On the quadrature exactness in hyperinterpolation

Abstract This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactn...
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Autor*in:	An, Congpei [verfasserIn] Wu, Hao-Ning

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2022

Schlagwörter:	Hyperinterpolation Quadrature Exactness Marcinkiewicz–Zygmund inequality

Anmerkung:	© The Author(s), under exclusive licence to Springer Nature B.V. 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Übergeordnetes Werk:	Enthalten in: BIT - Dordrecht [u.a.] : Springer Science + Business Media B.V, 1961, 62(2022), 4 vom: 05. Sept., Seite 1899-1919
Übergeordnetes Werk:	volume:62 ; year:2022 ; number:4 ; day:05 ; month:09 ; pages:1899-1919

Links:	Volltext

DOI / URN:	10.1007/s10543-022-00935-x

Katalog-ID:	SPR048662534

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520			\|a Abstract This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to %$n+k%$ with %$0<k\le n%$. Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the %$L^2%$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as %$n\rightarrow \infty %$ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs.
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912			\|a GBV_ILN_187
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allfields	10.1007/s10543-022-00935-x doi (DE-627)SPR048662534 (SPR)s10543-022-00935-x-e DE-627 ger DE-627 rakwb eng An, Congpei verfasserin (orcid)0000-0002-2392-8747 aut On the quadrature exactness in hyperinterpolation 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to %$n+k%$ with %$0<k\le n%$. Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the %$L^2%$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as %$n\rightarrow \infty %$ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs. Hyperinterpolation (dpeaa)DE-He213 Quadrature (dpeaa)DE-He213 Exactness (dpeaa)DE-He213 Marcinkiewicz–Zygmund inequality (dpeaa)DE-He213 Wu, Hao-Ning aut Enthalten in BIT Dordrecht [u.a.] : Springer Science + Business Media B.V, 1961 62(2022), 4 vom: 05. Sept., Seite 1899-1919 (DE-627)265778360 (DE-600)1465706-5 1572-9125 nnns volume:62 year:2022 number:4 day:05 month:09 pages:1899-1919 https://dx.doi.org/10.1007/s10543-022-00935-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 62 2022 4 05 09 1899-1919
spelling	10.1007/s10543-022-00935-x doi (DE-627)SPR048662534 (SPR)s10543-022-00935-x-e DE-627 ger DE-627 rakwb eng An, Congpei verfasserin (orcid)0000-0002-2392-8747 aut On the quadrature exactness in hyperinterpolation 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to %$n+k%$ with %$0<k\le n%$. Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the %$L^2%$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as %$n\rightarrow \infty %$ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs. Hyperinterpolation (dpeaa)DE-He213 Quadrature (dpeaa)DE-He213 Exactness (dpeaa)DE-He213 Marcinkiewicz–Zygmund inequality (dpeaa)DE-He213 Wu, Hao-Ning aut Enthalten in BIT Dordrecht [u.a.] : Springer Science + Business Media B.V, 1961 62(2022), 4 vom: 05. Sept., Seite 1899-1919 (DE-627)265778360 (DE-600)1465706-5 1572-9125 nnns volume:62 year:2022 number:4 day:05 month:09 pages:1899-1919 https://dx.doi.org/10.1007/s10543-022-00935-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 62 2022 4 05 09 1899-1919
allfields_unstemmed	10.1007/s10543-022-00935-x doi (DE-627)SPR048662534 (SPR)s10543-022-00935-x-e DE-627 ger DE-627 rakwb eng An, Congpei verfasserin (orcid)0000-0002-2392-8747 aut On the quadrature exactness in hyperinterpolation 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to %$n+k%$ with %$0<k\le n%$. Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the %$L^2%$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as %$n\rightarrow \infty %$ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs. Hyperinterpolation (dpeaa)DE-He213 Quadrature (dpeaa)DE-He213 Exactness (dpeaa)DE-He213 Marcinkiewicz–Zygmund inequality (dpeaa)DE-He213 Wu, Hao-Ning aut Enthalten in BIT Dordrecht [u.a.] : Springer Science + Business Media B.V, 1961 62(2022), 4 vom: 05. Sept., Seite 1899-1919 (DE-627)265778360 (DE-600)1465706-5 1572-9125 nnns volume:62 year:2022 number:4 day:05 month:09 pages:1899-1919 https://dx.doi.org/10.1007/s10543-022-00935-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 62 2022 4 05 09 1899-1919
allfieldsGer	10.1007/s10543-022-00935-x doi (DE-627)SPR048662534 (SPR)s10543-022-00935-x-e DE-627 ger DE-627 rakwb eng An, Congpei verfasserin (orcid)0000-0002-2392-8747 aut On the quadrature exactness in hyperinterpolation 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to %$n+k%$ with %$0<k\le n%$. Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the %$L^2%$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as %$n\rightarrow \infty %$ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs. Hyperinterpolation (dpeaa)DE-He213 Quadrature (dpeaa)DE-He213 Exactness (dpeaa)DE-He213 Marcinkiewicz–Zygmund inequality (dpeaa)DE-He213 Wu, Hao-Ning aut Enthalten in BIT Dordrecht [u.a.] : Springer Science + Business Media B.V, 1961 62(2022), 4 vom: 05. Sept., Seite 1899-1919 (DE-627)265778360 (DE-600)1465706-5 1572-9125 nnns volume:62 year:2022 number:4 day:05 month:09 pages:1899-1919 https://dx.doi.org/10.1007/s10543-022-00935-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 62 2022 4 05 09 1899-1919
