Two results on polynomially-generated wavelet frames and nonorthogonal polynomial frames

Redundancy of frames plays an important role in the context of erasures, as one might be able to reconstruct signals even if information is lost. The purpose of the paper is to present two results on overcomplete polynomially-generated frames. In the first part we show that bandlimited wavelet frames, for which the Fourier transform of the window $\psi$ has polynomial behavior in a neighborhood of zero, are remarkably stable towards erasures: given such a frame and any $N\in \mathbb{N}$, there exists an ordering of the frame elements having the property that for any $N\in \mathbb{N}$ the subfamily obtained by selecting each $N$th element itself a frame. The result is surprising, because it is known that band-limited wavelets for which $\widehat{\psi}$ vanishes on a neighborhood of zero never has this property. We illustrate the results with a number of concrete constructions, e.g., showing that it is even possible to construct a Parseval frame with the subsampling property. No such example has been identified in the literature so far. In the second part of the paper we introduce a method that allows to construct an overcomplete frame for Hilbert spaces of the form $L^2(-r,r)$ or $L^2(0,r)$, starting with a Riesz basis for the same space. When applied to standard orthogonal polynomials the construction yields nonorthogonal polynomial frames with attractive features: the frames are linearly independent, have infinite excess, the frame decomposition is simple, and the functions in the frame are ``very close'' to the functions in the given orthogonal system.