Walsh functions III: The convergence of the Walsh coefficients and pointwise convergence

In a previous post, we discussed the orthogonality properties of Walsh functions and showed that they form a complete orthonormal system in . In this post we discuss the rate of decay of the Walsh coefficients when the function has bounded variation of fractional order and we investigate pointwise convergence of the Walsh series and pointwise convergence of the Walsh series to the function. We consider only Walsh functions in base although the results can be generalized to Walsh functions over groups. Information on Walsh functions over groups can be found in this post. For the necessary background information see the previous post on Walsh functions here. A table of contents for the posts on Walsh functions can be found here.

Higher order scrambling

Recently I uploaded the paper

• J. Dick, Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands. ArXiv, 8 July 2010.

This paper deals with a generalization of Owen’s scrambling algorithm which improves on the convergence rate of the root mean square error for smooth integrands. The bound on the root mean square error is best possible (apart from the power of the factor) and this can also be observed from some simple numerical examples shown in the paper (note that the figures in the paper show the standard deviation (or root mean square error) and not the variance of the estimator). In this post you can also find Matlab programs which generate the quadrature points introduced in this paper and a program to generate the numerical results shown in the paper.