Poisson summation formulas and inversion theorems for an infinite continuous Legendre trans\-form (Q818490)

The classical Fourier transform and Fourier series are linked by the Poisson summation formula. First, the author presents a rather general version of the formula. Then he employs the general theorem to derive Poisson formulas for the Chebyshev polynomials of first and second kind, in order to show the mechanism of the general formula in these cases and for later comparison with the Legendre setting. The main goal of the present paper is to find an infinite continuous Legendre transform which complements Legendre series in a similar way as it is in the classical case. Now, the author extends the finite continuous Legendre transform due to \textit{P. L. Butzer}, \textit{R. L. Stens} and \textit{M. Wehrens} [Int. J. Math. Math. Sci. 3, 47--67 (1980; Zbl 0447.44001)] to an infinite transform. Finally he proves variants of the Poisson formula and an inversion theorem for the new Legendre transform.