Harmonic Analysis of the space BV

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Harmonic Analysis of the space BV

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

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

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.

Primary Subjects: 42C40, 46B70, 26B35, 42B25

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

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.rmi/1049123087
Mathematical Reviews number (MathSciNet): MR1993422
Zentralblatt MATH identifier: 02005274

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