CHARACTERIZATION OF SCALING FUNCTIONS IN A FRAME MULTIRESOLUTION ANALYSIS IN $H^{2}_{G}$

2008 ◽  
pp. 118-140 ◽  
Author(s):  
K. S. KAZARIAN ◽  
A. SAN ANTOLÍN
Keyword(s):  
Multiresolution Analysis ◽  
Scaling Functions ◽  
Frame Multiresolution Analysis

scholarly journals Frame Multiresolution Analysis on Local Fields of Positive Characteristic

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Firdous A. Shah
Keyword(s):  
Multiresolution Analysis ◽  
Positive Characteristic ◽  
Scaling Function ◽  
Local Fields ◽  
Wavelet Frames ◽  
Shift Invariant Spaces ◽  
Frame Multiresolution Analysis ◽  
Invariant Spaces ◽  
The Scaling Function

We present a notion of frame multiresolution analysis on local fields of positive characteristic based on the theory of shift-invariant spaces. In contrast to the standard setting, the associated subspace V0 of L2(K) has a frame, a collection of translates of the scaling function φ of the form φ(·-u(k)):k∈N0, where N0 is the set of nonnegative integers. We investigate certain properties of multiresolution subspaces which provides the quantitative criteria for the construction of frame multiresolution analysis (FMRA) on local fields of positive characteristic. Finally, we provide a characterization of wavelet frames associated with FMRA on local field K of positive characteristic using the shift-invariant space theory.

Multiscaling frame multiresolution analysis and associated wavelet frames

2020 ◽  
Vol 18 (03) ◽  
pp. 2050009
Author(s):  
Xiaojiang Yu
Keyword(s):  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
Frame Theory ◽  
Dual Frame ◽  
Scaling Functions ◽  
Wavelet Frames ◽  
Multiplicity Formula ◽  
Frame Multiresolution Analysis ◽  
Necessary And Sufficient

Frame multiresolution analysis (FMRA) in [Formula: see text] is an important topic in frame theory and its applications. In this paper, we consider the so-called multiscaling FMRA in [Formula: see text], which has matrix dilations and a finite number of scaling functions. This framework is a generalization of the theories both on monoscaling FMRA and on the classical MRA of multiplicity [Formula: see text]. We characterize wavelet frames and Parseval wavelet frames for [Formula: see text] under the circumstances that they can be associated with a multiscaling FMRA. We give two necessary and sufficient conditions for given functions [Formula: see text] in [Formula: see text] to be multiframe generators of [Formula: see text]. Especially, the second condition depends on the multiscaling FMRA and [Formula: see text] only, does not require the existence of other functions, and is relatively easier to verify. Moreover, for any finitely-generated frame of integer translates, we give explicitly the Fourier transforms of the generators of its canonical dual frame. We illustrate the implementation and an application of the theory with an example.

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.

scholarly journals Characterization of Frequency Domains of Bandlimited Frame Multiresolution Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1050
Author(s):  
Zhihua Zhang
Keyword(s):  
Frequency Domain ◽  
Multiresolution Analysis ◽  
Background Noise ◽  
Sparse Reconstruction ◽  
Time Frequency ◽  
Processing Data ◽  
Narrowband Signal ◽  
Frequency Domains ◽  
Frame Multiresolution Analysis

Framelets have been widely used in narrowband signal processing, data analysis, and sampling theory, due to their resilience to background noise, stability of sparse reconstruction, and ability to capture local time-frequency information. The well-known approach to construct framelets with useful properties is through frame multiresolution analysis (FMRA). In this article, we characterize the frequency domain of bandlimited FMRAs: there exists a bandlimited FMRA with the support of frequency domain G if and only if G satisfies G⊂2G, ⋃m2mG≅Rd, and G\G2⋂G2+2πν≅∅(ν∈Zd).

scholarly journals Characterization of scaling functions in a multiresolution analysis

2004 ◽  
Vol 133 (4) ◽  
pp. 1013-1023 ◽  
Author(s):  
P. Cifuentes ◽  
K. S. Kazarian ◽  
A. San Antolín
Keyword(s):  
Multiresolution Analysis ◽  
Scaling Functions

On scaling functions of non-uniform multiresolution analysis in L2(ℝ)

2019 ◽  
Vol 18 (02) ◽  
pp. 1950055
Author(s):  
Hari Krishan Malhotra ◽  
Lalit Kumar Vashisht
Keyword(s):  
Frequency Domain ◽  
Multiresolution Analysis ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Scaling Functions ◽  
Necessary And Sufficient

The main purpose of this paper is to provide a characterization of scaling functions for non-uniform multiresolution analysis (NUMRA, in short). Some necessary and sufficient conditions for scaling functions of wavelet NUMRA in the frequency domain are also obtained.

scholarly journals Local analysis of frame multiresolution analysis with a general dilation matrix

2003 ◽  
Vol 67 (2) ◽  
pp. 285-295 ◽  
Author(s):  
Hong Oh Kim ◽  
Rae Young Kim ◽  
Jae Kun Lim
Keyword(s):  
Multiresolution Analysis ◽  
Local Analysis ◽  
Scaling Functions ◽  
Invariant Space ◽  
Wavelet Set ◽  
Dilation Matrix ◽  
Frame Multiresolution Analysis ◽  
Orthogonal Frame ◽  
Shift Invariant Space

A multivariate semi-orthogonal frame multiresolution analysis with a general integer dilation matrix and multiple scaling functions is considered. We first derive the formulas of the lengths of the inital (central) shift-invariant space V0 and the next dilation space V1, and, using these formulas, we then address the problem of the number of the elements of a wavelet set, that is, the length of the shift-invariant space W0 := V1 ⊖ V0. Finally, we show that there does not exist a ‘genuine’ frame multiresolution analysis for which V0 and V1 are quasi-stable spaces satisfying the usual length condition.

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