The mathematical and numerical least squares solution of a general linear system of equations is discussed. Perturbation and differentiability theorems for pseudoinverses are given. Computational procedures for calculating least squares solutions using orthonormal transformations, multiplying matrices by a matrix of orthonormal basis vectors for the null-space of a given matrix, sequential proc...

New extremal properties of Daubechies 4-tap orthonormal filters are given: They maximize a certain functional, have the largest gain in (0, π/2), and allow maximum energy compaction in [0, π/2]. These properties do not carry over to Daubechies filters of arbitrary length. They complement what is known about Daubechies filters and highlight the specific role of the 4-tap filter. Moreover, we dem...

In a previous paper, one of us pointed out that the anomalous dimension matrices for all physical processes that have been calculated to date are complex symmetric, if stated in an orthonormal basis. In this paper we prove this fact and show that it is only true in a subset of all possible orthonormal bases, but that this subset is the natural one to use for physical calculations.

Recently we addressed the question of using comprehensive libraries of smooth orthonormal wavelets for signalbased wavelet design. Here we present the details of the algorithm for assembling a wavelet library from the continuum of compactly-supported orthonormal wavelets. The algorithm is based on a multi-level minimization, with local variation serving as the smoothness measure and the discret...

It is a well known fact that any orthonormal basis in L 2 can produce a \random density". If fng is an orthonormal basis and fang is a sequence of random variables such that a 2 n = 1 a.s., then f(x) = jann(x)j 2 is a random density. In this note we deene a random density via orthogonal bases of wavelets and explore some of its basic properties.