Basic relation between numerical range and Davis-Wielandt shell of an operator $A$ acting on a Hilbert space with orthonormal basis $xi={e_{i}|i in I}$ and its conjugate $bar{A}$ which is introduced in this paper are obtained. The results are used to study the relation between point spectrum, approximate spectrum and residual spectrum of $A$ and $bar{A}$. A necessary and sufficient condition fo...

A number of constructions in function field arithmetic involve extensions from linear objects using digit expansions. This technique is described here as a method of constructing orthonormal bases in spaces of continuous functions. We illustrate several examples of orthonormal bases from this viewpoint, and we also obtain a concrete model for the continuous functions on the integers of a local ...

It is shown that the discrete Calderón condition characterizes completeness of orthonormal wavelet systems, for arbitrary real dilations. That is, if a > 1, b > 0, and the system Ψ = {aψ(ax − bk) : j, k ∈ Z} is orthonormal in L(R), then Ψ is a basis for L(R) if and only if ∑ j∈Z |ψ̂(aξ)| = b for almost every ξ ∈ R. A new proof of the Second Oversampling Theorem is found, by similar methods.

This paper proposes a new subspace identification algorithm for continuous-time systems using generalized orthonormal basis functions. It is shown that a generalized orthonormal basis induces the transformation of continuoustime stochastic systems into discrete-time stochastic systems, and that the transformed noises have the ergodicity properties. With these basic observations, the standard su...

Methods from noncommutative harmonic analysis are used to develop an abstract theory of orthonormal wavelets. The relationship between the existence of an orthonormal wavelet and the existence of a multi-resolution is clariied, and four theorems guaranteeing the existence of wavelets are proved. As a special case of the fourth theorem, a generalization of known results on the existence of smoot...

Existing work on the representation of operators in one-dimensional, compactly-supported, orthonormal wavelet bases is extended to two dimensions. The non-standard form of the representation of operators is given in separable two-dimensional, periodic, orthonormal wavelet bases. The matrix representation of the partial differential operators ∂x and ∂y are constructed and a closed form formula f...

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...

It is very important for generating an orthonormal multiwavelet system to construct a conjugate quadrature filter (CQF). In this paper, a general method of constructing a length-J + 1 CQF with multiplicity r and scale a from a length-J CQF is obtained. As a special case, we study generally the construction of a length-J + 1 CQF with multiplicity 2 and scale 2 which can generate a compactly supp...

Sets K in d-dimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1 K is a single dyadic orthonormal wavelet. The construction is characterized by its generality in terms of a quantitative iterative procedure, by its computational implementation, and by its simplicity. The general case in which the inverse Fourier transfo...