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

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.