Revista Matemática Iberoamericana

Harmonic Analysis of the space BV

Albert Cohen, Wolfgang Dahmen, Ingrid Daubechies, and Ronald DeVore

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Abstract

We establish new results on the space BV of functions with bounded variation. While it is well known that this space admits no unconditional basis, we show that it is "almost" characterized by wavelet expansions in the following sense: if a function $f$ is in BV, its coefficient sequence in a BV normalized wavelet basis satisfies a class of weak-$\ell^1$ type estimates. These weak estimates can be employed to prove many interesting results. We use them to identify the interpolation spaces between BV and Sobolev or Besov spaces, and to derive new Gagliardo-Nirenberg-type inequalities.

Article information

Source
Rev. Mat. Iberoamericana, Volume 19, Number 1 (2003), 235-263.

Dates
First available in Project Euclid: 31 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1049123087

Mathematical Reviews number (MathSciNet)
MR1993422

Zentralblatt MATH identifier
1044.42028

Subjects
Primary: 42C40: Wavelets and other special systems 46B70: Interpolation between normed linear spaces [See also 46M35] 26B35: Special properties of functions of several variables, Hölder conditions, etc. 42B25: Maximal functions, Littlewood-Paley theory

Keywords
Bounded variation wavelet decompositions weak $\ell_1$ K-functionals interpolation Gagliardo-Nirenberg inequalities Besov spaces

Citation

Cohen, Albert; Dahmen, Wolfgang; Daubechies, Ingrid; DeVore, Ronald. Harmonic Analysis of the space BV. Rev. Mat. Iberoamericana 19 (2003), no. 1, 235--263. https://projecteuclid.org/euclid.rmi/1049123087


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