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The continous Legendre transform, its inverse transform, and applications

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  • P. L. Butzer
  • R. L. Stens
  • M. Wehrens

Abstract

This paper is concerned with the continuous Legendre transform, derived from the classical discrete Legendre transform by replacing the Legendre polynomial P k ( x ) by the function P λ ( x ) with λ real. Another approach to T.M. MacRobert's inversion formula is found; for this purpose an inverse Legendre transform, mapping L 1 ( ℠+ ) into L 2 ( − 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 .

Suggested Citation

  • P. L. Butzer & R. L. Stens & M. Wehrens, 1980. "The continous Legendre transform, its inverse transform, and applications," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 3, pages 1-21, January.
    • Handle: RePEc:hin:jijmms:935357
    DOI: 10.1155/S016117128000004X
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