I am interested in the mathematical justification for methods of approximating functions.
In $x \in (C[a, b], ||\cdot||_{\infty})$ we know that we can get an arbitrarily good approximation by using high enough order polynomials (Weierstrass Theorem).
Suppose that $x \in (C[a, b], ||\cdot||_{\infty})$. Let $y_n$ be defined by linearly interpolating $x$ on an uniform partition of $[a, b]$ (equidistant nodes). Is it true that \begin{equation} \lim_{n \to \infty} ||y_n - x||_{\infty} = 0? \end{equation}
Do we need to impose stronger conditions? For example \begin{equation} x(t) = \begin{cases} t \sin\left(\frac{1}{t}\right), & t \in (0, \pi] \\ 0, & t = 0 \end{cases} \end{equation} is in $C[0, 1]$, however it seems to me that we cannot get a good approximation near $t = 0$.
More generally, can anyone recommend a reference containing the theory of linear interpolation and splines? It would have to include conditions under which these approximation methods converge (in some metric) to the true function.