A convolution structure for Laguerre series

Abstract

Watson's product formula for Laguerre polynomials has been used by McCully [12] and Askey [1] to define a convolution structure. In comparison with the Jacobi expansion, the results in the Laguerre case are less satisfactory since the Laguerre translation operator is not positive. Nevertheless, other choices of the underlying spaces may lead to more convenient results. In the present paper we consider a set of weighted Lebesgue spaces L w p , 1 ⪯p⪯∞, which appear to be suitable for the Laguerre convolution in several respects. For these spaces there holds a convolution theorem such that for p = 1 a commutative Banach algebra is obtained. The Laguerre translation operator is “quasi-positive” on these spaces. In the applications of the convolution theorem to Cesàro and Abel-Poisson means of Laguerre series, it turns out that the supremum of the Lebesgue function is now attained at x=0, and that the summability conditions on L w p are quite similar to the corresponding conditions for Jacobi expansions. Moreover we discuss the convolution structure for the two-dimensional Laguerre translation introduced by Koornwinder [9].