## 515 – Plotting Féjer’s kernels

February 6, 2012

Féjer’s kernels are the functions that play the role of “approximations to the Dirac delta” in the computations we will use to obtain Weierstrass approximation theorem. The $n$th approximation is given by

$\displaystyle K_n(s)= \frac1{n+1}\left(\frac{\sin\frac{(n+1)s}2}{\sin\frac s2}\right)^2$ for $s\ne0$, $K_n(0)=n+1$.

## 515 – The fundamental theorem of calculus

February 6, 2012

Suppose that $f\in{\mathcal R}[a,b]$ and $f$ has an antiderivative $G$. Then

$\displaystyle G(b)-G(a)=\int_a^b f(t)dt.$