Intractability results for positive quadrature formulas and extremal problems for trigonometric polynomials

被引:12
作者
Novak, E [1 ]
机构
[1] Univ Erlangen Nurnberg, Inst Math, D-91054 Erlangen, Germany
关键词
D O I
10.1006/jcom.1999.0507
中图分类号
TP301 [理论、方法];
学科分类号
081202 ;
摘要
Lower bounds for the error of quadrature formulas with positive weights are proved. We get intractability results for quasi-Monte Carlo methods and, more generally, for positive formulas. We consider general classes of functions but concentrate on lower bounds for relatively small classes of trigonometric polynomials. We also conjecture that similar lower bounds hold for arbitrary quadrature formulas and state different equivalent conjectures concerning positive definiteness of certain matrices and certain extremal problems for trigonometric polynomials. We also study classes of functions with weighted norms where some Variables are "more important" than others. Positive quadrature formulas are then tractable iff the sum of the wrights is bounded. (C) 1999 Academic Press.
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收藏
页码:299 / 316
页数:18
相关论文
共 19 条
[1]  
BAKHVALOV NS, 1959, VESTNIK MOSK MMAFH, V4
[2]  
Bykovskii V.A., 1985, CORRECT ORDER ERROR
[3]  
FROLOV KK, 1976, SOV MATH DOKL, V17, P1665
[4]   A generalized discrepancy and quadrature error bound [J].
Hickernell, FJ .
MATHEMATICS OF COMPUTATION, 1998, 67 (221) :299-322
[5]  
Horn RA, 1990, P S APPL MATH, V40, P87, DOI DOI 10.1090/PSAPM/040/1059485
[6]  
LUENBERGER GG, 1969, OPTIMIZATION VECTOR
[7]  
Niederreiter H., 1992, RANDOM NUMBER GENERA
[8]   Tractability of tensor product linear operators [J].
Novak, E ;
Sloan, IH ;
Wozniakowski, H .
JOURNAL OF COMPLEXITY, 1997, 13 (04) :387-418
[9]  
NOVAK E, 1988, LECT NOTES MATH, V1349
[10]   An intractability result for multiple integration [J].
Sloan, IH ;
Wozniakowski, H .
MATHEMATICS OF COMPUTATION, 1997, 66 (219) :1119-1124