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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 $\mathcal{P}^\circ$ \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^\circn^M\ .$Reason: if$\tilde u _ n$denotes the piece-wise interpolation of the nodes$\mathcal{P} _ n^\circ$n^M$, 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 $k|t|^\alpha$.

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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 $\mathcal{P}^\circ$ 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 $t,s\in\mathcal{P} _ n^\circ\ .$

Reason: if $\tilde u _ n$ denotes the piece-wise interpolation of the nodes $\mathcal{P} _ n^\circ$, 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 $k|t|^\alpha$.