THE CONTINUOUS 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 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(R^+)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.

並列關鍵字

Continuous Legendre Transform ； Inverse Legendre Transform ； Inversion ormulae ； Shannon Sampling Theorem ； Truncation Error Estimate ； Behaviour of Legendre Functions Near Singular Points

國際替代計量

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