and applications", International Journal of Mathematics and Mathematical Sciences, vol. 3, Article ID 935357, 21 pages, 1980. https://doi.org/10.1155/S016117128000004X
The continous 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 by the function with real. Another approach to T.M. MacRobert's inversion formula is found; for this purpose an inverse Legendre transform, mapping into , 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 .
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