Need for holder continuous functions

Question

I am working on an optimization problem related to Hölder continuous functions:

A function $f:I\subset \mathbb{R}\to\mathbb{R}$ is Hölder continuous if there are constants $\alpha$ and $K$ such that $$|f(x)-f(y)|\leq K|x-y|^\alpha$$ for all $x,y\in I$.

For my algorithm to work the constant $K$ must be known in advance however this isn't always possible for my case .So I'm looking for a method to estimate this constant. I have written my first attempt at this and hope to test this algorithm so I need some test functions that are Hölder continuous with known $K$ and $\alpha$ . Can someone point me to a reference where I could find such functions to test my algorithm on ? Am also hoping to solve this for multivariable functions and any examples would be welcomed .

Answer 1

I guess if you take a function $f(x)$ which has a Taylor expansion $\sum a_n(x-x_0)^n$ in $x_0$, then $f$ is Hölder continuous in $x_0$ with constant $K=a_n$ and exponent $n$ for the smallest $\alpha=n>0$ so that $a_n\not=0$.

Especially any function $|x|^\alpha$ is Hölder continuous in $x=0$ with $K=1$ and $\alpha>0$ chosen arbitrarily.

Answer 2

If you know your function $f(x)$, then you can simply rewrite the Holder Condition as $\frac{|f(x)-f(y)|}{|x-y|^\alpha}\leq K$. Now since you know your function, you can compute any arbitrary $|f(x)-f(y)|$ and $|x-y|$. For $\alpha$ you can toy around with different values to see what will work for a constant $K$.

For a multivariable function, simply change absolute value to norm and proceed.

Here is another useful tip. If a function is $C^1([a,b])$ and the derivative is bounded, then the function is Lipschitz, and in particular 1-Holder. So first try to work with Lipschitz continuity and then expand out to the more general Holder continuity.