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Mathematics of Computation

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A uniform distribution question related to numerical analysis
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by Harald Niederreiter and Charles F. Osgood PDF
Math. Comp. 30 (1976), 366-370 Request permission


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.
    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, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0419394
  • H. Niederreiter and Walter Philipp, Berry-Esseen bounds and a theorem of Erdős and Turán on uniform distribution $\textrm {mod}\ 1$, Duke Math. J. 40 (1973), 633–649. MR 337873
  • J. G. van der Corput, Zahlentheoretische Abschätzungen, Math. Ann. 84 (1921), no. 1-2, 53–79 (German). MR 1512020, DOI 10.1007/BF01458693
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Math. Comp. 30 (1976), 366-370
  • MSC: Primary 65D30; Secondary 10K05
  • DOI:
  • MathSciNet review: 0398067