Optimal algorithms for doubly weighted approximation of univariate functions

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• 作者：F.Y. Kuo^a ; ^{f.kuo@unsw.edu.au" class="auth_mail" title="E-mail the corresponding author} ; L. Plaskota^b ; ^{leszekp@mimuw.edu.pl" class="auth_mail" title="E-mail the corresponding author} ; G.W. Wasilkowski^c ; ^{greg@cs.uky.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Function approximation ; Unbounded domains ; Optimal algorithms
• 刊名：Journal of Approximation Theory
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：201
• 期：Complete
• 页码：30-47
• 全文大小：286 K

文摘

We consider a g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si4.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=8d8d5cb9f5ea261b577d69918d8377bf" title="Click to view the MathML source">ϱgif" overflow="scroll">ϱ-weighted g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si5.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=dbc475a96eafc5e980bffcdbeb6f9b94" title="Click to view the MathML source">L_qgif" overflow="scroll">Lq approximation in the space of univariate functions g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si6.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=103007f43e7030f0941a4d08553e1ef0" title="Click to view the MathML source">f:R₊→Rgif" overflow="scroll">f:R+→R with finite g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si7.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=98dba9e60be04b83934b3e0c7ff84b8a" title="Click to view the MathML source">‖f^(r)ψ‖_Lpgif" overflow="scroll">‖f(r)ψ‖Lp. Let g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si8.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=5205338e98f38cc54035118995b443c2" title="Click to view the MathML source">α=r−1/p+1/qgif" overflow="scroll">α=r−1/p+1/q and g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si9.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=501059ebee377fe8696b18afbe6bfab3" title="Click to view the MathML source">ω=ϱ/ψgif" overflow="scroll">ω=ϱ/ψ. Assuming that g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si10.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=a5370b2f11084c9b1a8cbd5eb6aa7f4b" title="Click to view the MathML source">ψgif" overflow="scroll">ψ and g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si11.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=4161dd0ca10a71d1053f28fdb2dea829" title="Click to view the MathML source">ωgif" overflow="scroll">ω are non-increasing and the quasi-norm g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si12.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=c2f957312e1c01b3b2edb8d45a21d6ca" title="Click to view the MathML source">‖ω‖_L1/αgif" overflow="scroll">‖ω‖L1/α is finite, we construct algorithms using function/derivatives evaluations at g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si13.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=48bc9f2e6ed4a821a44cad274731a661" title="Click to view the MathML source">ngif" overflow="scroll">n points with the worst case errors proportional to g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si14.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=2203acaa90a61f4817e6299655f994bc" title="Click to view the MathML source">‖ω‖_L1/αn^{−r+(1/p−1/q)₊}gif" overflow="scroll">‖ω‖L1/αn−r+(1/p−1/q)+. In addition we show that this bound is sharp; in particular, if g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si15.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=36b4e0e2a708ed4ca536f51af932d510" title="Click to view the MathML source">‖ω‖_L1/α=∞gif" overflow="scroll">‖ω‖L1/α=∞ then the rate g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si16.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=bdf7538232d849c735c25ead96567fb9" title="Click to view the MathML source">n^{−r+(1/p−1/q)₊}gif" overflow="scroll">n−r+(1/p−1/q)+ cannot be achieved. Our results generalize known results for bounded domains such as g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si17.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=f7dbe23bb4351b956b4ee529219284c2" title="Click to view the MathML source">[0,1]gif" overflow="scroll">[0,1] and g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904515001318&_mathId=si18.gif&_user=111111111&_pii=S0021904515001318&_rdoc=1&_issn=00219045&md5=f1a70e57a06d797a5db4b683335a2cad" title="Click to view the MathML source">ϱ=ψ≡1gif" overflow="scroll">ϱ=ψ≡1. We also provide a numerical illustration.