Block Cholesky factorization of infinite matrices and orthonormalization of vectors of functions

The following results on the block Cholesky factorization of bi-infinite and semi-infinite matrices are obtained. A method is proposed for computing the LDMT- and block Cholesky factors of a bi-infinite banded block Toeplitz matrix. An equivalence relation is introduced to describe when two semi-infinite matrices with entries A(ij) coincide exponentially as i,j,i f j --> infinity. If two equivalent bi-infinite matrices have block Cholesky factorizations, then their black Cholesky factors and their inverses are equivalent. If a bi-infinite block matrix A has a block Cholesky factorization whose lower triangular factor L and its lower triangular inverse decay exponentially away fi om the diagonal, then the semi-infinite truncation of A has a lower triangular block Cholesky factor whose elements approach those of L exponentially. These results are then applied to studying the asymptotic behavior of vectors of functions obtained by orthonormalizing a large finite set of integer translates of an exponentially decaying vector of functions.