I need some help with explaining the differences, and or similarities between these two versions of the Weierstrass Approximation theorem.
On the one hand I have Wikipedia, WolframMathworld and many others stating that the theorem is the following:
- If $f$ is a continuous real-valued function on $[a, b]$ and if any $\epsilon>0$ is given, then there exists a polynomial $p$ on $[a, b]$ such that $$ |f(x)-P(x)|<\epsilon $$ for all $x \in[a, b]$. In words, any continuous function on a closed and bounded interval can be uniformly approximated on that interval by polynomials to any degree of accuracy.
While in one of my analysis books (Invitation to Classical Analysis by Peter Duren) it is stated as the following:
- Let $q(x)$ be continuous on the interval $[-\pi , \pi ]$, with $q(- \pi)=q(\pi)$. Then for each $\varepsilon>0$ there is a trigonometric polynomial $T(x)$ such that $|q(x)-T(x)|<\varepsilon$ for $- \pi \leq x \leq \pi $.