Riesz Bases and Multiresolution Analyses

Department of Mathematics, Auburn University, Auburn, Alabama, 36849-5310, f2zalik@math.auburn.eduf2
Applied and Computational Harmonic Analysis (Impact Factor: 2.04). 11/1999; 7(3):315-331. DOI: 10.1006/acha.1999.0274


Recently we found a family of nearly orthonormal affine Riesz bases of compact support and arbitrary degrees of smoothness, obtained by perturbing the one-dimensional Haar mother wavelet using B-splines. The mother wavelets thus obtained are symmetric and given in closed form, features which are generally lacking in the orthogonal case. We also showed that for an important subfamily the wavelet coefficients can be calculated in O(n) steps, just as for orthogonal wavelets. It was conjectured by Aldroubi, and independently by the author, that these bases cannot be obtained by a multiresolution analysis. Here we prove this conjecture. The work is divided into four sections. The first section is introductory. The main feature of the second is simple necessary and sufficient conditions for an affine Riesz basis to be generated by a multiresolution analysis, valid for a large class of mother wavelets. In the third section we apply the results of the second section to several examples. In the last section we show that our bases cannot be obtained by a multiresolution analysis.


Available from: R. A. Zalik, Jul 15, 2014
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    • "Over the last years, there has been many achievements in the applications of Riesz bases [12] [25], and a lot of research results about the characterizations and stability of Riesz bases have arisen [6] [11] [24]. Riesz bases, which are equivalent to exact frames in Hilbert spaces, have become a powerful theoretical tool to research signal analysis. "
    [Show abstract] [Hide abstract] ABSTRACT: G-frames, which were considered recently as generalized frames in Hilbert spaces, have many properties similar to those of frames, but not all the properties are similar. For example, exact frames are equivalent to Riesz bases, but exact g-frames are not equivalent to g-Riesz bases. In this paper, we firstly give a characterization of an exact g-frame in a complex Hilbert space. We also obtain an equivalent relation between an exact g-frame and a g-Riesz basis under some conditions. Lastly we consider the stability of an exact g-frame for a Hilbert space under perturbation. These properties of exact g-frames for Hilbert spaces are not similar to those of exact frames.
    Preview · Article · Feb 2011 · Journal of Mathematical Analysis and Applications
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    • "This requires two steps. The theorem in [5] that provides the basis for Theorem 2.1, only shows in general that the the constructed wavelet is obtained from the GMRA determined by φ, in the sense defined by Zalik [18]. This means only that the wavelet ψ ∈ V 1 . "
    [Show abstract] [Hide abstract] ABSTRACT: A generalized filter construction is used to build non-MRA Par-seval wavelets for dilation by 2 in L 2 (R 2). These examples have the same multiplicity functions as wavelet sets, yet can be made to be C r with C r Fourier transform for any fixed positive integer r.
    Preview · Article · Jan 2008
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    • "It was shown in [5] that ψ ∈ V 1 , where {V j } is the GMRA determined by φ 1 and φ 2 as in Proposition 1. This says that ψ is obtained from the GMRA in the sense defined by Zalik [26]. We will now show that in this particular case, ψ is associated with the GRMA {V j } in the sense that V j is the closed linear span of {ψ l,k } l<j . "
    [Show abstract] [Hide abstract] ABSTRACT: A generalized filter construction is used to build an example of a non-MRA normalized tight frame wavelet for dilation by 2 in $L^2(\mathbb R)$. This example has the same multiplicity function as the Journ\'e wavelet, yet has a $C^{\infty}$ Fourier transform and can be made to be $C^r$ for any fixed positive integer $r$.
    Full-text · Article · May 2005 · Acta Applicandae Mathematicae
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