Revista Matemática Iberoamericana

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Volume 19, Issue 1, 2003, pp. 235–263
DOI: 10.4171/RMI/345

Published online: 2003-04-30

Harmonic Analysis of the space BV

Albert Cohen[1], Wolfgang Dahmen[2], Ingrid Daubechies[3] and Ronald A. DeVore[4]

(1) Université Pierre et Marie Curie, Paris, France
(2) Technische Hochschule Aachen, Germany
(3) Duke University, Durham, USA
(4) Texas A&M University, College Station, USA

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.

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

Cohen Albert, Dahmen Wolfgang, Daubechies Ingrid, DeVore Ronald: Harmonic Analysis of the space BV. Rev. Mat. Iberoamericana 19 (2003), 235-263. doi: 10.4171/RMI/345