The deconvolution of signals is studied with thresholding estimators that decompose signals in an orthonormal basis and threshold the resulting coefficients. A general criterion is established to choose the orthonormal basis in order to minimize the estimation risk. Wavelet bases are highly sub-optimal to restore signals and images blurred by a low-pass filter whose transfer function vanishes a...

In the present paper we find new constructions of orthonormal multiresolution analyses on the triangle ∆. In the first one, we describe a direct method to define an orthonormal multiresolution analysis R which is adapted for the study of the Sobolev spaces H 0(∆) (s ∈ N). In the second one, we add boundary conditions for constructing an orthonormal multiresolution analysis which is adapted for ...

We prove that any Parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the Baumslag-Solitar group BS(1, 2) = 〈u, t | utu = t〉. We give a precise description of this representation in some special cases, and show that for wavelet sets, it is related to symbolic dynamics (Theorem 3.14). We show that the structure of the representation depends on the ana...

In the slice Hardy space over unit ball of quaternions, we introduce hyperbolic backward shift operator $$\mathcal S_a$$ with decomposition process $$\begin{aligned} f=e_a\langle f, e_a\rangle +B_{a}*\mathcal S_a \end{aligned}$$ where $$e_a$$ denotes normalized Szegö kernel and $$ B_a Blaschke factor. Iterating above process, a corresponding maximal selection principle gives rise to adaptive Fo...

We solve the Gauss law of SU(2) lattice gauge theory using the harmonic oscillator prepotential formulation. We construct a generating function of a manifestly gauge invariant and orthonormal basis in the physical Hilbert space of (d+1) dimensional SU(2) lattice gauge theory. The resulting orthonormal physical states are given in closed form. The generalization to SU(N) gauge group is discussed.

A systematic construction method of orthonormal basis on self-similar sets is given by the use of representations of the Cuntz algebras. We introduce two kinds of orthonormal basis of self-similar sets including explicit description for the Cantor set, the Sierpiński gasket and the Sierpiński carpet.

In applications, many states given for a system can be expressed by orthonormal elements, called “state elements”, taken in separable Hilbert space (called space”). The exact nature of the depends on system; example, state position and momentum is square-integrable functions. symmetries quantum represented class unitary operators that act space. ladder have effect lowering or raising energy sta...