A uniform distribution question related to numerical analysis
Authors:
Harald Niederreiter and Charles F. Osgood
Journal:
Math. Comp. 30 (1976), 366370
MSC:
Primary 65D30; Secondary 10K05
DOI:
https://doi.org/10.1090/S00255718197603980677
MathSciNet review:
0398067
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Abstract: Using the theory of uniform distribution modulo one, it is shown that under certain conditions on the realvalued functions $\alpha (x)$ and $g(x)$ on [0,1 ], \[ h\;\sum \limits _{\gamma = 1}^{[{h^{  1}}]} {{{\{ {h^{  1}}\alpha (\gamma h)\} }^m}g(\gamma h) = {{(m + 1)}^{  1}}\int _0^1 {g(x)\;dx + o\left ( {{h^{1/3}}\log \frac {1}{h}} \right )} \quad {\text {as}}\;h \to 0 + ,} \] where $m > 0$ and x denotes the fractional part of x. The conditions are as follows: $\alpha ”(x)$ exists for all but finitely many points in [0, 1], changes sign at most finitely often, and is bounded away in absolute value from both 0 and $\infty$, whereas $g(x)$ is of bounded variation on [0,1]. Also, under these conditions on $\alpha (x)$, \[ h\;\sum \limits _{\gamma = 1}^{[{h^{  1}}]} {{{\{ {h^{  1}}\alpha (\gamma h)\} }^m} = {{(m + 1)}^{  1}} + o({h^{1/3}})\quad {\text {as}}\;h \to 0 + .} \] These results, which are, in fact, deduced from somewhat more general propositions, answer questions of Feldstein connected with discretization methods for differential equations.

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 H. Niederreiter and Walter Philipp, BerryEsseen bounds and a theorem of Erdős and Turán on uniform distribution ${\rm mod}\ 1$, Duke Math. J. 40 (1973), 633–649. MR 337873
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Article copyright:
© Copyright 1976
American Mathematical Society