Fibonacci sets and symmetrization in discrepancy theory

Bilyk, Dmitriy; Temlyakov, Vladimir; Yu, Rui

doi:10.1016/j.jco.2011.07.001

Cited by 20 publications

(24 citation statements)

References 20 publications

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“…, σ n ) ⊤ is an element of Z n 2 . We obtain a digit shifted digital net by altering the fourth step in the construction scheme of digital nets to C 2 r + σ =: (y (2) r,1 , . .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We apply the notation of [8] in this paper and use several results and ideas from there. The third paper which inspired this work is by Bilyk, Temlyakov and Yu [2], who computed the Fourier coefficients of the discrepancy function of the symmetrized Fibonacci lattice exactly in order to find an exact formula for its L 2 discrepancy. We do the same for a class of digital (0, n, 2)-net with the difference that we compute the Haar coefficients instead of the Fourier coefficients, since Haar functions fit the structure of digital nets much better than harmonic functions.…”

Section: Remarkmentioning

confidence: 99%

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Finding exact formulas for the $L_2$ discrepancy of digital $(0,n,2)$-nets via Haar functions

Kritzinger

2019

Acta Arith.

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We use the Haar function system in order to study the L 2 discrepancy of a class of digital (0, n, 2)-nets. Our approach yields exact formulas for this quantity, which measures the irregularities of distribution of a set of points in the unit interval. We will obtain such formulas not only for the classical digital nets, but also for shifted and symmetrized versions thereof. The basic idea of our proofs is to calculate all Haar coefficents of the discrepancy function exactly and insert them into Parseval's identity. We will also discuss reasons why certain (symmetrized) digital nets fail to achieve the optimal order of L 2 discrepancy and use the Littlewood-Paley inequality in order to obtain results on the L p discrepancy for all p ∈ (1, ∞).

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“…, σ n ) ⊤ is an element of Z n 2 . We obtain a digit shifted digital net by altering the fourth step in the construction scheme of digital nets to C 2 r + σ =: (y (2) r,1 , . .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Finding exact formulas for the $L_2$ discrepancy of digital $(0,n,2)$-nets via Haar functions

Kritzinger

2019

Acta Arith.

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“…Note that

Φ_{n} (f)

represents a special Korobov type integration formula. The idea to use Fibonacci numbers goes back to and was later used by Temlyakov to study integration in spaces with mixed smoothness (see also the recent contribution ). We will first focus on periodic functions and extend the results later to the non‐periodic situation.…”

Section: Integration On the Fibonacci Latticementioning

confidence: 99%

“…By using simple embedding properties, our results below directly imply Temlyakov's earlier results [, Thm. IV.2.1], [, Thm. 1.1] on Sobolev spaces

W_{p}^{r} (T^{2})

.…”

Section: Integration On the Fibonacci Latticementioning

confidence: 99%

Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

Dũng

Ullrich²

2014

Math. Nachr.

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We prove lower bounds for the error of optimal cubature formulae for d-variate functions from Besov spaces of mixed smoothness B α p,θ (G d ) in the case 0 < p, θ ≤ ∞ and α > 1/p, whereWe prove upper bounds for QMC methods of integration on the Fibonacci lattice for bivariate periodic functions from B α p,θ (T 2 ) in the case 1 ≤ p ≤ ∞, 0 < θ ≤ ∞, α > 1/p. A non-periodic modification of this classical formula yields upper bounds for B α p,θ (I 2 ) if 1/p < α < 1 + 1/p. In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from B α p,θ (G 2 ) and indicate that a corresponding result is most likely also true in case d > 2. This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids is never optimal for the general setting.

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“…for some positive constant C(α) depending only on α. The purpose of this paper is to find the precise order of magnitude of the left hand side of (4), where A is as in (3). We will work with a weaker assumption than α being badly approximable, however: we will assume that the continued fraction representation α = [a 0 ; a 1 , a 2 , .…”

mentioning

confidence: 99%

On the theorem of Davenport and generalized Dedekind sums

Borda

2017

Journal of Number Theory

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A symmetrized lattice of 2n points in terms of an irrational real number α is considered in the unit square, as in the theorem of Davenport. If α is a quadratic irrational, the square of the L 2 discrepancy is found to be c(α) log n + O (log log n) for a computable positive constant c(α). For the golden ratio ϕ, the value c(ϕ) log n yields the smallest L 2 discrepancy of any sequence of explicitly constructed finite point sets in the unit square. If the partial quotients a k of α grow at most polynomially fast, the L 2 discrepancy is found in terms of a k up to an explicitly bounded error term. It is also shown that certain generalized Dedekind sums can be approximated using the same methods. For a special generalized Dedekind sum with arguments a, b an asymptotic formula in terms of the partial quotients of a b is proved.

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Fibonacci sets and symmetrization in discrepancy theory

Cited by 20 publications

References 20 publications

Finding exact formulas for the $L_2$ discrepancy of digital $(0,n,2)$-nets via Haar functions

Finding exact formulas for the $L_2$ discrepancy of digital $(0,n,2)$-nets via Haar functions

Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

On the theorem of Davenport and generalized Dedekind sums

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