On the Cholesky Factorization of the Gram Matrix of Multivariate Functions

We study the Cholesky factorization of certain biinnnite matrices and related nite matrices. These results are applied to show that if the uniform translates of a suitably decaying multivariate function are orthonormali-zed by the Gram-Schmidt process over certain increasing nite sets, then the resulting functions converge to translates of a xed function which is obtained by a global orthonormalization procedure. This convergence is also illustrated numerically. 1. Introduction Motivated by the use of orthogonal spline functions in various applications, e.g. 7] and 6], the authors showed in 3] that when uniform translates of a compactly supported function are orthonormalized by the Gram-Schmidt process over increasing nite intervals, then the resulting functions converge exponentially to translates of a xed function which is obtained by an orthonormalization procedure over the whole line. This global orthonormalization is derived from the Cholesky factorization of the biinnnite Gram matrix of the translates of the original function , while the Gram-Schmidt process is derived from the Cholesky factorization of nite Gram matrices which approximate sections of the biinnnite matrix. The convergence result is then derived from a more general result about convergence of Cholesky factors of positive deenite matrices whose elements decay exponentially away from the diagonal. In 5] these results were extended to the case where the original function need not have compact support but decays exponentially. In this paper we shall extend the above results in two directions: to functions in more than one dimension which may decay at a slower rate than exponential. The extensions both in dimension and in decay rate require signiicant changes in technique and lead to a much richer theory. In Section 2 we consider the Cholesky factorization of the biinnnite Gram matrix which, since the matrix is Toeplitz, is equivalent to the spectral factorization of its symbol. This factorization is gained by writing the logarithm of the symbol as an appropriate sum of two terms and then taking the exponential, a procedure justiied by applying a technique of Newman 8]. In electrical engineering this is called the cepstral method 1, 9]. In