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Mathematics of Computation

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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 ], \[ h\;\sum \limits _{\gamma = 1}^{[{h^{ - 1}}]} {{{\{ {h^{ - 1}}\alpha (\gamma h)\} }^m}g(\gamma h) = {{(m + 1)}^{ - 1}}\int _0^1 {g(x)\;dx + o\left ( {{h^{1/3}}\log \frac {1}{h}} \right )} \quad {\text {as}}\;h \to 0 + ,} \] 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.

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

    M. A. FELDSTEIN, Discretization Methods for Retarded Ordinary Differential Equations, Ph.D. Dissertation, University of California, Los Angeles, 1964. E. W. HOBSON, The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, Vol. I, 3rd ed., Cambridge Univ. Press, London, 1927.
  • L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0419394
  • H. Niederreiter and Walter Philipp, Berry-Esseen 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