Roth’s Orthogonal Function Method in Discrepancy Theory and Some New Connections

Bilyk, Dmitriy

doi:10.1007/978-3-319-04696-9_2

Cited by 44 publications

(93 citation statements)

References 86 publications

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“…For further historical discussion we refer the reader to surveys [31], [5], [14], and books [4], [19], [42].…”

Section: A Brief History Of Results On Classical Discrepancymentioning

confidence: 99%

Connections between numerical integration, discrepancy, dispersion, and universal discretization

Temlyakov¹

2019

The SMAI Journal of Computational Mathematics

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The main goal of this paper is to provide a brief survey of recent results which connect together results from different areas of research. It is well known that numerical integration of functions with mixed smoothness is closely related to the discrepancy theory. We discuss this connection in detail and provide a general view of this connection. It was established recently that the new concept of fixed volume discrepancy is very useful in proving the upper bounds for the dispersion. Also, it was understood recently that point sets with small dispersion are very good for the universal discretization of the uniform norm of trigonometric polynomials.

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“…For further historical discussion we refer the reader to surveys [31], [5], [14], and books [4], [19], [42].…”

Section: A Brief History Of Results On Classical Discrepancymentioning

confidence: 99%

Connections between numerical integration, discrepancy, dispersion, and universal discretization

Temlyakov¹

2019

The SMAI Journal of Computational Mathematics

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“…This paper is devoted to the study of a discrepancy-type characteristic -the fixed volume discrepancy -of a point set in the unit square Ω 2 := [0, 1) 2 . We refer the reader to the following books and survey papers on discrepancy theory and numerical integration [2], [7], [8], [19], [3] [5], [15], and [20]. Recently, an important new observation was made in [16].…”

Section: Introductionmentioning

confidence: 99%

On the fixed volume discrepancy of the Fibonacci sets in the integral norms

Temlyakov

Ullrich

2020

Journal of Complexity

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This paper is devoted to the study of a discrepancy-type characteristic -the fixed volume discrepancy -of the Fibonacci point set in the unit square. It was observed recently that this new characteristic allows us to obtain optimal rate of dispersion from numerical integration results. This observation motivates us to thoroughly study this new version of discrepancy, which seems to be interesting by itself. The new ingredient of this paper is the use of the average over the shifts of hat functions instead of taking the supremum over the shifts. We show that this change in the setting results in an improvement of the upper bound for the smooth fixed volume discrepancy, similarly to the well-known results for the usual L p -discrepancy. Interestingly, this shows that "bad boxes" for the usual discrepancy cannot be "too small". The known results on smooth discrepancy show that the obtained bounds cannot be improved in a certain sense.

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“…for infinitely many N and some ε d > 0 depending only on d. There is no consensus on what the sharp result should be: the two main conjectures (we refer to [5]) are that for any sequence there are infinitely many N such that…”

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confidence: 99%