Function Spaces and Wavelets on Domains

Triebel, Hans

doi:10.4171/019

Cited by 195 publications

(300 citation statements)

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“…More details on the adaptation of wavelet bases for periodic domains can be found in [11,Section 1.3]. In fact, we follow the notation of that book.…”

Section: A Daubechies Wavelets On the Circlementioning

confidence: 99%

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Compressibility of symmetric-α-stable processes

Ward

Fageot

Unser

2015

2015 International Conference on Sampling Theory and Applications (SampTA)

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Abstract-Within a deterministic framework, it is well known that n-term wavelet approximation rates of functions can be deduced from their Besov regularity. We use this principle to determine approximation rates for symmetric-α-stable (SαS) stochastic processes. First, we characterize the Besov regularity of SαS processes. Then the n-term approximation rates follow. To capture the local smoothness behavior, we consider sparse processes defined on the circle that are solutions of stochastic differential equations.

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“…More details on the adaptation of wavelet bases for periodic domains can be found in [11,Section 1.3]. In fact, we follow the notation of that book.…”

Section: A Daubechies Wavelets On the Circlementioning

confidence: 99%

“…Criteria for unconditional convergence and many additional properties can be derived. We refer the interested reader to [11,Section 1.3].…”

Section: B Besov Spaces Of Zero Meanmentioning

confidence: 99%

Compressibility of symmetric-α-stable processes

Ward

Fageot

Unser

2015

2015 International Conference on Sampling Theory and Applications (SampTA)

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“…Roughly speaking, if f ∈ B s pq then we could say that 'derivatives up to order s are in L p ', see e.g. [13,14]. The parameter q is for fine tuning, and as the solution is not very sensitive to this parameter we choose here p = q, which leads to a simpler form for the norm.…”

Section: Wavelet-based Besov Space Priorsmentioning

confidence: 99%

“…We give here only a brief description of wavelets; more details can be found, e.g., in [13,14]. The wavelet expansion for a function f : R n → R is given by…”

Section: Wavelet-based Besov Space Priorsmentioning

confidence: 99%

Evaluation of the areal material distribution of paper from its optical transmission image

Takalo

Timonen²,

Sampo³

et al. 2011

Eur. Phys. J. Appl. Phys.

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The goal of this study was to evaluate the areal mass distribution (defined as the x-ray transmission image) of paper from its optical transmission image. A Bayesian inversion framework was used in the related deconvolution process so as to combine indirect optical information with a priori knowledge about the type of paper imaged. The a priori knowledge was expressed in the form of an empirical Besov space prior distribution constructed in a computationally effective way using the wavelet transform. The estimation process took the form of a large-scale optimization problem, which was in turn solved using the gradient descent method of Barzilai and Borwein. It was demonstrated that optical transmission images can indeed be transformed so as to fairly closely resemble the ones that reflect the true areal distribution of mass. Furthermore, the Besov space prior was found to give better results than the classical Gaussian smoothness prior (here equivalent to Tikhonov regularization).

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“…Fractional extension results are essential to improve some fractional embedding theorems and have been discussed by many people such as Nezza-Palatulli-Valdinoci [6], Shvartsman [17], Triebel [19] and Zhou [21]. It is well-known that the space…”

Section: Preliminariesmentioning

confidence: 99%