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).
Posted in Quasi-Monte Carlo, Research
Tagged cyclic digital nets, digital net, discrepancy, discrepancy theory, duality theory, Duality Theory for Digital Nets, fast component by component, geometric discrepancy, Higher order digital net, higher order digital sequence, higher order polynomial lattice rule, higher order Sobol sequence, hyperplane nets, Niederreiter sequence, numerical integration, polynomial lattice rule, Propagation Rule, quasi-Monte Carlo, randomised quasi-Monte Carlo, Sobol sequence, uniform distribution, Walsh function
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
In this post I summarize some useful properties of Walsh functions. These functions were introduced by Joseph Walsh in
- 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
- 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 and dimension .
We write for the set of natural numbers and for the set of nonnegative integers .
A table of contents for the posts on Walsh functions can be found here.
Definition of Walsh functions Continue reading