A Taylor theorem for Hölder continuous function?

Question

Let $ C^{m,s}_{b} $ be the space of bounded function $ u: \mathbb{R} \rightarrow \mathbb{R} $ which satisfies \begin{alignat*}{2} \bigg| u^{(m)}(x) - u^{(m)}(y) \bigg| \leq C | x - y |^{s}. \end{alignat*} I'm looking for a Taylor - type theorem which could bound the remainder of the following polynomial $$ u(x) = \sum_{k \geq N} \frac{(x-y)^{k}}{k!} u^{(k)}(y) + R_{N}(x) $$ by $$| R_{N}(x) | \leq |x-y|^{m+s}.$$ In the simple case that $ N=1 $ I know that $$ u(x) = u(y) + u'(y)(x-y) + \int_{0}^{1} \Big( u'\big(y + t(x-y) \big) - u'(y) \Big)dt (x-y) .$$ But I have yet been successful in finding the right expression for the arbitrary case. Could anyone please shed some light on the problem? Thanks in advance!

Answer 1

Take the Taylor formula of order $m-1$ with the Lagrange remainder $$ u(x) = \sum_{k=0}^{m-1} \frac{(x-y)^{k}}{k!} u^{(k)}(y) + \frac{(x-y)^{m}}{m!} u^{(m)}(\xi), $$ where $\xi$ is between $y$ and $x$, and rewrite it as $$ u(x) = \sum_{k=0}^{m} \frac{(x-y)^{k}}{k!} u^{(k)}(y) + \frac{(x-y)^{m}(u^{(m)}(\xi)-u^{(m)}(y))}{m!}. $$ Using the Holder condition for the remainder one has $$ |R_m(x)|=\left|\frac{(x-y)^{m}(u^{(m)}(\xi)-u^{(m)}(y))}{m!} \right|\le C\frac{|x-y|^{m+s}}{m!}. $$