A general Laguerre transform and a related distance between probability measures

Abstract

A Laguerre transform will be exhibited which maps both functions in L2( x1, x) and finite signed measures into two-sided square summable sequences, extending previous results of Keilson, Nunn, and Sumita [4, 5. 141. The norm natural to the associated Hilbert space will be shown to provide a theoretically simple and numerically quantifiable measure for the distance between two probability distributions. A practical tool for measuring the convergence of iterative procedures. and a stopping criteria for such procedures are thereby obtained. In previous papers [4, 5, 143 an algorithmic framework was developed for the computer evaluation of repeated combinations of multiple convolutions, differentiation, integration, multiplication by polynomials, and other operations on the full real continuum R. The procedure enables one to evaluate distributions of interest in engineering, statistics, applied probability, and operations research, which have been available only formally behind the “Laplacian curtain.” The power and precision of the procedure have been demonstrated through a variety of applications [4, 5. 6, 7, 8, 9, 12, 13, 14, 15-j. The success of the procedure rests in large part on the availability of operational properties mapping the continuum operations into Lattice operations, and vice versa. The transform based on generalized Fourier series employs as a basis the union of the set of Laguerre functions I,(x) =e-1!2”L,(x) on the positive continuum, and a complementary set for the negative continuum. Here L,(x) is the Laguerre polynomial of degree n defined, for example, by Rodrigues’ formula L,(x) = (l/n!) e”(d/dx)“(x”e-” ). A brief description of the procedure and notation is given in the Appendix. That notation will be employed throughout the paper. The restriction of the domain of the Laguerre transform to functions in L,( c~j, a) has limited, to some extent, the easy applicability of the 288 0022-247X/86 $3.00