On a theorem of Piatetsky-Shapiro and approximation of multiple integrals
Seymour Haber and Charles F. Osgood
Math. Comp. 23 (1969), 165-168
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Abstract: Let be a function of real variables which is of period in each variable, and let the integral of over the unit cube in -space be approximated by (where is a point in -space). For certain classes of 's, defined by conditions on their Fourier coefficients, it is shown using methods of N. M. Korobov, that 's can be found for which error bounds of the form will be true. However, for the class of all 's with absolutely convergent Fourier series, it is shown that there are no 's for which a bound of the form will hold, for any which approaches zero as goes to infinity.
N. M. Korobov, Number-Theoretic Methods of Approximate Analysis, Fizmatgiz, Moscow, 1963, p. 85. (Russian) MR 28 #716.
F. Simmons, Introduction to topology and modern analysis,
McGraw-Hill Book Co., Inc., New York-San Francisco, Calif.-Toronto-London,
0146625 (26 #4145)
- N. M. Korobov, Number-Theoretic Methods of Approximate Analysis, Fizmatgiz, Moscow, 1963, p. 85. (Russian) MR 28 #716.
- G. F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, New York, 1965, p. 239. MR 0146625 (26:4145)
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