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

Li, Yun‐Zhang; Zhang, Jianping

doi:10.1080/00036811.2017.1298745

Cited by 12 publications

(6 citation statements)

References 44 publications

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“…Since our theory is derived from the wavelet multilevel sampling, it is necessary to introduce some mathematical materials about wavelet before proceeding further. Readers can refer to Li and Jia, Li and Zhang, and Li and Lan for more details on wavelets in Sobolev spaces.…”

Section: Recovery Of Riesz Transform By Box Spline Multilevel Samplingmentioning

confidence: 99%

Recovery of Riesz transform of functions in Sobolev space by wavelet multilevel sampling

Yang

Shang³

et al. 2018

Math Methods in App Sciences

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The explicit formula of the Riesz transform of the box spline B2(x, y) : = B2(x)B2(y) is given, where B2 is the cardinal B‐spline of order 2. By using the wavelet multilevel method, a sampling recovery scheme derived from B2 is established to recover the Riesz transform of the functions in Sobolev space Hsfalse(R2false) with s > 1. For any fixed level, our recovery is involved with a finite sum series. Since the Riesz transforms of some functions are continuous but scriptRB2 has numerical singularity at some points, it is necessary to eliminate the numerical singularity. We first establish the shift‐perturbation error estimate of the multilevel sampling approximation, derived from B2, to the functions in Hsfalse(R2false). By the perturbed approximation system, we give a method to eliminate the numerical singularity. Numerical simulations are conducted to test the recovery efficiency.

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Section: Recovery Of Riesz Transform By Box Spline Multilevel Samplingmentioning

confidence: 99%

Recovery of Riesz transform of functions in Sobolev space by wavelet multilevel sampling

Yang

Shang³

et al. 2018

Math Methods in App Sciences

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“…Li and Zhang in [17] generalized Proposition 4 to Sobolev space pairs ðH s ðℝ d Þ, H −s ðR d ÞÞ for nonhomogeneous dual wavelet frames: Proposition 5. Given s ∈ R, let X s ðψ 0 , ΨÞ and X −s ðψ 0 , ΨÞ be Bessel sequences in H s ðℝ d Þ and H −s ðℝ d Þ, respectively.…”

Section: Introductionmentioning

confidence: 99%

A Class of Weak Dual Wavelet Frames for Reducing Subspaces of Sobolev Spaces

Zhang

2022

Journal of Function Spaces

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In recent years, dual wavelet frames derived from a pair of refinable functions have been widely studied by many researchers. However, the requirement of the Bessel property of wavelet systems is always required, which is too technical and artificial. In present paper, we will relax this restriction and only require the integer translation of the wavelet functions (or refinable functions) to form Bessel sequences. For this purpose, we introduce the notion of weak dual wavelet frames. And for generality, we work under the setting of reducing subspaces of Sobolev spaces, we characterize a pair of weak dual wavelet frames, and by using this characterization, we obtain a mixed oblique extension principle for such weak dual wavelet frames.

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“…They in [20] also characterized the Sobolev space by using non-stationary tight wavelet frames for L 2 (R). Li and Zhang in [26] characterized the nonhomogeneous dual wavelet frames in Sobolev space and derived the mixed oblique extension principle. Li and Jia in [24] investigated the properties of weak nonhomogeneous wavelet bi-frames(WNWBF) in the reducing subspaces of a pair of dual Sobolev spaces and constructed the WNWBF .…”

Section: Introductionmentioning

confidence: 99%

Irregular wavelet/Gabor frames in Sobolev space

Tian

2022

Preprint

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The theory of regular wavelet/Gabor frames has been extensively investigated, including the sufficient and necessary conditions for regular wavelet/Gabor systems to be frames and the perturbation theorem. This paper addresses the irregular wavelet/Gabor systems in Sobolev space Hs(R). The sufficient and necessary conditions for irregular wavelet/Gabor systems to be frames are presented. As corollaries, under certain restrictions on the support, we also give the characterization of irregular wavelet/Gabor systems to be frames. Finally, we discuss the perturbation theorem of irregular wavelet/Gabor frames.

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

Cited by 12 publications

References 44 publications

Recovery of Riesz transform of functions in Sobolev space by wavelet multilevel sampling

Recovery of Riesz transform of functions in Sobolev space by wavelet multilevel sampling

A Class of Weak Dual Wavelet Frames for Reducing Subspaces of Sobolev Spaces

Irregular wavelet/Gabor frames in Sobolev space

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