Affine Systems in L2(Rd): The Analysis of the Analysis Operator.

Abstract

Discrete a ne systems are obtained by applying dilations to a given shift-invariant system. The complicated structure of the a ne system is due, rst and foremost, to the fact that it is not invariant under shifts. A ne frames carry the additional di culty that they are \global" in nature: it is the entire interaction between the various dilation levels that determines whether the system is a frame, and not the behaviour of the system within one dilation level. We completely unravel the structure of the a ne system with the aid of two new notions: the a ne product, and a quasi-a ne system. This leads to a characterization of a ne frames; the induced characterization of tight a ne frames is in terms of exact orthogonality relations that the wavelets should satisfy on the Fourier domain. Several results, such as a general oversampling theorem follow from these characterizations. Most importantly, the a ne product can be factored during a multiresolution analysis construction, and this leads to a complete characterization of all tight frames that can be constructed by such methods. Moreover, this characterization suggests very simple su cient conditions for constructing tight frames from multiresolution. Of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions, and that the freedom o ered by redundancy can be fully exploited in these constructions.

The Method of Virtual Components in the Multivariate Setting

“…5. For convenience, let us use the expression in (16). We note that the first component of the vector Y e = [y 1 , .…”

“…That is, and q j 4 (η, θ ) = q j 1 (η + π, θ + π). Thus, by using the UEP in [16] we have Theorem 6.1 Let Q j (η, θ ) = q j 1 (η, θ ) and define ψ (j ) (x, y) in terms of Fourier transform by…”

Small Support Spline Riesz Wavelets in Low Dimensions

“…Another class of small support wavelet systems that are widely used in applications are spline tight wavelet frame systems constructed by the unitary extension principle of [42]. The spline wavelet tight frame systems are redundant and self dual systems with small support.…”

The Study of Periodic Tight Framelets and Wavelet Frame Packets and Applications

“…Basically, Duffin and Schaeffer abstracted the fundamental notion of Gabor for studying signal processing [2]. These ideas did not seem to generate much general interest outside of nonharmonic Fourier series however (see Young's [3]) until the landmark paper of Daubechies, Grossmann, and Meyer [4] in 1986. After this groundbreaking work, the theory of frames began to be more widely studied both in theory and in applications [5,6], such as signal processing, image processing, data compression, sampling theory.…”