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

DOI:
https://doi.org/10.1090/S0025-5718-1976-0398067-7

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 and on [0,1 ],

*x*denotes the fractional part of

*x*. The conditions are as follows: 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 , whereas is of bounded variation on [0,1]. Also, under these conditions on ,

**[1]**M. A. FELDSTEIN,*Discretization Methods for Retarded Ordinary Differential Equations*, Ph.D. Dissertation, University of California, Los Angeles, 1964.**[2]**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.**[3]**L. KUIPERS & H. NIEDERREITER,*Uniform Distribution of Sequences*, Wiley, New York, 1974. MR**0419394 (54:7415)****[4]**H. NIEDERREITER & W. PHILIPP, "Berry-Esseen bounds and a theorem of Erdös and Turán on uniform distribution ,"*Duke Math J.*, v. 40, 1973, pp. 633-649. MR**49**#2642. MR**0337873 (49:2642)****[5]**J. G. VAN DER CORPUT, "Zahlentheoretische Abschätzungen,"*Math. Ann.*, v. 84, 1921, pp. 53-79. MR**1512020**

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DOI:
https://doi.org/10.1090/S0025-5718-1976-0398067-7

Article copyright:
© Copyright 1976
American Mathematical Society