On the Cholesky factorization of the gram matrix of multivariate functions

We study the Cholesky factorization of certain bi-infinite matrices and related finite matrices. These results are applied to show that if the uniform translates of a suitably decaying multivariate function are orthonormalized by the Gram-Schmidt process over certain increasing finite sets, then the resulting functions converge to translates of a fixed function which is obtained by a global orthonormalization procedure. This convergence is also illustrated numerically.