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

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 \[ Q(f) = \frac {1} {N}\sum \limits _{r = 1}^N {f(r} {\text {x)}}\] (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 $\left | {I(f) - Q(f)} \right | < K(f){N^{ - p}}$ 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 $\left | {I(f) - Q(f)} \right | = O(F(N))$ will hold, for any $F(N)$ which approaches zero as $N$ goes to infinity.