The Laguerre Transform and a Family of Functions with Nonnegative Laguerre Coefficients

Abstract
The Laguerre transform induced in Keilson and Nunn (Keilson, J., W. R. Nunn. 1979. Laguerre transformation as a tool for the numerical solution of integral equations of convolution type. Appl. Math. Comp. 5 313–359.) and Keilson, Nunn and Sumita (Keilson, J., W. R. Nunn, U. Sumita. 1981. The bilateral Laguerre transform. Appl. Math. Comp. 8 137–174.) maps functions in L2 into discrete sequences and permits rapid numerical calculation of basic continuum operations such as differentiation, integration and convolution. In this paper we study a family of functions 𝒫 with nonnegative Fourier–Laguerre coefficients. A uniform bound for truncation error can be easily found for this family. It will be shown that the family 𝒫 is closed under positive scalar multiplication, convex mixing, and multiplication by completely monotone functions.

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