Following the approach of Chui and Quak, we investigate semi-orthogonal spline wavelets on the unit interval [0, 1]. We give a slightly different construction of boundary wavelets. As a result, we are able to prove that the inner wavelets and the newly constructed boundary wavelets together constitute a Riesz basis for the wavelet space at each level with the Riesz bounds being level-independen...

In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonharmonic series given by Duffin and Schaefer in [6] and used recently in some applications (see [3]). In the case of an orthonormal basis our estimate reduces to Kadec’ optimal 1/4 re...

In this paper it is investigated how to find a matrix representation of operators on a Hilbert space H with Bessel sequences, frames and Riesz bases. In many applications these sequences are often preferable to orthonormal bases (ONBs). Therefore it is useful to extend the known method of matrix representation by using these sequences instead of ONBs for these application areas. We will give ba...

We construct bivariate local trigonometric bases on a \two-overlapping" triangular grid. A main result is the description of various trigonometric bases on triangles, satisfying parity conditions at the edges. Moreover, we introduce folding operators for the triangular grid. From these results we derive assertions on Riesz stability and the bi-orthogonal basis.

Extending band-limited constructions of orthonormal refinable functions, a special class of periodic functions is used to generate a family of band-limited refinable functions. Characterizations of Riesz bases and frames formed by integer shifts of these refinable functions are obtained. Such families of refinable functions are employed to construct band-limited biorthogonal wavelet bases and b...

In this paper we present an overview of how geometric methods can be successfully used to solve problems in Analysis. We will focus on selfsimilar objects and use their structure to construct frames, Riesz bases and wavelet bases in R with a single generator function. Further, we show that the generating functions for these systems are dense in L2(R).

On the n-dimensional hypercube, for given k ∈ N, wavelet Riesz bases are constructed for the subspace of divergence-free vector fields of the Sobolev space Hk((0, 1)n)n with general homogeneous Dirichlet boundary conditions, including slip or no-slip boundary conditions. Both primal and suitable dual wavelets can be constructed to be locally supported. The construction of the isotropic wavelet ...

This paper is concerned with the construction of biorthogonal wavelet bases on n-dimensional cubes which provide Riesz bases for Sobolev and Besov spaces with homogeneous Dirichlet boundary conditions on part of the boundary. The essential point is that the primal and dual wavelets satisfy certain corresponding complementary boundary conditions. These results form the key ingredients of the con...

On the n-dimensional hypercube, for given k ∈ N, biorthogonal wavelet Riesz bases are constructed for the subspace of divergence-free vector fields of the Sobolev space Hk((0, 1)n)n with general homogeneous Dirichlet boundary conditions, including slip or no-slip boundary conditions. Both primal and dual wavelets can be constructed to be locally supported. The construction of the isotropic wave...

Sequences of unit vectors for which the Kaczmarz algorithm always converges in Hilbert space can be characterized in frame theory by tight frames with constant 1. We generalize this result to the context of frames and bases. In particular, we show that the only effective sequences which are Riesz bases are orthonormal bases. Moreover, we consider the infinite system of linear algebraic equation...