Tag Archives: Walsh function

Digital Nets and Sequences Preprint

A complete preprint of the book

is now available here. The final published version can be obtained directly from Cambridge University Press here.

The preprint version differs of course from the final version, for instance, the page numbers are different. However, the numbering of Chapters, Sections, Theorems, Lemmas, Corollaries, Definitions and Examples is the same in both versions. The list of corrections is for the published version. We do not have a separate list for the preprint version (though the corrections for the published version also apply to the preprint version).

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 L_2([0,1]). In this post we discuss the rate of decay of the Walsh coefficients when the function has bounded variation of fractional order 0 \textless \alpha \le 1 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 2, 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. Continue reading

Walsh functions II: Walsh functions in general base and Walsh functions over groups

In a previous post we introduced Walsh functions. We showed that this set of functions is orthogonal and complete. In this post we generalize the Walsh function system such that the orthonormality and completeness of the new Walsh function system still holds. A table of contents for the posts on Walsh functions can be found here. Continue reading

Walsh functions I. Orthonormality and completeness

In this post I summarize some useful properties of Walsh functions. These functions were introduced by Joseph Walsh in

  1. J. L. Walsh, A closed set of normal orthogonal functions. Amer. J. Math., 45, 5-24, 1923.

Another paper where many ideas can be found is by Nathan Fine

  1. N. J. Fine, On the Walsh functions. Trans. Amer. Math. Soc., 65, 372-414, 1949.

In this exposition here we only concentrate on the simplest case of base b = 2 and dimension s = 1.

We write \mathbb{N} for the set of natural numbers 1,2,3, \ldots and \mathbb{N}_0 for the set of nonnegative integers 0,1,2,\ldots.

A table of contents for the posts on Walsh functions can be found here.

Definition of Walsh functions Continue reading