0. Table of Contents

6. Approximation of Functions in 1D

6.1 Introduction

Approximation of functions: Generic view

• Given: a function $f : D \subset \mathbb{R}^n \to \mathbb{R}^d, \, n,d \in \mathbb{N},$ often in procedural form, e.g. for $n = d = 1,$ as double f(double)
• Goal: find a simple function $\tilde{f} : D \to \mathbb{R}^d$ such that the approximation error $f - \tilde{f}$ is small.

We define an abstract concept for the sake of clarity: When in this chapter we talk about an "approximation scheme" we refer to a mapping $A : X \to V,$ where $X$ and $V$ are spaces of functions $I \to \mathbb{K}, \, I \subset \mathbb{R}$ an interval.

Examples are:

• $X = C^k(I),$ the spaces of functions $I \to \mathbb{K}$ that are $k$ times continuously differentiable.
• $V = \mathcal{P}_m(I),$ the space of polynomials of degree $\leq m$.
• $V = \mathcal{S}_{d, \, \mathcal{M}},$ the space of splines of degree $d$ on the knot set $\mathcal{M} \subset I$.
• $V = \mathcal{P}_{2n}^T$, the space of trigonometric polynomials of degree $2n$.

Every interpolation scheme spawns a corresponding approximation scheme:

Interpolation scheme + sampling → approximation scheme

$$ f : I \subset \mathbb{R} \to \mathbb{K} \rightarrow_{sampling} (t_i, y_i := f(t_i)){i = 0}^m \rightarrow{interpolation} \tilde{f} := I_{\mathcal{T}}y \,\,\, (\tilde{f}(t_i) = y_i). $$