Minimally Supported Frequency (MSF) d-Dilation Wavelets

In this paper, we provide a geometric construction of a symmetric 2n-interval minimally supported frequency (MSF) d-dilation wavelet set with d∈(1,∞) and characterize all symmetric d-dilation wavelet sets. We also provide two special kinds of symmetric d-dilation wavelet sets, one of which has 4m-intervals whereas the other has (4m+2)-intervals, for m∈N. In addition, we construct a family of d-dilation wavelet sets that has an infinite number of components.

On wavelet induced isomorphisms for reducing subspaces

In this paper, we adapt the notion of a wavelet induced isomorphism of [Formula: see text] associated with a wavelet set, introduced in [E. J. Ionascu, A new construction of wavelet sets, Real Anal. Exchange 28(2) (2002/03) 593–610], to the case of an [Formula: see text]-wavelet set, where [Formula: see text] is a reducing subspace [X. Dai and S. Lu, Wavelets in subspaces, Michigan Math. J. 43 (1996) 81–98]. We characterize all these wavelet induced isomorphisms similar to those given in Ionascu paper and provide specific examples of this theory in the case of symmetric [Formula: see text]-wavelet sets. Examples when [Formula: see text] is the classical Hardy space are also considered.

On fixed point sets of wavelet induced isomorphisms and frame induced monomorphisms

In this paper, we study fixed point sets of wavelet induced isomorphisms for symmetric wavelet sets. Also, introducing the notion of frame induced monomorphism for a frame wavelet set, as a generalization of the wavelet induced isomorphism for a wavelet set, we provide a construction of frame wavelet sets in [Formula: see text]. We also study fixed point sets of these maps.

On wavelet induced isomorphisms for joint (d, -d)-dilation wavelet and multiwavelet sets

For a dyadic wavelet set W, Ionascu [A new construction of wavelet sets, Real Anal. Exchange28 (2002) 593–610] obtained a measurable self-bijection on the interval [0, 1), called the wavelet induced isomorphism of [0, 1), denoted by [Formula: see text]. Extending the result for a d-dilation wavelet set, we characterize a joint (d, -d)-dilation wavelet set, where |d| is an integer greater than 1, in terms of wavelet induced isomorphisms. Its analogue for a joint (d, -d)-dilation multiwavelet set has also been provided. In addition, denoting by [Formula: see text], the wavelet induced isomorphism associated with a d-dilation wavelet set W, we show that for a joint (d, -d)-dilation wavelet set W, the measures of the fixed point sets of [Formula: see text] and [Formula: see text] are equal almost everywhere.

SMOOTH PARSEVAL FRAMES FOR L2(ℝ) AND GENERALIZATIONS TO L2(ℝd)

Wavelet set wavelets were the first examples of wavelets that may not have associated multiresolution analyses. Furthermore, they provided examples of complete orthonormal wavelet systems in L2(ℝd) which only require a single generating wavelet. Although work had been done to smooth these wavelets, which are by definition discontinuous on the frequency domain, nothing had been explicitly done over ℝd, d > 1. This paper, along with another one cowritten by the author, finally addresses this issue. Smoothing does not work as expected in higher dimensions. For example, Bin Han's proof of existence of Schwartz class functions which are Parseval frame wavelets and approximate Parseval frame wavelet set wavelets does not easily generalize to higher dimensions. However, a construction of wavelet sets in [Formula: see text] which may be smoothed is presented. Finally, it is shown that a commonly used class of functions cannot be the result of convolutional smoothing of a wavelet set wavelet.

The Existences of Frame Wavelet Set

In this paper, Let be a real expansive matrix, it mainly discusses the existences of frame wavelet set, we discuss the characterization of frame wavelet sets in, and several examples are presented, in order to deepen the understanding of frame wavelet set, which gives the two related theorems; we try to use an equivalent condition to describe frame scale sets, and give an equivalent description about a normalized frame scale set.

NON-MSF A-WAVELETS FROM A-WAVELET SETS

Generalizing the result of Bownik and Speegle [Approximation Theory X: Wavelets, Splines and Applications, Vanderbilt University Press, pp. 63–85, 2002], we provide plenty of non-MSF A-wavelets with the help of a given A-wavelet set. Further, by showing that the dimension function of the non-MSF A-wavelet constructed through an A-wavelet set W coincides with the dimension function of W, we conclude that the non-MSF A-wavelet and the A-wavelet set through which it is constructed possess the same nature as far as the multiresolution analysis is concerned. Some examples of non-MSF d-wavelets and non-MSF A-wavelets are also provided. As an illustration we exhibit a pathwise connected class of non-MSF non-MRA wavelets sharing the same wavelet dimension function.

CONSTRUCTING NON-MSF WAVELETS FROM GENERALIZED JOURNÉ WAVELET SETS

Dai and Larson [Mem. Amer. Math. Soc.134 (1998), no. 640] obtained a family of wavelet sets using the Journé wavelet set. In this paper, we expand this family and call its members to be generalized Journé wavelet sets. Furthermore, with the help of these wavelet sets, we provide a class of non-MSF wavelets which includes the one constructed by Vyas [Bull. Polish Acad. Sci. Math.57 (2009) 33–40]. Most of these non-MSF wavelets are found to be non-MRA.