scholarly journals Minimum-Energy Bivariate Wavelet Frame with Arbitrary Dilation Matrix

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Fengjuan Zhu ◽  
Qiufu Li ◽  
Yongdong Huang
Keyword(s):  
Scaling Function ◽  
Minimum Energy ◽  
Sufficient Conditions ◽  
Reconstruction Algorithm ◽  
Wavelet Function ◽  
Necessary Condition ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Numerical Examples ◽  
Dilation Matrix

In order to characterize the bivariate signals, minimum-energy bivariate wavelet frames with arbitrary dilation matrix are studied, which are based on superiority of the minimum-energy frame and the significant properties of bivariate wavelet. Firstly, the concept of minimum-energy bivariate wavelet frame is defined, and its equivalent characterizations and a necessary condition are presented. Secondly, based on polyphase form of symbol functions of scaling function and wavelet function, two sufficient conditions and an explicit constructed method are given. Finally, the decomposition algorithm, reconstruction algorithm, and numerical examples are designed.

scholarly journals The necessary and sufficient conditions for wavelet frames in Sobolev space over local fields

2021 ◽  
Vol 39 (3) ◽  
pp. 81-92
Author(s):  
Ashish Pathak ◽  
Dileep Kumar ◽  
Guru P. Singh
Keyword(s):  
Sobolev Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Local Fields ◽  
Necessary Condition ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Necessary And Sufficient

In this paper we construct wavelet frame on Sobolev space. A necessary condition and sufficient conditions for wavelet frames in Sobolev space are given.

Necessary condition and sufficient conditions for nonuniform wavelet frames in L2(K)

2018 ◽  
Vol 16 (01) ◽  
pp. 1850005 ◽  
Author(s):  
M. Younus Bhat
Keyword(s):  
Multiresolution Analysis ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Necessary Condition ◽  
Wavelet Basis ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Constructive Algorithm ◽  
Spectral Pairs ◽  
Funct Anal

A constructive algorithm based on the theory of spectral pairs for constructing nonuniform wavelet basis in [Formula: see text] was considered by Gabardo and Nashed [Nonuniform multiresolution analysis and spectral pairs, J. Funct. Anal. 158 (1998) 209–241]. In this setting, the associated translation set is a spectrum [Formula: see text] which is not necessarily a group nor a uniform discrete set, given [Formula: see text] where [Formula: see text] (an integer) and [Formula: see text] is an odd integer with [Formula: see text] such that [Formula: see text] and [Formula: see text] are relatively prime and [Formula: see text] is the set of all integers. The objective of this paper is to construct nonuniform wavelet frame on local fields. A necessary condition and four sufficient conditions for nonuniform wavelet frame on local fields are given.

Constructing totally interpolating wavelet frame systems

2015 ◽  
Vol 13 (03) ◽  
pp. 1550017
Author(s):  
Baobin Li
Keyword(s):  
Filter Banks ◽  
Scaling Function ◽  
Tight Frame ◽  
Special Structure ◽  
Dual Frame ◽  
Wavelet Frame ◽  
Continuous Linear ◽  
Wavelet Frames ◽  
Frame Systems ◽  
The Scaling Function

The system of totally interpolating wavelet frames is discussed in this paper, in which both the scaling function and one of wavelet functions are interpolating. It will be shown that corresponding filter banks possess the special structure, and the parametrization of filter banks is present. Moreover, we show that when considering tight frame systems with two generators, the Ron–Shen's continuous-linear-spline-based tight frame is the only one with totally interpolating property and symmetry. But in the dual frame context, more good examples of bi-frames with symmetric/antisymmetric property can be obtained and constructed, which in particular, include frames with the uniform symmetry.

