1,443 research outputs found
On Whitney type inequalities for local anisotropic polynomial approximation
We prove a multivariate Whitney type theorem for the local anisotropic
polynomial approximation in with . Here is a
-parallelepiped in \RR^d with sides parallel to the coordinate axes. We
consider the error of best approximation of a function by algebraic
polynomials of fixed degree at most in variable ,
and relate it to a so-called total mixed modulus of smoothness appropriate to
characterizing the convergence rate of the approximation error. This theorem is
derived from a Johnen type theorem on equivalence between a certain
K-functional and the total mixed modulus of smoothness which is proved in the
present paper.Comment: 12 pages; the proofs of Theorems 1.2 and 2.2 and Lemma 3.1 are
revised; typos are corrected; Acknowledgments are added; the results are
unchange
The Alternative Daugavet Property of -algebras and -triples
A Banach space is said to have the alternative Daugavet property if for
every (bounded and linear) rank-one operator there
exists a modulus one scalar such that .
We give geometric characterizations of this property in the setting of
-algebras, -triples and their isometric preduals
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
We consider linear inverse problems where the solution is assumed to have a
sparse expansion on an arbitrary pre-assigned orthonormal basis. We prove that
replacing the usual quadratic regularizing penalties by weighted l^p-penalties
on the coefficients of such expansions, with 1 < or = p < or =2, still
regularizes the problem. If p < 2, regularized solutions of such l^p-penalized
problems will have sparser expansions, with respect to the basis under
consideration. To compute the corresponding regularized solutions we propose an
iterative algorithm that amounts to a Landweber iteration with thresholding (or
nonlinear shrinkage) applied at each iteration step. We prove that this
algorithm converges in norm. We also review some potential applications of this
method.Comment: 30 pages, 3 figures; this is version 2 - changes with respect to v1:
small correction in proof (but not statement of) lemma 3.15; description of
Besov spaces in intro and app A clarified (and corrected); smaller pointsize
(making 30 instead of 38 pages
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