A characterization of higher order nets using Weyl sums and its applications.

*(English)*Zbl 1249.11071The 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.

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.

Reviewer: Vassil Grozdanov (Blagoevgrad)