# Families of functions used to approximate local neighbourhoods of non-smooth functions?

For local function approximation perhaps the most famous example how to do this is probably to use a basis of monomial functions $$\{1,x,x^2,\cdots,x^k,\cdots,x^n\}$$

In the most local setting, this can be done by Taylor expansion, where the coefficients in the linear combination are determined by the derivatives of the function:

$$f(x) \approx p_n(x) = \sum_{i=0}^n\underset{\text{Taylor coefficient #i}}{\underbrace{\frac{1}{i!}\cdot\frac{\partial^i f}{\partial x^i}(x_0)}}\cdot(x-x_0)^i$$

This is all nice and well if our function is differentiable $n$ times at the point where we want to make the expansion. But if for example we want to approximate a function which is not differentiable so many times (or at least we are not able to determine derivatives with any accuracy), then we won't be able to make this work for us.

So the question is, what methods and bases of functions can we use for approximating less regular functions (in some neighbourhood)?

If this would seem too broad of a question we can reduce it to consider we can measure mean in integral sense of all derivatives over in some neighbourhood $x\in [x-\delta,x+ \delta]$. In other words, we know $$m_i = \int_{x-\delta}^{x+\delta} f^{(i)}(x)dx$$ and that this neighbourhood is the same as we want to perform approximation on.

There is the finite element method: There functions are approximated by piecewise smooth functios, that are polyomials on small subsets of $\mathbb R^n$ (intervals, triangles, cubes etc).