allfieldsSound	10.1007/s10543-022-00935-x doi (DE-627)SPR048662534 (SPR)s10543-022-00935-x-e DE-627 ger DE-627 rakwb eng An, Congpei verfasserin (orcid)0000-0002-2392-8747 aut On the quadrature exactness in hyperinterpolation 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to %$n+k%$ with %$0<k\le n%$. Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the %$L^2%$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as %$n\rightarrow \infty %$ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs. Hyperinterpolation (dpeaa)DE-He213 Quadrature (dpeaa)DE-He213 Exactness (dpeaa)DE-He213 Marcinkiewicz–Zygmund inequality (dpeaa)DE-He213 Wu, Hao-Ning aut Enthalten in BIT Dordrecht [u.a.] : Springer Science + Business Media B.V, 1961 62(2022), 4 vom: 05. Sept., Seite 1899-1919 (DE-627)265778360 (DE-600)1465706-5 1572-9125 nnns volume:62 year:2022 number:4 day:05 month:09 pages:1899-1919 https://dx.doi.org/10.1007/s10543-022-00935-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 62 2022 4 05 09 1899-1919
language	English
source	Enthalten in BIT 62(2022), 4 vom: 05. Sept., Seite 1899-1919 volume:62 year:2022 number:4 day:05 month:09 pages:1899-1919
sourceStr	Enthalten in BIT 62(2022), 4 vom: 05. Sept., Seite 1899-1919 volume:62 year:2022 number:4 day:05 month:09 pages:1899-1919
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Hyperinterpolation Quadrature Exactness Marcinkiewicz–Zygmund inequality
isfreeaccess_bool	false
container_title	BIT
authorswithroles_txt_mv	An, Congpei @@aut@@ Wu, Hao-Ning @@aut@@
publishDateDaySort_date	2022-09-05T00:00:00Z
hierarchy_top_id	265778360
id	SPR048662534
language_de	englisch
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Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to %$n+k%$ with %$0<k\le n%$. 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author	An, Congpei
spellingShingle	An, Congpei misc Hyperinterpolation misc Quadrature misc Exactness misc Marcinkiewicz–Zygmund inequality On the quadrature exactness in hyperinterpolation
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topic_title	On the quadrature exactness in hyperinterpolation Hyperinterpolation (dpeaa)DE-He213 Quadrature (dpeaa)DE-He213 Exactness (dpeaa)DE-He213 Marcinkiewicz–Zygmund inequality (dpeaa)DE-He213
topic	misc Hyperinterpolation misc Quadrature misc Exactness misc Marcinkiewicz–Zygmund inequality
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title	On the quadrature exactness in hyperinterpolation
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title_full	On the quadrature exactness in hyperinterpolation
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title_sort	on the quadrature exactness in hyperinterpolation
title_auth	On the quadrature exactness in hyperinterpolation
abstract	Abstract This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to %$n+k%$ with %$0<k\le n%$. Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the %$L^2%$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as %$n\rightarrow \infty %$ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs. © The Author(s), under exclusive licence to Springer Nature B.V. 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
abstractGer	Abstract This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to %$n+k%$ with %$0<k\le n%$. Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the %$L^2%$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as %$n\rightarrow \infty %$ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs. © The Author(s), under exclusive licence to Springer Nature B.V. 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
abstract_unstemmed	Abstract This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to %$n+k%$ with %$0<k\le n%$. Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the %$L^2%$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as %$n\rightarrow \infty %$ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs. © The Author(s), under exclusive licence to Springer Nature B.V. 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
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title_short	On the quadrature exactness in hyperinterpolation
url	https://dx.doi.org/10.1007/s10543-022-00935-x
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Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to %$n+k%$ with %$0<k\le n%$. Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the %$L^2%$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as %$n\rightarrow \infty %$ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Hyperinterpolation</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Quadrature</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Exactness</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Marcinkiewicz–Zygmund inequality</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Wu, Hao-Ning</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">BIT</subfield><subfield code="d">Dordrecht [u.a.] : Springer Science + Business Media B.V, 1961</subfield><subfield code="g">62(2022), 4 vom: 05. Sept., Seite 1899-1919</subfield><subfield code="w">(DE-627)265778360</subfield><subfield code="w">(DE-600)1465706-5</subfield><subfield code="x">1572-9125</subfield><subfield code="7">nnns</subfield></datafield><datafield tag="773" ind1="1" ind2="8"><subfield code="g">volume:62</subfield><subfield code="g">year:2022</subfield><subfield code="g">number:4</subfield><subfield code="g">day:05</subfield><subfield code="g">month:09</subfield><subfield code="g">pages:1899-1919</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="u">https://dx.doi.org/10.1007/s10543-022-00935-x</subfield><subfield code="z">lizenzpflichtig</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_USEFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SYSFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_SPRINGER</subfield></datafield><datafield 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On the quadrature exactness in hyperinterpolation

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