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A characterization of higher order nets using Weyl sums and its applications. (English) Zbl 1249.11071
The low discrepancy nets and sequences are very important objects and have many applications in science, especially they are often used in quasi-Monte Carlo methods for numerical integration.
In the present paper the authors consider the Weyl sum based on the Walsh functions in base $$b \geq 2$$ as a tool for a characterization of the $$(t,\alpha ,\beta ,n,m,s)$$-nets.
In the Introduction a small survey of the concept of the $$(t,\alpha ,\beta ,n,m,s)$$-nets and their applications in the theory of the quasi-Monte Carlo integration is realized.
In section 2 the definition and the construction of the $$(t,\alpha ,\beta ,n,m,s)$$-nets are provided. The concept of the Weyl sum based on the Walsh functions is recalled and the importance of these sums as a tool for studying the uniform distribution of sequences is discussed. The definition of the weighted Korobov space $$W_{\alpha ,s,\gamma }$$ is recalled.
In section 3 the characterization of the $$(t,\alpha ,\beta ,n,m,s)$$-nets in terms of the Weyl sums is given. In Lemma 1 the value of the Weyl sum based on the Walsh functions is obtained. The main result here is Theorem 1. This theorem provides a necessary and sufficient condition that a finite sequence of $$b^{m}$$ points in the $$s$$-dimensional unit cube $$[0,1)^{s}$$ to be a $$(t,\alpha ,\beta ,n,m,s)$$-net.
In section 4 an application of the characterization of the $$(t,\alpha ,\beta ,n,m,s)$$-nets to the numerical integration in the space $$W_{\alpha ,s,\gamma }$$ is realized. It is shown that $$(t,\alpha ,\beta ,n,m,s)$$-nets can exploit the smoothness $$\alpha$$ of the functions from the space $$W_{\alpha ,s,\gamma }$$. Theorem 2 shows that the error of the integration in the space $$W_{\alpha ,s,\gamma }$$ by using an arbitrary $$(t,\alpha ,\beta ,n,m,s)$$-net has an order $$N^{-(\alpha -1)}$$ multiplied by a power of a $$\log N$$ factor.
In section 5 an analytical process of a randomization using a digital shift of higher order nets is proposed, and these randomized point sets are applied to the numerical integration. The concept of the one-dimensional and the multi-dimensional digital shift is developed. In Proposition 1 the uniform distribution of the randomized point set is shown. Proposition 2 states the fact that the introduced digital shift preserves the net property of the $$(t,\alpha ,\beta ,n,m,s)$$-nets. In Theorem 3 it is shown that the root mean square worst-case error of the integration in the space $$W_{\alpha ,s,\gamma }$$ has an order $$N^{-(\alpha -{1\over 2})}$$ multiplied by a power of a $$\log N$$ factor.
In section 6 the $$(u,u+v)$$-construction from coding theory is generalized to a construction of $$(t,\alpha ,\beta ,n,m,s)$$-nets. The concept of the $$(u,u+v)$$-construction of $$(t,\alpha ,\beta ,n,m,s)$$-nets is developed. In Theorem 4 the parameters of the new constructed $$(t,\alpha ,\beta ,n,m,s)$$-net are given. To prove this theorem essentially the characterization of $$(t,\alpha ,\beta ,n,m,s)$$-nets is used.

##### MSC:
 11K31 Special sequences 11K38 Irregularities of distribution, discrepancy 11L15 Weyl sums
##### Keywords:
Weyl sum; Walsh function; numerical integration; randomization