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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A uniform distribution question related to numerical analysis


Authors: Harald Niederreiter and Charles F. Osgood
Journal: Math. Comp. 30 (1976), 366-370
MSC: Primary 65D30; Secondary 10K05
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 real-valued functions $ \alpha (x)$ and $ g(x)$ on [0,1 ],

$\displaystyle h\;\sum\limits_{\gamma = 1}^{[{h^{ - 1}}]} {{{\{ {h^{ - 1}}\alpha... ...+ 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)$,

$\displaystyle h\;\sum\limits_{\gamma = 1}^{[{h^{ - 1}}]} {{{\{ {h^{ - 1}}\alpha... ...a 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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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1976-0398067-7
PII: S 0025-5718(1976)0398067-7
Article copyright: © Copyright 1976 American Mathematical Society