The \(p\)-adic diaphony of the Halton sequence. (English) Zbl 1282.11087

Pillichshammer, Friedrich

Funct. Approximatio, Comment. Math. 49, No. 1, 91-102 (2013).

Let \(\omega=(\mathbf{x}_n)\) be a sequence in \([0,1)^s\). Let \(\mathbb{P}\) denote the set of prime numbers. For \(\mathbf{p}=(p_1, \dots, p_s)\in \mathbb{P}^s\) the \(\mathbf{p}\)-adic diaphony \(F_N(\omega)\) of the sequence \(\omega\) was defined by P. Hellekalek [Acta Arith. 145, No. 3, 273–284 (2010; Zbl 1243.11082)]. The worst-case integration error in a function space \(\mathcal{H}\) on \([0,1)^s\) with norm \(\|\cdot\|\) is given by \[ e(\mathcal{H},\omega)=\sup_{f\in\mathcal{H}, \|f\|\leq 1}\left|\int_{[0,1]^s}f(\mathbf{x})\,d\mathbf{x}-\frac{1}{N}\sum_{n=0}^{N-1}f(\mathbf{x}_n)\right|. \] In this paper it is shown that the worst-case integration error in a certain reproducing kernel Hilbert space \(\mathcal{H}_{\mathbf{p},s}\) and the \(\mathbf{p}\)-adic diaphony \(F_N(\omega)\) of the sequence \(\omega\) are related by \(e(\mathcal{H}_{\mathbf{p},s},\omega)=\sqrt{\sigma_{\mathbf{p}}-1}F_N(\omega)\) where \(\sigma_{\mathbf{p}}=\prod_{i=1}^s(p_i+1)\). The main result of this paper is that the \(\mathbf{p}\)-adic diaphony \(F_N(\omega)\) of the Halton sequence \(\omega\) in pairwise different prime base \(\mathbf{p}=(p_1, \dots, p_s)\) satisfies \(F_N(\omega)=O((\log N)^{s/2}/N)\).

Reviewer: Yukio Ohkubo (Kagoshima)

MSC:

11K06	General theory of distribution modulo \(1\)
11K38	Irregularities of distribution, discrepancy
11K41	Continuous, \(p\)-adic and abstract analogues

Keywords:

irregularity of distribution; diaphony; Halton sequence; quasi-Monte Carlo

Citations:

Zbl 1243.11082

References:

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