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On a theorem of Piatetsky-Shapiro and approximation of multiple integrals

Authors: Seymour Haber and Charles F. Osgood
Journal: Math. Comp. 23 (1969), 165-168
MSC: Primary 65.55
MathSciNet review: 0239758
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Abstract: Let $ f$ be a function of $ s$ real variables which is of period $ 1$ in each variable, and let the integral $ I$ of $ f$ over the unit cube in $ s$-space be approximated by

$\displaystyle Q(f) = \frac{1} {N}\sum\limits_{r = 1}^N {f(r} {\text{x)}}$

(where $ {\text{x = x(N)}}$ is a point in $ s$-space). For certain classes of $ f'{\text{s}}$'s, defined by conditions on their Fourier coefficients, it is shown using methods of N. M. Korobov, that $ {\text{x's}}$'s can be found for which error bounds of the form $ \left\vert {I(f) - Q(f)} \right\vert < K(f){N^{ - p}}$ will be true. However, for the class of all $ f'{\text{s}}$'s with absolutely convergent Fourier series, it is shown that there are no $ {\text{x}}'{\text{s}}$'s for which a bound of the form $ \left\vert {I(f) - Q(f)} \right\vert = O(F(N))$ will hold, for any $ F(N)$ which approaches zero as $ N$ goes to infinity.

References [Enhancements On Off] (What's this?)

  • [1] N. M. Korobov, Number-Theoretic Methods of Approximate Analysis, Fizmatgiz, Moscow, 1963, p. 85. (Russian) MR 28 #716.
  • [2] George F. Simmons, Introduction to topology and modern analysis, McGraw-Hill Book Co., Inc., New York-San Francisco, Calif.-Toronto-London, 1963. MR 0146625

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Article copyright: © Copyright 1969 American Mathematical Society

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