# Optimal free parameters in orthonormal approximations

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Optimal Free Parameters In Orthonormal Approximations - Signal Processing, IEEE Transactions on IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 8, AUGUST 1998 2081 Optimal Free Parameters in Orthonormal Approximations Albertus C. den Brinker and Harm J. W. Belt, Member, IEEE Abstract— We consider orthonormal expansions where the basis functions are governed by some free parameters. If the basis functions adhere to a certain differential or difference equation, then an expression can be given for a specific en- forced convergence rate criterion as well as an upper bound for the quadratic truncation error. This expression is a function of the free parameters and some simple signal measurements. Restrictions on the differential or difference equation that make this possible are given. Minimization of either the upper bound or the enforced convergence criterion as a function of the free parameters yields the same optimal parameters, which are of a simple form. This method is applied to several continuous-time and discrete-time orthonormal expansions that are all related to classical orthogonal polynomials. Index Terms— Orthogonal functions, parameter estimation, polynomials, transforms. I. INTRODUCTION APPROXIMATIONS of functions by a set of orthonormalfunctions is an often applied technique. Examples are the use of orthogonal polynomials [1], [2]; the use of Laguerre, Hermite, and Kautz functions in system identification ([3] and references therein), and signal coding [4]–[6]. In many cases, the set of orthogonal functions depends on one or more free parameters. In that case, it is of interest to establish the optimal free parameters in the case of a limited number of expansion terms. Since the free parameters appear in the error criterion in a nonlinear way, they are usually hard to find: Many local minima may occur in the error surface. Further, one would like to establish these parameters before actually engaging the orthonormal expansion. Therefore, the question is c

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