The Dirichlet Kernel

The Dirichlet kernel arises in the context of Fourier series whose orthonormal basis functions on $[0,2\pi ]$ are

$$\{ u_k (x)\} = \left\{ \frac{1}{\sqrt{2\pi}},\frac{1}{\sqrt{\pi}}\cos nx, \frac{1}{\sqrt{\pi}}\sin nx\right\}\,.$$

Consider the $N^{\textrm{th}}$ partial sum $S_N$ of $f$ , a function integrable on the interval $[0,2\pi ]$

$$S_N = \frac{a_0}{2}+\sum^N_{n=1} a_n\cos nx +\sum^N_{n=1} b_n \sin nx$$

$$S_N(x) = \frac{1}{\pi}\int^{2\pi}_0 \left[ \frac{1}{2} +\sum^N_{n=1}\cos nx\cos nt+\sum^N_{n=1}\sin nx\sin nt\right] f(t)dt$$

$$= \int^{2\pi}_0 \frac{1}{\pi} \left[\frac{1}{2}+\sum^N_{n=1}\cos n(x-t) \right] f(t)dt$$

$$\equiv \int^{2\pi}_0 \delta_N(x-t) f(t)dt\,.$$

This is the (optimal) least squares approximation of $f$ .

Definition: (Dirichlet kernel $=$ ``periodic finite impulse function'') The function

$$\delta_N(u)=\frac{1}{\pi}\left[\frac{1}{2}+\sum^N_{n=1}\cos nu\right] = \frac{1}{2\pi}\sum^N_{n=-N} e^{inu}~~\qquad~~\hbox{with}~~u=x-t$$ (21)

is called the Dirichlet kernel and it is also given by

$$\frac{1}{2\pi}\,\frac{e^{-iNu}-e^{i(N+1)u}}{1-e^{iu}} = \frac{1}{... ... \,\frac{\sin \left(N+\frac{1}{2}\right) u}{\sin \frac{u}{2}} = \delta_N (u)\,.$$ (22)

Lecture 11