Convergence Rates of Approximation by Translates
In this paper we consider the problem of approximating a function belonging to some funtion space Φ by a linear comination of n translates of a given function G. Ussing a lemma by Jones (1990) and Barron (1991) we show that it is possible to define function spaces and functions G for which the rate...
| Main Authors: | , |
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| Language: | en_US |
| Published: |
2004
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| Online Access: | http://hdl.handle.net/1721.1/7316 |
| _version_ | 1811080520276115456 |
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| author | Girosi, Federico Anzellotti, Gabriele |
| author_facet | Girosi, Federico Anzellotti, Gabriele |
| author_sort | Girosi, Federico |
| collection | MIT |
| description | In this paper we consider the problem of approximating a function belonging to some funtion space Φ by a linear comination of n translates of a given function G. Ussing a lemma by Jones (1990) and Barron (1991) we show that it is possible to define function spaces and functions G for which the rate of convergence to zero of the erro is 0(1/n) in any number of dimensions. The apparent avoidance of the "curse of dimensionality" is due to the fact that these function spaces are more and more constrained as the dimension increases. Examples include spaces of the Sobolev tpe, in which the number of weak derivatives is required to be larger than the number of dimensions. We give results both for approximation in the L2 norm and in the Lc norm. The interesting feature of these results is that, thanks to the constructive nature of Jones" and Barron"s lemma, an iterative procedure is defined that can achieve this rate. |
| first_indexed | 2024-09-23T11:33:00Z |
| id | mit-1721.1/7316 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:33:00Z |
| publishDate | 2004 |
| record_format | dspace |
| spelling | mit-1721.1/73162019-04-12T08:34:38Z Convergence Rates of Approximation by Translates Girosi, Federico Anzellotti, Gabriele In this paper we consider the problem of approximating a function belonging to some funtion space Φ by a linear comination of n translates of a given function G. Ussing a lemma by Jones (1990) and Barron (1991) we show that it is possible to define function spaces and functions G for which the rate of convergence to zero of the erro is 0(1/n) in any number of dimensions. The apparent avoidance of the "curse of dimensionality" is due to the fact that these function spaces are more and more constrained as the dimension increases. Examples include spaces of the Sobolev tpe, in which the number of weak derivatives is required to be larger than the number of dimensions. We give results both for approximation in the L2 norm and in the Lc norm. The interesting feature of these results is that, thanks to the constructive nature of Jones" and Barron"s lemma, an iterative procedure is defined that can achieve this rate. 2004-11-04T16:53:30Z 2004-11-04T16:53:30Z 1992-03-01 AIM-1288 http://hdl.handle.net/1721.1/7316 en_US AIM-1288 77663 bytes 329320 bytes application/octet-stream application/pdf application/octet-stream application/pdf |
| spellingShingle | Girosi, Federico Anzellotti, Gabriele Convergence Rates of Approximation by Translates |
| title | Convergence Rates of Approximation by Translates |
| title_full | Convergence Rates of Approximation by Translates |
| title_fullStr | Convergence Rates of Approximation by Translates |
| title_full_unstemmed | Convergence Rates of Approximation by Translates |
| title_short | Convergence Rates of Approximation by Translates |
| title_sort | convergence rates of approximation by translates |
| url | http://hdl.handle.net/1721.1/7316 |
| work_keys_str_mv | AT girosifederico convergenceratesofapproximationbytranslates AT anzellottigabriele convergenceratesofapproximationbytranslates |