Hölder continuity of uniform limit of piecewise constant functions

2012-05-23T21:17:12Z

Consider a piecewise constant function $v: [a,b] \rightarrow \mathbb{R}$ defined by a finite partition $a=t_0 < t_1 < t_2 < ... < t_s=b$ 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].

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.

Does anyone know of a condition that characterizes when such a function $u$ is $\alpha$-Hölder continuous for some $0<\alpha<1$?

E(up)lio.

Answer by Pietro Majer for Hölder continuity of uniform limit of piecewise constant functions

2012-05-23T22:32:45Z

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 $I\in \mathcal{P}$.

Assume that

1. $\|\mathcal{P _ n}\|\to0\ ;$

2. ${u _ n} _{|\mathcal{P _ n} }$ 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 $t,s\in\mathcal{P} _ n^M\ .$

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 $k|t|^\alpha$.

Hölder continuity of uniform limit of piecewise constant functions - MathOverflow

Hölder continuity of uniform limit of piecewise constant functions

Answer by Pietro Majer for Hölder continuity of uniform limit of piecewise constant functions