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Greedy approximation of characteristic functions. (English) Zbl 1207.41014

Proc. Steklov Inst. Math. 269, 247-258 (2010) and Trudy Mat. Inst. Steklova 269, 254-264 (2010).
Author’s abstract: The problem of sparse representation of domains in \(\mathbb R^d\) is discussed. It is demonstrated how the recently developed general theory of greedy approximation in Banach spaces can be used in this problem. The use of greedy approximation has two important advantages: (1) it works for an arbitrary dictionary of sets used for sparse representation and (2) the method of approximation does not depend on smoothness properties of the domains and automatically provides a near optimal rate of approximation for domains with different smoothness properties. Some lower estimates of the approximation error are given and a specific greedy algorithm for approximation of convex domains in \(\mathbb R^2\) is discussed.

MSC:

41A30 Approximation by other special function classes
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