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Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group G ∗ . These results can further be generalized to arbitrary locally compact abelian groups. [ABSTRACT FROM AUTHOR]

Let be a expansive integral matrix with . This paper investigates matrix Fourier multipliers for -dilation Parseval multi-wavelet frames, which are matrices with function entries, map -dilation Parseval multi-wavelet frames of length to -dilation Parseval multi-wavelet frames of length , where . We completely characterize all matrix Fourier multipliers for -dilation Parseval multi-wavelet frames and construct several numerical examples. As Fourier wavelet frame multiplier, matrix Fourier multipliers can be used to derive new -dilation Parseval multi-wavelet frames and can help us better understand the basic properties of frame theory. [ABSTRACT FROM AUTHOR]

We determine two-interval normalized tight frame wavelet sets for real dilation d ∈ (1, ∞), and characterize all symmetric normalized tight frame wavelet sets. We also construct a normalized tight frame wavelet set which has an infinite number of components accumulating at the origin. These normalized tight frame wavelet sets and their closures possess the same measure. Finally an example of a normalized tight frame wavelet set is provided whose measure is strictly less than the measure of its closure. [ABSTRACT FROM AUTHOR]

In this paper, we show that [½, 1) is not a fixed point set of any wavelet induced isomorphism. The same holds for [0, ½). This settles Problem 1 posed in the paper " On wavelet induced isomorphisms" by Singh in Int. J. Wavelets Multiresolut. Inf. Process. 8 (2010) 359-371. Further, we obtain that every other subinterval of [0, 1) of measure ½ is a fixed point set of some wavelet induced isomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we obtain that the space W2 of minimally supported frequency wavelets, the supports of whose Fourier transforms consist of two intervals, is path-connected [ABSTRACT FROM AUTHOR]

For a natural number n, selecting a 2n-interval symmetric wavelet set by making use of a result of Arcozzi, Behera and Madan [J. Geom. Anal. 13 (2003), 557-579], we construct a family of symmetric wavelet sets accumulating at the origin. Such a family of wavelet sets is also obtained from a family of symmetric six-interval wavelet sets provided by them in the same paper. Three-interval wavelet sets are employed in having a family of wavelet sets accumulating at the origin which are non-symmetric. Further, we obtain a larger class of H²-wavelet sets accumulating at the origin, which include the one given by Behera in [Bull. Polish Acad. Sci. Math. 52 (2004), 169-178]. Finally, non-MSF non-MRA wavelets are obtained through the selected family of 2n-interval symmetric wavelet sets. [ABSTRACT FROM AUTHOR]

While constructing a dyadic wavelet set through an approach which is purely set-theoretic, Ionascu observed that a dyadic one-dimensional wavelet set W gives rise to a specific measurable, bijective, piecewise increasing selfmap $\tilde{h}_{W}$ on [0, 1) and termed it to be a wavelet induced isomorphism. Further, he found that such maps provide wavelet sets which, in turn, characterize wavelet sets. In this paper, we consider two-interval, three-interval and symmetric four-interval wavelet sets and determine their wavelet induced isomorphisms. Also, fixed point sets of $\tilde{h}_{W}$ are determined for these wavelet sets. [ABSTRACT FROM AUTHOR]

In this paper, we characterize scaling sets consisting of three intervals. In addition, we provide a procedure to obtain scaling sets possessing finitely many intervals. [ABSTRACT FROM AUTHOR]

We construct a familiy of band-limited wavelets with Fourier transform discontinuous at the origin. [ABSTRACT FROM AUTHOR]

Given a function ψ in ${\cal L}^2({\Bbb R}^d),$ the affine (wavelet) system generated by ψ, associated to an invertible matrix a and a lattice Γ, is the collection of functions $\{|{\rm det}\, a|^{j/2} \psi(a^jx-\gamma): j \in {\Bbb Z}, \gamma \in {\Gamma}\}.$ In this paper we prove that the set of functions generating affine systems that are a Riesz basis of ${\cal L}^2({\Bbb R}^d).$ ${\cal L}^2({\Bbb R}^d)$ is dense in We also prove that a stronger result is true for affine systems that are a frame of ${\cal L}^2({\Bbb R}^d).$ In this case we show that the generators associated to a fixed but arbitrary dilation are a dense set. Furthermore, we analyze the orthogonal case in which we prove that the set of generators of orthogonal (not necessarily complete) affine systems, that are compactly supported in frequency, are dense in the unit sphere of ${\cal L}^2({\Bbb R}^d)$ with the induced metric. As a byproduct we introduce the p-Grammian of a function and prove a convergence result of this Grammian as a function of the lattice. This result gives insight in the problem of oversampling of affine systems. [ABSTRACT FROM AUTHOR]

W. C. Lang determined wavelets on Cantor dyadic group by using Multiresolution analysis method. In this paper we have given characterization of wavelet sets on Cantor dyadic group which provides another method for the construction of wavelets. All the wavelets originating from wavelet sets are not necessarily associated with a multiresolution analysis. We have also established relation between multiresolution analysis and wavelets determined from wavelet sets. [ABSTRACT FROM AUTHOR]

This note corrects three typographical errors in the paper Wavelet Sets Accumulating at the Origin, Real Anal. Exchange 35(2), 463-478. The corrections are listed according to the page of the original article. [ABSTRACT FROM AUTHOR]