Are piecewise linear curves dense among Hölder curves? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T03:20:56Z http://mathoverflow.net/feeds/question/91469 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91469/are-piecewise-linear-curves-dense-among-holder-curves Are piecewise linear curves dense among Hölder curves? Pablo Lessa 2012-03-17T14:44:21Z 2012-03-18T10:09:27Z <p>Consider for some $0 &lt; \alpha \le 1$ the space functions $x:[0,1] \to \mathbb{R}^n$ such that $x(0) = 0$ and $\sup_{s,t} \frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$ is finite.</p> <p>There are at least two reasonable norms defined on this space. The first is the Hölder norm which is just the supremum above. Another is the $1/\alpha$-variation which is the supremum over all partitions $t_0 = 0 \le t_1 \le \cdots \le t_r = 1$ of $\left(\sum_{i=0}^{r-1}|f(t_{i+1}) - f(t_i)|^{1/\alpha}\right)^\alpha$.</p> <p>Let us fix $\alpha= \frac{1}{2}$ and $x(t) = \sqrt{t}$ and suppose $y:[0,1] \to \mathbb{R}$ is piecewise linear with $y(0) = 0$. It follows easily that $\lim_{t\to 0}\frac{\|x(t)-y(t)\|}{\sqrt{t}} = 1$.</p> <p>This implies that there is no sequence of piecewise linear approximations to $x$ in Hölder norm.</p> <p>However, it's not too hard to show that $x$ can be approximated in $2$-variation by piecewise linear functions.</p> <p>My question is the following: Are piecewise linear functions dense among $1/2$-Hölder functions in the $2$-variation sense?</p> <p>I'm also interested in the same question replacing piecewise linear functions by smooth functions.</p> http://mathoverflow.net/questions/91469/are-piecewise-linear-curves-dense-among-holder-curves/91519#91519 Answer by pgassiat for Are piecewise linear curves dense among Hölder curves? pgassiat 2012-03-18T10:09:27Z 2012-03-18T10:09:27Z <p>To expand on fedja's comment :</p> <p>For $p>1$, a function $f$ on $[0,1]$ is in the $p$-variation closure of smooth functions $C^{0,p-var}$ iff</p> <p>$$\lim_{\delta \rightarrow 0}\;\;\; \sup_{\substack{0=t_0&lt;\ldots&lt; t_m=1 \\ |t_{i+1}-t_i|\leq\delta}} \sum (f(t_{i+1})-f(t_i))^p = 0. \label{rel}$$ </p> <p>Then the function $g(x) = \sum_{i \geq 1} c^{-i/p} \sin(c^i x)$ is $(1/p)$-Hölder, but does not satisfy this relation (for $c$ large enough).</p> <p>Note that any continuous function of finite $q$-variation, for some $q&lt; p$ (such as your square root example), is in $C^{0,p-var}$.</p> <p>For $p=1$, $C^{0,1-var}$ is the space of absolutely continuous functions.</p> <p>You can find these results e.g. in Subsection 5.3.3 of Friz&amp;Victoir "Multidimensional stochastic processes as rough paths" (<a href="http://page.math.tu-berlin.de/~friz/master4_May6th.pdf" rel="nofollow" title="example">pdf</a>)</p>