Narrow Results

In this paper, the wavelet Riesz bases in the space ℓ²(ℤ) are studied. The main tool employed is discrete splines, which have received a considerable attention by many authors. The discrete splines are used for multiresolution analysis of the space ℓ²(ℤ).

We consider perturbation of frames and frame sequences in a Hilbert space ℋ. It is known that small perturbations of a frame give rise to another frame. We show that the canonical dual of the perturbed sequence is a perturbation of the canonical dual of the original one and estimate the error in the approximation of functions belonging to the perturbed space. We then construct perturbations of irregular translates of a bandlimited function in L²(ℝ^d). We give conditions for the perturbed sequence to inherit the property of being Riesz or frame sequence. For this case we again calculate the error in the approximation of functions that belong to the perturbed space and compare it with our previous estimation error for general Hilbert spaces.

Sufficient frame conditions for periodic dual wavelet are established. These conditions are given in terms of the Fourier coefficients. A way of constructing dual wavelet Riesz bases starting with trigonometric polynomials is provided.

In this paper, continuous piecewise quadratic finite element wavelets are constructed on general polygons in ${ℝ}^2$. The wavelets are stable in $H^s$ for $\left|s\right|<\frac{3}{2}$ and have two vanishing moments. Each wavelet is a linear combination of 11 or 13 nodal basis functions. Numerically computed condition numbers for $s\in \{-1,0,1\}$ are provided for the unit square.