Article
Riesz Bases and Multiresolution Analyses
Department of Mathematics, Auburn University, Auburn, Alabama, 368495310, f2zalik@math.auburn.eduf2
Applied and Computational Harmonic Analysis (Impact Factor: 2.04). 11/1999; 7(3):315331. DOI: 10.1006/acha.1999.0274 Get notified about updates to this publication Follow publication 
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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. "
Article: Exact gframes in Hilbert spaces
[Show abstract] [Hide abstract] ABSTRACT: Gframes, 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 gframes are not equivalent to gRiesz bases. In this paper, we firstly give a characterization of an exact gframe in a complex Hilbert space. We also obtain an equivalent relation between an exact gframe and a gRiesz basis under some conditions. Lastly we consider the stability of an exact gframe for a Hilbert space under perturbation. These properties of exact gframes for Hilbert spaces are not similar to those of exact frames. 
 "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 nonMRA Parseval 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. 
 "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 nonMRA 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 $r$.