scholarly journals Nonhomogeneous Wavelet Dual Frames and Extension Principles in Reducing Subspaces

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Jianping Zhang ◽  
Huifang Jia
Keyword(s):  
Wavelet Frames ◽  
Dual Frames ◽  
Reducing Subspaces ◽  
Extension Principles

It can be seen from the literature that nonhomogeneous wavelet frames are much simpler to characterize and construct than homogeneous ones. In this work, we address such problems in reducing subspaces of L2ℝd. A characterization of nonhomogeneous wavelet dual frames is obtained, and by using the characterization, an MOEP and an MEP are derived under general assumptions for such wavelet dual frames.

Weak nonhomogeneous wavelet dual frames for Walsh reducing subspace of L2(ℝ+)

2021 ◽  
pp. 2150040
Author(s):  
Yan Zhang ◽  
Yun-Zhang Li
Keyword(s):  
Extension Principle ◽  
Transform Domain ◽  
Wavelet Frames ◽  
Dual Frames ◽  
Reducing Subspace ◽  
Reducing Subspaces ◽  
Extension Principles ◽  
Dual Wavelet Frames ◽  
Dual Wavelet ◽  
Fourier Transform Domain

In wavelet analysis, refinable functions are the bases of extension principles for constructing (weak) dual wavelet frames for [Formula: see text] and its reducing subspaces. This paper addresses refinable function-based dual wavelet frames construction in Walsh reducing subspaces of [Formula: see text]. We obtain a Walsh–Fourier transform domain characterization for weak [Formula: see text]-adic nonhomogeneous dual wavelet frames; and present a mixed oblique extension principle for constructing weak [Formula: see text]-adic nonhomogeneous dual wavelet frames in Walsh reducing subspaces of [Formula: see text].

scholarly journals A Characterization of Multi-Wavelet Dual Frames in Sobolev Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Jianping Zhang ◽  
Jiajia Li
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Wavelet Frames ◽  
Dual Frames ◽  
The Past

For the past few years, wavelet and multi-wavelet frames have attracted interest from researchers. In this paper, we address some of these problems in the setting of the Sobolev space, and characterize of multi-wavelet dual frames in these spaces by using a pair of equations.

scholarly journals On Parseval Wavelet Frames via Multiresolution Analyses in

2019 ◽  
Vol 63 (1) ◽  
pp. 157-172
Author(s):  
A. San Antolín
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Extension Principle ◽  
Refinable Functions ◽  
Wavelet Frames ◽  
Approximate Continuity ◽  
Reducing Subspaces ◽  
The Fourier Transform ◽  
Frame Multiresolution Analysis

AbstractWe give a characterization of all Parseval wavelet frames arising from a given frame multiresolution analysis. As a consequence, we obtain a description of all Parseval wavelet frames associated with a frame multiresolution analysis. These results are based on a version of Oblique Extension Principle with the assumption that the origin is a point of approximate continuity of the Fourier transform of the involved refinable functions. Our results are written for reducing subspaces.

TIGHT FRAME CHARACTERIZATION OF MULTIWAVELET VECTOR FUNCTIONS IN TERMS OF THE POLYPHASE MATRIX

2009 ◽  
Vol 07 (01) ◽  
pp. 9-21 ◽  
Author(s):  
JENS KROMMWEH
Keyword(s):  
Tight Frame ◽  
Extension Principle ◽  
Wavelet Frames ◽  
Unitary Extension ◽  
Directional Wavelets ◽  
Extension Principles ◽  
Polyphase Matrix ◽  
Common Extension ◽  
Polyphase Representation

The extension principles play an important role in characterizing and constructing of wavelet frames. The common extension principles, the unitary extension principle (UEP) or the oblique extension principle (OEP), are based on the unitarity of the modulation matrix. In this paper, we state the UEP and OEP for refinable function vectors in the polyphase representation. Finally, we apply our results to directional wavelets on triangles which we have constructed in a previous work. We will show that the wavelet system generates a tight frame for L2(ℝ2).

Characterization of matrix Fourier multipliers for A-dilation Parseval multi-wavelet frames

2015 ◽  
Vol 13 (06) ◽  
pp. 1550051
Author(s):  
Yongdong Huang ◽  
Fengjuan Zhu
Keyword(s):  
Fourier Multipliers ◽  
Frame Theory ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Integral Matrix ◽  
Numerical Examples ◽  
Basic Properties ◽  
New Formula

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

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