Numerical integration of H枚lder continuous, absolutely convergent Fourier, Fourier cosine, and Walsh series

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• 作者：Josef Dick ^{josef.dick@unsw.edu.au" class="auth_mail}
• 关键词：65D30 ; 65D32 ; 65C05 ; 65C10
• 刊名：Journal of Approximation Theory
• 出版年：July 2014
• 年：2014
• 卷：183
• 期：Complete
• 页码：14-30
• 全文大小：254 K

文摘

We introduce quasi-Monte Carlo rules for the numerical integration of functions n id="mmlsi1" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000598&_mathId=si1.gif&_user=111111111&_pii=S0021904514000598&_rdoc=1&_issn=00219045&md5=9897a3631e00fc0914f01b51f71bef2b" title="Click to view the MathML source">fn>n class="mathContainer hidden">n class="mathCode">$f$n>n>n> defined on n id="mmlsi2" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000598&_mathId=si2.gif&_user=111111111&_pii=S0021904514000598&_rdoc=1&_issn=00219045&md5=f4cf97ee0efb91821706e6d0cfb66a30" title="Click to view the MathML source">[0,1]^sn>n class="mathContainer hidden">n class="mathCode">n>n>n>, n id="mmlsi3" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000598&_mathId=si3.gif&_user=111111111&_pii=S0021904514000598&_rdoc=1&_issn=00219045&md5=94605a753e002716b1b4fdd617e786c9" title="Click to view the MathML source">s≥1n>n class="mathContainer hidden">n class="mathCode">n>n>n>, which satisfy the following properties: the Fourier, Fourier cosine or Walsh coefficients of n id="mmlsi4" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000598&_mathId=si4.gif&_user=111111111&_pii=S0021904514000598&_rdoc=1&_issn=00219045&md5=3f3c3e8c6a2516ac6ce2e39ac709f4ea" title="Click to view the MathML source">fn>n class="mathContainer hidden">n class="mathCode">$f$n>n>n> are absolutely summable and n id="mmlsi5" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000598&_mathId=si5.gif&_user=111111111&_pii=S0021904514000598&_rdoc=1&_issn=00219045&md5=3d78823c42635d6116b7554037f99e4b" title="Click to view the MathML source">fn>n class="mathContainer hidden">n class="mathCode">$f$n>n>n> satisfies a Hölder condition of order n id="mmlsi6" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000598&_mathId=si6.gif&_user=111111111&_pii=S0021904514000598&_rdoc=1&_issn=00219045&md5=7b0f7cc44146d252b8effe7bb802cc81" title="Click to view the MathML source">伪n>n class="mathContainer hidden">n class="mathCode">$伪$n>n>n>, for some n id="mmlsi7" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000598&_mathId=si7.gif&_user=111111111&_pii=S0021904514000598&_rdoc=1&_issn=00219045&md5=f074bd0a4447505dac5ea099f5033c1e" title="Click to view the MathML source">0<伪&le;1n>n class="mathContainer hidden">n class="mathCode">n>n>n>. We show a convergence rate of the integration error of order n id="mmlsi8" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000598&_mathId=si8.gif&_user=111111111&_pii=S0021904514000598&_rdoc=1&_issn=00219045&md5=816bc8f0caffab9c81ccb27d1bbe5269" title="Click to view the MathML source">max((s&minus;1)N^&minus;1/2,s^伪/2N^&minus;伪)n>n class="mathContainer hidden">n class="mathCode">n>n>n>. The construction of the quadrature points is explicit and is based on Weil sums.