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.

scholarly journals Frames associated with shift invariant spaces on positive half line

2021 ◽  
Vol 13 (1) ◽  
pp. 23-44
Author(s):  
Owais Ahmad ◽  
Mobin Ahmad ◽  
Neyaz Ahmad
Keyword(s):  
Sufficient Conditions ◽  
Gabor Frames ◽  
Half Line ◽  
Necessary Condition ◽  
Wavelet Frames ◽  
Invariant Space ◽  
Unified Approach ◽  
Shift Invariant Spaces ◽  
Invariant Spaces ◽  
Positive Half

Abstract In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

scholarly journals A Short Note on Wavelet Frames Based on FMRA on Local Fields

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
M. Younus Bhat
Keyword(s):  
Multiresolution Analysis ◽  
Positive Characteristic ◽  
Minimum Energy ◽  
Short Note ◽  
Explicit Construction ◽  
Local Fields ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Frame Multiresolution Analysis ◽  
Field Of Positive Characteristic

The concept of frame multiresolution analysis (FMRA) on local fields of positive characteristic was given by Shah in his paper, Frame Multiresolution Analysis on Local Fields published by Journal of Operators. The author has studied the concept of minimum-energy wavelet frames on these prime characteristic fields. We continued the studies based on frame multiresolution analysis and minimum-energy wavelet frames on local fields of positive characteristic. In this paper, we introduce the notion of the construction of minimum-energy wavelet frames based on FMRA on local fields of positive characteristic. We provide a constructive algorithm for the existence of the minimum-energy wavelet frame on the local field of positive characteristic. An explicit construction of the frames and bases is given. In the end, we exhibit an example to illustrate our algorithm.

A Study of Binary Minimum-Energy Shortly Supported Wavelet Frames Associated with a Scaling Function

2011 ◽  
Vol 219-220 ◽  
pp. 500-503
Author(s):  
Qing Jiang Chen ◽  
Gai Hu
Keyword(s):  
Explicit Formula ◽  
Multiresolution Analysis ◽  
Scaling Function ◽  
Minimum Energy ◽  
Research Field ◽  
Sufficient Condition ◽  
Wavelet Frames ◽  
Existence Criterion ◽  
Generalized Multiresolution Analysis ◽  
Active Research

Frames have become the focus of active research field, both in theory and in applications. In the article, the binary minimum-energy wavelet frames and frame multiresolution resolution are introduced. A precise existence criterion for minimum-energy frames in terms of an ineqity conditi- -on on the Laurent poly-nomial symbols of the filter functions is provided. An explicit formula for designing minimum-energy frames is also established. The sufficient condition for the existence of affine pseudoframes is obtained by virtue of a generalized multiresolution analysis. The pyramid de -composition scheme is established based on such a generalized multiresolution structure.

scholarly journals Minimum-Energy Multiwavelet Frames with Arbitrary Integer Dilation Factor

2012 ◽  
Vol 2012 ◽  
pp. 1-37 ◽  
Author(s):  
Yongdong Huang ◽  
Qiufu Li ◽  
Ming Li
Keyword(s):  
Necessary Conditions ◽  
Minimum Energy ◽  
Sufficient Conditions ◽  
Spline Functions ◽  
Arbitrary Integer ◽  
B Spline ◽  
Numerical Examples ◽  
Dilation Factor ◽  
Organically Combine

In order to organically combine the minimum-energy frame with the significant properties of multiwavelets, minimum-energy multiwavelet frames with arbitrary integer dilation factor are studied. Firstly, we define the concept of minimum-energy multiwavelet frame with arbitrary dilation factor and present its equivalent characterizations. Secondly, some necessary conditions and sufficient conditions for minimum-energy multiwavelet frame are given. Thirdly, the decomposition and reconstruction formulas of minimum-energy multiwavelet frame with arbitrary integer dilation factor are deduced. Finally, we give several numerical examples based on B-spline functions.

Fractional wavelet frames in L2(ℝ)

2018 ◽  
Vol 21 (2) ◽  
pp. 399-422 ◽  
Author(s):  
Firdous A. Shah ◽  
Lokenath Debnath
Keyword(s):  
Fourier Transform ◽  
Fractional Fourier Transform ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Wavelet Frames

Abstract The objective of this paper is to construct fractional wavelet frames in L2(ℝ). A necessary condition and four sufficient conditions for fractional wavelet frames are given by virtue of fractional Fourier transform. The proposed inequalities generalize all the classical wavelet inequalities when θ = π/2. An example is presented at the end.

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