scholarly journals Tight wavelet frames in Lebesgue and Sobolev spaces

2004 ◽  
Vol 2 (3) ◽  
pp. 227-252 ◽  
Author(s):  
L. Borup ◽  
R. Gribonval ◽  
M. Nielsen
Keyword(s):  
Sobolev Space ◽  
Jackson Inequality ◽  
Wavelet Frame ◽  
Approximation Rate ◽  
Wavelet Frames ◽  
Atomic Decompositions ◽  
Approximation Spaces ◽  
Tight Wavelet Frame ◽  
Frame Systems ◽  
Term Approximation

We study tight wavelet frame systems inLp(ℝd)and prove that such systems (under mild hypotheses) give atomic decompositions ofLp(ℝd)for1≺p≺∞. We also characterizeLp(ℝd)and Sobolev space norms by the analysis coefficients for the frame. We consider Jackson inequalities for bestm-term approximation with the systems inLp(ℝd)and prove that such inequalities exist. Moreover, it is proved that the approximation rate given by the Jackson inequality can be realized by thresholding the frame coefficients. Finally, we show that in certain restricted cases, the approximation spaces, for bestm-term approximation, associated with tight wavelet frames can be characterized in terms of (essentially) Besov spaces.

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.

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.

On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle

2014 ◽  
Vol 57 (2) ◽  
pp. 254-263 ◽  
Author(s):  
Ole Christensen ◽  
Hong Oh Kim ◽  
Rae Young Kim
Keyword(s):  
Extension Principle ◽  
Sufficient Condition ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Wavelet System ◽  
Number Of Generators ◽  
Unitary Extension

AbstractThe unitary extension principle (UEP) by A. Ron and Z. Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.

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 Feichtinger Conjecture for Wavelet Frames, Gabor Frames and Frames of Translates

2006 ◽  
Vol 58 (6) ◽  
pp. 1121-1143 ◽  
Author(s):  
Marcin Bownik ◽  
Darrin Speegle
Keyword(s):  
Regularity Condition ◽  
Finite Union ◽  
Gabor Frame ◽  
Gabor Frames ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Wavelet System ◽  
Weak Regularity

AbstractThe Feichtinger conjecture is considered for three special families of frames. It is shown that if a wavelet frame satisfies a certain weak regularity condition, then it can be written as the finite union of Riesz basic sequences each of which is a wavelet system. Moreover, the above is not true for general wavelet frames. It is also shown that a sup-adjoint Gabor frame can be written as the finite union of Riesz basic sequences. Finally, we show how existing techniques can be applied to determine whether frames of translates can be written as the finite union of Riesz basic sequences. We end by giving an example of a frame of translates such that any Riesz basic subsequence must consist of highly irregular translates.

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 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.

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