Poisson Summation Formulas and Inversion Theorems for an Infinite Continuous Legendre Transform

摘要

The classical Fourier transform and Fourier series are linked by the Poisson summation formula. The goal of this article is to find an infinite continuous Legendre transform which complements Legendre series in a similar way. To this end, the finite continuous Legendre transform due to Butzer/Stens/Wehrens is extended to an infinite transform. We show that for the new Legendre transform variants of Poisson’s formula and inversion theorems hold.