Affine dual frames and Extension Principles

Atreas, Nikolaos; Melas, Antonios D.; Stavropoulos, Theodoros

doi:10.1016/j.acha.2013.02.003

Cited by 29 publications

(30 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We also prove that the Mixed Oblique Extension Principle characterizes dual multiwavelet frames. Our results extend recent characterizations of affine dual frames derived from scalar refinable functions obtained in [3].…”

supporting

confidence: 66%

“…He also established a connection between the Mixed Oblique Extension Principle and the former type of frames. Note that an equivalence between nonhomogeneous Parseval wavelet frames and their homogeneous counterparts in L 2 (R) for a scalar refinable function ϕ = φ was first established in [59, Theorem 2.3] and then was extended in [3]. Our main result, Theorem 1 stated below provides characterizations of affine dual frames generalizing these aforementioned results.…”

Section: Introductionmentioning

confidence: 56%

“…In the cases considered in both of these papers the Fundamental function is identically equal to one. Our viewpoint is in line with [3]. We focus on extending the results of [3] for refinable function vectors, thus underscoring the geometric role of the Fundamental and Mixed Fundamental functions.…”

Section: Introductionmentioning

confidence: 85%

See 2 more Smart Citations

Extension Principles for Dual Multiwavelet Frames of $$L_2(\mathbb {R}^s)$$ L 2 ( R s ) constructed from Multirefinable Generators

Atreas

Papadakis

Stavropoulos

2015

J Fourier Anal Appl

Self Cite

View full text Add to dashboard Cite

In this work we prove that any pair of homogeneous dual multiwavelet frames of L 2 (R s ) constructed from a pair of refinable function vectors gives rise to a pair of nonhomogeneous dual multiwavelet frames and vice versa. We also prove that the Mixed Oblique Extension Principle characterizes dual multiwavelet frames. Our results extend recent characterizations of affine dual frames derived from scalar refinable functions obtained in [3].

show abstract

supporting

confidence: 66%

Section: Introductionmentioning

confidence: 56%

See 1 more Smart Citation

Extension Principles for Dual Multiwavelet Frames of $$L_2(\mathbb {R}^s)$$ L 2 ( R s ) constructed from Multirefinable Generators

Atreas

Papadakis

Stavropoulos

2015

J Fourier Anal Appl

Self Cite

View full text Add to dashboard Cite

show abstract

“…They date back to [5,6,9,10,[12][13][14][15][16]21,22] and references therein. Due to their application potential in signal processing, the study of various extension principles has been attracting many researchers [23][24][25][26][27][28][29][30][31][32].…”

Section: An Overview Of Dual Wavelet Framesmentioning

confidence: 99%

Nonhomogeneous dual wavelet frames and mixed oblique extension principles in Sobolev spaces

Zhang

2017

Applicable Analysis

View full text Add to dashboard Cite

From the literature, it is known that nonhomogeneous dual wavelet frames admit fast wavelet transform as homogeneous ones, and have more designing freedom than homogeneous ones. In this paper, we give a characterization of nonhomogeneous dual wavelet frames in (H s (R d ), H −s (R d )), and using this characterization we derive a mixed oblique extension principle for such dual wavelet frames. ARTICLE HISTORY

show abstract

“…Due to many desirable properties of nonhomogeneous affine systems, it is of interest in both theory and application to investigate when a homogeneous affine system can be linked to a nonhomogeneous affine system. Along this direction, [2,3] and [22,Section 4.5] have studied the connections of a homogeneous tight/dual framelet to a nonhomogeneous tight/dual framelet, while the connection of a homogeneous wavelet with a multiresolution analysis has been discussed in [6,22,26,30,31,37,38,47] and references therein. In this paper we shall comprehensively study the connections of homogeneous wavelets and framelets with nonhomogeneous wavelets and framelets and their connections to the refinable structure.…”

Section: Introduction and Motivationsmentioning

confidence: 99%

Homogeneous wavelets and framelets with the refinable structure

Han¹

2017

Sci. China Math.

View full text Add to dashboard Cite

Homogeneous wavelets and framelets have been extensively investigated in the classical theory of wavelets and they are often constructed from refinable functions via the multiresolution analysis. On the other hand, nonhomogeneous wavelets and framelets enjoy many desirable theoretical properties and are often intrinsically linked to the refinable structure and multiresolution analysis. In this paper we shall provide a comprehensive study on connecting homogeneous wavelets and framelets to nonhomogeneous ones with the refinable structure. This allows us to understand better the structure of homogeneous wavelets and framelets as well as their connections to the refinable structure and multiresolution analysis.2010 Mathematics Subject Classification. 42A40, 42C15, 41A30. Key words and phrases. Homogeneous wavelets and framelets, nonhomogeneous wavelets and framelets, refinable structure, shift-invariant spaces, multiresolution analysis, Schur decomposition for Hermite matrices of measurable functions, singular value decomposition for matrices of measurable functions.

show abstract

Affine dual frames and Extension Principles

Cited by 29 publications

References 20 publications

Extension Principles for Dual Multiwavelet Frames of $$L_2(\mathbb {R}^s)$$ L 2 ( R s ) constructed from Multirefinable Generators

Extension Principles for Dual Multiwavelet Frames of $$L_2(\mathbb {R}^s)$$ L 2 ( R s ) constructed from Multirefinable Generators

Nonhomogeneous dual wavelet frames and mixed oblique extension principles in Sobolev spaces

Homogeneous wavelets and framelets with the refinable structure

Contact Info

Product

Resources

About