On Power Functions and Error Estimates for Radial Basis Function Interpolation
This paper discusses approximation errors for interpolation in a variational setting which may be obtained from the analysis given by Golomb and Weinberger. We show how this analysis may be used to derive the power function estimate of the error as introduced by Schaback and Powell. A simple error t...
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| Published in | Journal of approximation theory Vol. 92; no. 2; pp. 245 - 266 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
01.02.1998
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| Online Access | Get full text |
| ISSN | 0021-9045 1096-0430 |
| DOI | 10.1006/jath.1997.3118 |
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| Abstract | This paper discusses approximation errors for interpolation in a variational setting which may be obtained from the analysis given by Golomb and Weinberger. We show how this analysis may be used to derive the power function estimate of the error as introduced by Schaback and Powell. A simple error tool for the power function is presented, which is similar to one appearing in the work of Madych and Nelson. It is then shown that this tool is adequate to reproducing the original error analysis presented by Duchon. An interesting consequence of our work is that no explicit use is made of the polynomial reproduction properties of the interpolation operator. |
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| AbstractList | This paper discusses approximation errors for interpolation in a variational setting which may be obtained from the analysis given by Golomb and Weinberger. We show how this analysis may be used to derive the power function estimate of the error as introduced by Schaback and Powell. A simple error tool for the power function is presented, which is similar to one appearing in the work of Madych and Nelson. It is then shown that this tool is adequate to reproducing the original error analysis presented by Duchon. An interesting consequence of our work is that no explicit use is made of the polynomial reproduction properties of the interpolation operator. |
| Author | Wayne, Henry Light, Will |
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| CitedBy_id | crossref_primary_10_1016_j_fss_2022_05_004 crossref_primary_10_1007_s00211_007_0092_z crossref_primary_10_1214_09_AOAS243 crossref_primary_10_1006_jath_2001_3635 crossref_primary_10_1016_j_aml_2022_108258 crossref_primary_10_1016_j_jat_2004_04_006 crossref_primary_10_1016_j_jat_2004_04_005 crossref_primary_10_1007_s00211_004_0543_8 crossref_primary_10_1007_s11075_004_3618_9 crossref_primary_10_1007_s11075_007_9103_5 crossref_primary_10_1016_j_jat_2004_03_005 crossref_primary_10_1007_s10092_010_0024_7 crossref_primary_10_1007_s10444_005_9001_0 crossref_primary_10_1137_060650428 crossref_primary_10_1002_mma_2906 |
| Cites_doi | 10.5802/aif.55 10.1090/S0025-5718-1990-0993931-7 10.1007/BF01601941 10.1007/s002110050051 10.1051/m2an/1978120403251 |
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| References | J. Meinguet, Multivariate interpolation at arbitrary points made simple, Z. Angew. Math. Phys. 30, 292, 304 Apostol (AT973118RF1) 1974 Golomb, Weinberger (AT973118RF6) 1959 Schaback (AT973118RF12) 1993 Deny, Lions (AT973118RF2) 1954; 5 Duchon (AT973118RF4) 1977; 571 Rudin (AT973118RF11) 1973 Ciarlet (AT973118RF5) 1978 Duchon (AT973118RF3) 1978; 12 Powell (AT973118RF10) 1994; 68 Madych, Nelson (AT973118RF8) 1988; 4 W. R. Madych, S. A. Nelson, Multivariate interpolation and conditionally positive definite functions, II, Math. Comp. 54, 211, 230 Rudin (10.1006/jath.1997.3118_AT973118RF11) 1973 10.1006/jath.1997.3118_AT973118RF9 Duchon (10.1006/jath.1997.3118_AT973118RF4) 1977; 571 Ciarlet (10.1006/jath.1997.3118_AT973118RF5) 1978 Duchon (10.1006/jath.1997.3118_AT973118RF3) 1978; 12 10.1006/jath.1997.3118_AT973118RF7 Madych (10.1006/jath.1997.3118_AT973118RF8) 1988; 4 Apostol (10.1006/jath.1997.3118_AT973118RF1) 1974 Deny (10.1006/jath.1997.3118_AT973118RF2) 1954; 5 Schaback (10.1006/jath.1997.3118_AT973118RF12) 1993 Golomb (10.1006/jath.1997.3118_AT973118RF6) 1959 Powell (10.1006/jath.1997.3118_AT973118RF10) 1994; 68 |
