Maximal functions measuring smoothness

DeVore, Ronald A.; Sharpley, Robert

doi:10.1090/memo/0293

Cited by 198 publications

(156 citation statements)

References 8 publications

Supporting

Mentioning

150

Contrasting

Unclassified

Order By: Relevance

“…The first one allows to draw a comparison between E D (t) and D −2α via a term that does not depend on t ,α,2 anymore but on t ,α,p(α) . It is the discrete analogue of a particular case of Theorem 4.3. of [13].…”

Section: Proof Of Theorem 61: the Main Linesmentioning

confidence: 95%

Estimating a discrete distributionviahistogram selection

Akakpo

2011

ESAIM: PS

View full text Add to dashboard Cite

Abstract. Our aim is to estimate the joint distribution of a finite sequence of independent categorical variables. We consider the collection of partitions into dyadic intervals and the associated histograms, and we select from the data the best histogram by minimizing a penalized least-squares criterion. The choice of the collection of partitions is inspired from approximation results due to DeVore and Yu. Our estimator satisfies a nonasymptotic oracle-type inequality and adaptivity properties in the minimax sense. Moreover, its computational complexity is only linear in the length of the sequence. We also use that estimator during the preliminary stage of a hybrid procedure for detecting multiple change-points in the joint distribution of the sequence. That second procedure still satisfies adaptivity properties and can be implemented efficiently. We provide a simulation study and apply the hybrid procedure to the segmentation of a DNA sequence.Mathematics Subject Classification. 62G05, 62C20, 41A17.

show abstract

Section: Proof Of Theorem 61: the Main Linesmentioning

confidence: 95%

Estimating a discrete distributionviahistogram selection

Akakpo

2011

ESAIM: PS

View full text Add to dashboard Cite

show abstract

“…We shall define an extension operator If (similar to that introduced in [4]) which extends each function / £ Lp(Si) to all of Rd and has the property that if / £ B^(LP(Q)), then g'f £ B^(Lp(Rd)) (with suitable restrictions on a, p, q , and Í2). We assume at the outset that £2 is a Lipschitz graph domain and treat more general domains in the next section.…”

Section: Extension Operators Local Approximation and Modulimentioning

confidence: 99%

“…Similarly we denote by Fc the Whitney decomposition of Q.c\dQ. Then, According to [7, p. 170 Properties (i)-(iv) and (vi) are proved in [7], while a proof of (v) can be found in [4]. Here m is an arbitrary integer and c depends only on d, Q, and m .…”

Section: Extension Operators Local Approximation and Modulimentioning

confidence: 99%

“…We speak of Qs as being the cube symmetric to ß across d£l. The symmetric cubes Of were introduced in [4, p. 77] and we recall now some of their properties proved in [4]. While ß and Qs need not have the same size, they are comparable; i.e.…”

Section: Extension Operators Local Approximation and Modulimentioning

confidence: 99%

“…While ß and Qs need not have the same size, they are comparable; i.e. there is a constant C > 0 for which there holds (for a proof see [4]). …”

Section: Extension Operators Local Approximation and Modulimentioning

confidence: 99%

See 2 more Smart Citations

Besov spaces on domains in ${\bf R}\sp d$

DeVore¹,

Sharpley²

1993

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

Adaptive Wavelet Techniques in Numerical Simulation

Cohen

Dahmen

DeVore

2004

Encyclopedia of Computational Mechanics

View full text Add to dashboard Cite

This chapter highlights recent developments concerning adaptive wavelet methods for time dependent and stationary problems. The first problem class focusses on hyperbolic conservation laws where wavelet concepts exploit sparse representations of the conserved variables. Regarding the second problem class, we begin with matrix compression in the context of boundary integral equations where the key issue is now to obtain sparse representations of (global) operators like singular integral operators in wavelet coordinates. In the remainder of the chapter a new fully adaptive algorithmic paradigm along with some analysis concepts are outlined which, in particular, works for nonlinear problems and where the sparsity of both, functions and operators, is exploited.

show abstract

Maximal functions measuring smoothness

Cited by 198 publications

References 8 publications

Estimating a discrete distributionviahistogram selection

Estimating a discrete distributionviahistogram selection

Besov spaces on domains in ${\bf R}\sp d$

Adaptive Wavelet Techniques in Numerical Simulation

Contact Info

Product

Resources

About