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Greedy wavelet projections are bounded on BV. (English) Zbl 1134.42022

This paper gives an affirmative answer of a conjecture of Y. Meyer [Oscillating patterns in image processing and nonlinear evolution equations. University Lecture Series. 22 (2001; Zbl 0987.35003)]. Let \(\varphi\) be a compactly supported univariate scaling function in the space \(BV(\mathbb{R})\) (of functions of bounded variations on \(\mathbb{R}\)) which generates the compactly supported orthogonal wavelet \(\psi\). For \(d \geq 2\), we consider the multivariate orthogonal wavelet system \((\psi_\lambda)_{\lambda \in \Delta}\) obtained from \(\varphi\) and \(\psi\), and normalized in \(BV(\mathbb{R}^d)\) (i.e., \(|\psi_\lambda|_{BV(\mathbb{R}^d}=1\)). Then this wavelet system has the following \(BV\) stability property. If \(f \in BV(\mathbb{R}^d)\), \(d \geq 2\), let \[ f = \sum_{\lambda \in \Delta} c_\lambda(f)\psi_\lambda \] be the wavelet expansion of \(f\). Let for any \(N\), \(\Lambda_N(f)\) be the set of \(N\) indices \(\lambda \in \Delta\) for which \(|c_\lambda(f)|\) are largest. Then the nonlinear operator \[ G_{N}(f):= \sum_{\lambda \in \Lambda_N(f)} c_\lambda(f)\psi_\lambda \] satisfies \[ |G_N(f)|_{BV(\mathbb{R}^d)} \leq C(\varphi,d) |f|_{BV(\mathbb{R}^d)}\;. \] As a consequence of this theorem, it is shown that \(G_N(f)\) (which is called a greedy approximation to \(f\)) is a near minimizer of the \(K\)-functional of the pair \((L_{d^\star}(\mathbb{R}^d),BV(\mathbb{R}^d)), d^\star=\frac{d}{d-1}\).

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
26A45 Functions of bounded variation, generalizations

Citations:

Zbl 0987.35003
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References:

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