# Tag Archives: variation

## 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

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