Hölder continuity of uniform limit of piecewise constant functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T16:44:33Z http://mathoverflow.net/feeds/question/97793 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97793/holder-continuity-of-uniform-limit-of-piecewise-constant-functions Hölder continuity of uniform limit of piecewise constant functions Euplio M. 2012-05-23T21:17:12Z 2012-05-25T20:57:34Z <p>Consider a piecewise constant function $v: [a,b] \rightarrow \mathbb{R}$ defined by a finite partition $a=t_0 &lt; t_1 &lt; t_2 &lt; ... &lt; t_s=b$ of the interval $[a,b]$, and constants $m_1,m_2,...,m_s$ with $v(t)=m_i$ whenever $t \in I_i := [t_{i-1},t_i)$ for $i=1,\dots,s$. Let $\mathcal{P}$ denote the collection of intervals $I_i$ that make up the partition of [a,b].</p> <p>Now, let $u: [a,b] \rightarrow \mathbb{R}$ be a regulated function (that is, $u$ is the uniform limit of a sequence of piecewise constant functions $u_n: [a,b] \rightarrow \mathbb{R}$) each $u_n$ of which is defined via a corresponding partition $\mathcal{P}_n$ of $[a,b]$ as described above.</p> <p>Does anyone know of a condition that characterizes when such a function $u$ is $\alpha$-Hölder continuous for some $0&lt;\alpha&lt;1$?</p> <p>E(up)lio.</p> http://mathoverflow.net/questions/97793/holder-continuity-of-uniform-limit-of-piecewise-constant-functions/97798#97798 Answer by Pietro Majer for Hölder continuity of uniform limit of piecewise constant functions Pietro Majer 2012-05-23T22:32:45Z 2012-05-25T20:57:34Z <p>I can't think of a characterization which is not too close to a tautology; a sufficient condition is the following. Denote the modulus of the subdivision $\mathcal{P}$ by $\|\mathcal{P}\|:=\max _ {1\le i\le s} (t _ i-t _ {i-1})$, and by $\mathcal{P}^M$ the set of mid-points of the intervals $I\in \mathcal{P}$.</p> <p>Assume that</p> <p><strong>1.</strong> $\|\mathcal{P _ n}\|\to0\ ;$ </p> <p><strong>2.</strong> ${u _ n} _{|\mathcal{P _ n} }$ are uniformly $\alpha$-Hölder, that is there is $k\ge0$ such that $|u_n(t) - u _ n(s)|\le k|t-s|^\alpha$ holds for any $n\in\mathbb{N}$ and for any $t,s\in\mathcal{P} _ n^M\ .$</p> <p>Reason: if $\tilde u _ n$ denotes the piece-wise interpolation of the nodes $\mathcal{P} _ n^M$, then by concavity $\tilde u _ n$ has modulus of continuity $k|t|^\alpha$ on $[a,b]$, and $\| u _ n - \tilde u _ n\| _ \infty\le k\|\mathcal{P} _ n\|^\alpha=o(1)$ as $n\to\infty$. Therefore $u$ has the same modulus of continuity $k|t|^\alpha$. </p>