# Tag Archives: bounded variation

## QMC rules over R^s: Matlab code and numerical example

In this post you can find a Matlab code for constructing digital nets on $\mathbb{R}^s$ which was recently proposed in J. Dick, Quasi-Monte Carlo numerical integration on $\mathbb{R}^s$: digital nets and worst-case error. Submitted, 2010. See the previous post where an explanation of the method and a link to the paper can be found. In the numerical example we consider a simple three-dimensional integral. In this example the computation time with the new method is reduced by a factor of ten and additionally the integration error is also reduced. The numerical result in the paper shows that, for the example considered there, that the computation time can be reduced from two and a half minutes to less than two seconds for a certain given error level. Continue reading

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
We now discuss curves and their lengths in more detail. Let $n\in\mathbb{N}$ be an arbitrary natural number which is fixed throughout this post. Note that in general one needs to distinguish between a parameterised curve, which is a continuous mapping $\boldsymbol{c}:[a,b]\to\mathbb{R}^n,$ and the curve $\mathcal{C},$ which is the image of $\boldsymbol{c}$ given by $\{\boldsymbol{c}(t) \in \mathbb{R}^n: t\in [a,b]\}.$ Here we shall discuss parameterised curves. Hence, for instance, the parameterised curve $\boldsymbol{c}(t)= \cos t \widehat{\boldsymbol{i}} + \sin t \widehat{\boldsymbol{j}}$ with $0 \le t \le 4\pi$ is a circle traversed twice and has therefore length $4\pi,$ whereas its image is just a circle which has length $2\pi.$ Continue reading