Uniform Lipshitz continuity implies Continuous Differentiability

Question

Mallat in his book on wavelets makes the following definition - A function $f : [a,b] \to \mathbb{R}$ is $(C, \alpha)$-Lipschitz at $v \in [a, b]$ if there is a polynomial $p_v$ of degree at most $\lfloor \alpha \rfloor$ such that $|f(x) - p_v(x)| \leq C|x - v|^\alpha$ for any $x \in [a, b]$. It is uniformly Lipschitz if it is Lipschitz for all $v \in [a ,b]$ with a constant that is independent of $v$. Continuous differentiability implies Lipschitz continuity as above, since one can use the Taylor polynomial of the function.

Question: Apparently, the converse is also true - uniform $(C, \alpha)$-Lipschitz continuity implies that the function is $\lfloor \alpha \rfloor$ times continuously differentiable. I'm having difficulty proving this. It is clear that if $\alpha > 1$ then the function is differentiable. I'm guessing the derivative is uniformly $\alpha-1$ Lipschitz which would complete the proof by induction, but I'm having trouble showing this. Any hints would be appreciated!

Answer 1

There exist difference formulas $$ L_u(p)=\frac1h\sum_{k\in I}w_kp(u+kh) $$ for some fixed finite index set $I\subset\Bbb Z$ (usually $I=0,1,...,n$) and constants $w_k$ that are yield $$ L_u(p)=p'(u) $$ for all polynomials up to degree $n$. Especially from the formula for $p\equiv1$ we get $$\sum_{k\in I}w_k=0.$$ Then as \begin{align} |p_v(x)-p_u(x)|&\le|p_v(x)-f(x)|+|f(x)-p_u(x)|\le C|x-v|^α+C|x-u|^α \end{align} we get \begin{align} |p_v'(u)-p_u'(u)|&=|L_u(p_v)-L_u(p_u)| \\ &\le\frac1h\sum_{k\in I}|w_k|\,|p_v(u+kh)-p_u(u+kh)| \\ &\le\frac1h\sum_{k\in I}|w_k|\,C(|u+kh-v|^α+|kh|^α) \end{align} Now $p_u'(u)=f'(u)$ and for fixed $v$ $$ p_v'(u)=L_u(p_v)=\frac1h\sum_{k\in I}w_kp_v(u+kh)) =\sum_{k\in I}w_k\frac{p_v(u+kh)-p_v(u)}{h} $$ is a polynomial in $u$ on the left side and thus also on the right, all occurrences of $h$ cancel, including the division. To get the desired bound, set $h$ to a multiple of $v-u$, to get a reasonably small constant, set $nh=v-u$ so that you get $$ |f'(u)-p_v'(u)|\le C_1|u-v|^{α-1} $$ with $$ C_1=nC\sum_{k\in I}|w_k|\,(|1-\tfrac{k}{n}|^α+|\tfrac{k}n|^α). $$ This establishes the assumptions for $f'$ with Hölder/Lipschitz index $α-1$.