| References_xml | – volume: 5 start-page: 305 year: 1954 end-page: 370 ident: AT973118RF2 article-title: Les espaces du type de Beppo Levi publication-title: Ann. Inst. Fourier – year: 1978 ident: AT973118RF5 publication-title: The finite element method for elliptic problems – reference: W. R. Madych, S. A. Nelson, Multivariate interpolation and conditionally positive definite functions, II, Math. Comp. 54, 211, 230 – start-page: 117 year: 1959 end-page: 190 ident: AT973118RF6 article-title: Optimal approximation and error bounds publication-title: On Numerical Approximation – year: 1973 ident: AT973118RF11 publication-title: Functional Analysis – volume: 68 start-page: 107 year: 1994 end-page: 128 ident: AT973118RF10 article-title: The uniform convergence of thin plate spline interpolation in two dimensions publication-title: Numer. Math. – volume: 571 start-page: 85 year: 1977 end-page: 100 ident: AT973118RF4 article-title: Splines minimising rotation-invariant seminorms in Sobolev spaces publication-title: Constructive theory of Functions of Several Variables – start-page: 293 year: 1993 end-page: 305 ident: AT973118RF12 article-title: A comparison of radial basis function interpolants publication-title: Multivariate approximation—from CAGD to wavelets – volume: 12 start-page: 325 year: 1978 end-page: 334 ident: AT973118RF3 article-title: Sur l'erreur d'interpolation des fonctions de plusieurs variables par les publication-title: RAIRO Anal. Numer. – volume: 4 start-page: 77 year: 1988 end-page: 89 ident: AT973118RF8 article-title: Multivariate interpolation and conditionally positive definite functions publication-title: J. Approx. Theory Appl. – year: 1974 ident: AT973118RF1 publication-title: Mathematical Analysis – reference: J. Meinguet, Multivariate interpolation at arbitrary points made simple, Z. Angew. Math. Phys. 30, 292, 304 – start-page: 293 year: 1993 ident: 10.1006/jath.1997.3118_AT973118RF12 article-title: A comparison of radial basis function interpolants – volume: 5 start-page: 305 year: 1954 ident: 10.1006/jath.1997.3118_AT973118RF2 article-title: Les espaces du type de Beppo Levi publication-title: Ann. Inst. Fourier doi: 10.5802/aif.55 – ident: 10.1006/jath.1997.3118_AT973118RF7 doi: 10.1090/S0025-5718-1990-0993931-7 – ident: 10.1006/jath.1997.3118_AT973118RF9 doi: 10.1007/BF01601941 – year: 1974 ident: 10.1006/jath.1997.3118_AT973118RF1 – volume: 571 start-page: 85 year: 1977 ident: 10.1006/jath.1997.3118_AT973118RF4 article-title: Splines minimising rotation-invariant seminorms in Sobolev spaces – volume: 4 start-page: 77 year: 1988 ident: 10.1006/jath.1997.3118_AT973118RF8 article-title: Multivariate interpolation and conditionally positive definite functions publication-title: J. Approx. Theory Appl. – start-page: 117 year: 1959 ident: 10.1006/jath.1997.3118_AT973118RF6 article-title: Optimal approximation and error bounds – year: 1978 ident: 10.1006/jath.1997.3118_AT973118RF5 – volume: 68 start-page: 107 year: 1994 ident: 10.1006/jath.1997.3118_AT973118RF10 article-title: The uniform convergence of thin plate spline interpolation in two dimensions publication-title: Numer. Math. doi: 10.1007/s002110050051 – volume: 12 start-page: 325 year: 1978 ident: 10.1006/jath.1997.3118_AT973118RF3 article-title: Sur l'erreur d'interpolation des fonctions de plusieurs variables par lesDm-splines publication-title: RAIRO Anal. Numer. doi: 10.1051/m2an/1978120403251 – year: 1973 ident: 10.1006/jath.1997.3118_AT973118RF11 |
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