A Note on Summability of Multiple Laguerre Expansions

A simple structure of the multiple Laguerre polynomial expansions is used to study the Cesàro summability above the critical index for the convolution type Laguerre expansions. The multiple Laguerre polynomial expansion of an `1-radial function f0(|x|) is shown to be an `1-radial function that coincides with the Laguerre polynomial expansion of f0, which allows us to settle the problem of summability below the critical index for the `1-radial functions. Introduction The aim of this note is to use a simple structure of the multiple Laguerre expansions to study their Cesàro summability. Let Ln, α > −1, denote the Laguerre polynomial of degree n on the half line R+ = [0,∞). The polynomials Ln form an orthogonal system in L(R+, xαe−xdx). The multiple Laguerre polynomials are products of Li ni that form an orthogonal system in L (R+,xedx), where α = (α1, . . . , αd) and |x| = x1+· · ·+xd. By considering the Laguerre functions Ln(x) = Ln(x)ex and their other varieties, Ln(x/2)x or Ln(x)(2x), one can form orthogonal systems in L(R+, dx) and L(R+, xdx). Consequently, there are at least four types of Laguerre expansions on R+ studied in the literature and each has its extension in the multiple setting (see [5] for the definitions). An excellent reference on the summability of Laguerre expansion is the monograph [6] by Thangavelu; see also the references in [6] for various other studies. Our point of view is from that of orthogonal polynomials. Thus, we study the case that the multiple Laguerre polynomials form an orthogonal basis in L(R+,xedx). Recently, we studied the summability of classical orthogonal polynomial expansions on the unit ball and on the standard simplex of R in [9], [10], and found that they are quite different from that of classical orthogonal polynomial expansions on the unit cube, and are in fact easier to deal with. The reason lies in the fact that the orthogonal structures on the ball and on the simplex are related to the orthogonal structure on the unit sphere, and the classical orthogonal polynomials are related to a special case of Dunkl’s theory of spherical harmonic Received by the editors February 5, 1999. 1991 Mathematics Subject Classification. Primary 42C05, 33C50, 41A63.