Canonical Sets of Best L1-Approximation
In mathematics, the term approximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns ou...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2012/435945 |
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| author | Dimiter Dryanov Petar Petrov |
| author_facet | Dimiter Dryanov Petar Petrov |
| author_sort | Dimiter Dryanov |
| collection | DOAJ |
| description | In mathematics, the term approximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are called canonical sets of best approximation. The present paper summarizes results on canonical sets of best L1-approximation with emphasis on multivariate interpolation and best L1-approximation by blending functions. The best L1-approximants are characterized as transfinite interpolants on canonical sets. The notion of a Haar-Chebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariate L1-approximation by sums of univariate functions. Explicit constructions of best one-sided L1-approximants give rise to well-known and new inequalities. |
| format | Article |
| id | doaj-art-8b90c05869ad4000a9e6b076705e9238 |
| institution | Kabale University |
| issn | 0972-6802 1758-4965 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-8b90c05869ad4000a9e6b076705e92382025-02-03T06:00:42ZengWileyJournal of Function Spaces and Applications0972-68021758-49652012-01-01201210.1155/2012/435945435945Canonical Sets of Best L1-ApproximationDimiter Dryanov0Petar Petrov1Department of Mathematics and Statistics, Concordia University, Montreal, QC, H3G 1M8, CanadaNumerical Modeling Department, Leibniz Institute for Crystal Growth, Max-Born-Street 2, D-12489 Berlin, GermanyIn mathematics, the term approximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are called canonical sets of best approximation. The present paper summarizes results on canonical sets of best L1-approximation with emphasis on multivariate interpolation and best L1-approximation by blending functions. The best L1-approximants are characterized as transfinite interpolants on canonical sets. The notion of a Haar-Chebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariate L1-approximation by sums of univariate functions. Explicit constructions of best one-sided L1-approximants give rise to well-known and new inequalities.http://dx.doi.org/10.1155/2012/435945 |
| spellingShingle | Dimiter Dryanov Petar Petrov Canonical Sets of Best L1-Approximation Journal of Function Spaces and Applications |
| title | Canonical Sets of Best L1-Approximation |
| title_full | Canonical Sets of Best L1-Approximation |
| title_fullStr | Canonical Sets of Best L1-Approximation |
| title_full_unstemmed | Canonical Sets of Best L1-Approximation |
| title_short | Canonical Sets of Best L1-Approximation |
| title_sort | canonical sets of best l1 approximation |
| url | http://dx.doi.org/10.1155/2012/435945 |
| work_keys_str_mv | AT dimiterdryanov canonicalsetsofbestl1approximation AT petarpetrov canonicalsetsofbestl1approximation |