Frame Wavelet Sets Research Articles

Frame wavelet set and frequency frame wavelet in $$L^2({\mathbb {R}}^n)$$

Frame wavelet set and frequency frame wavelet in $$L^2({\mathbb {R}}^n)$$

Construction of three interval frame scaling sets

In this paper, a construction of three interval frame scaling sets is described and certain classes of normalized tight frame wavelet sets are also constructed with the help of these three interval frame scaling sets. We also generalize this construction to obtain [Formula: see text]-interval frame scaling sets.

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 Normalized Tight Frame Wavelet Sets

We determine two-interval normalized tight frame wavelet sets for real dilation d 2 (1;1), and characterize all symmetric normalized tight frame wavelet sets. We also construct a normalized tight frame wavelet set which has an innite 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.

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.

Smooth Functions Associated with Wavelet Sets on ℝ d , d≥1, and Frame Bound Gaps

The theme is to smooth characteristic functions of Parseval frame wavelet sets by convolution in order to obtain implementable, computationally viable, smooth wavelet frames. We introduce the following: a new method to improve frame bound estimation; a shrinking technique to construct frames; and a nascent theory concerning frame bound gaps. The phenomenon of a frame bound gap occurs when certain sequences of functions, converging in L 2 to a Parseval frame wavelet, generate systems with frame bounds that are uniformly bounded away from 1. We prove that smoothing a Parseval frame wavelet set wavelet on the frequency domain by convolution with elements of an approximate identity produces a frame bound gap. Furthermore, the frame bound gap for such frame wavelets in L 2 (R d ) increases and converges as d increases.

Frame scaling function sets and frame wavelet sets in [formula omitted

In this paper, we classify frame wavelet sets and frame scaling function sets in higher dimensions. Firstly, we obtain a necessary condition for a set to be the frame wavelet sets. Then, we present a necessary and sufficient condition for a set to be a frame scaling function set. We give a property of frame scaling function sets, too. Some corresponding examples are given to prove our theory in each section.

Frame wavelets with matrix dilations in L2( Rn)

In this paper, we obtain a necessary condition and a sufficient condition for a general wavelet with a matrix dilation to be a frame in L 2( R n ). We extend the concept of frame wavelet sets and give their constructions. Several examples are presented and compared with some known results.

Frame wavelets with frame set support in the frequency domain

In this paper, we study a special class of frame wavelets. The supports of the Fourier transforms of these frame wavelets are the so called frame wavelet sets. Our results provide ways to construct frame wavelets not known previously. In particular, for any given positive number $\alpha$, we show that there exist various frame wavelets whose Fourier transforms are supported by a set of measure $\alpha$.

Frame wavelet sets in [formula omitted

In this paper, we discuss the characterization of frame wavelet sets. We extend some results obtained earlier in the one-dimensional case. More specifically, we completely characterize tight frame wavelet sets in higher dimensions and obtain some necessary conditions and sufficient conditions for a set E to be a frame wavelet set in R d . Several examples are presented and compared with those in the one-dimensional case. Using our results, one can easily construct various frame wavelet sets.