The Generalized Discrete Legendre Transformation

The discrete transformation LF. x.sf x.s `ns0 2nq1.r2.Pn x.F n., where the Pn x.’s denote the well-known Legendre polynomials, is an isomorphism between the space L N0. of the complex-valued sequences of rapid descent and the space L y1, 1.of all infinitely differentiable complex-valued functions defined on y1-x-1, of slow growth in the end points y1 and 1. This result is extended to the corresponding dual spaces. A space of multipliers for these spaces is considered and the translation operator and the convolution are investigated on them. The operational calculus generated involves certain finite-difference operators and is applied to find the solution of some ordinary and partial finitedifference equations.