This seems like something one might find in a book so I would be grateful for any references you think may be helpful.


I am interested in the rate at which of a function integrated against the $N$th Fejer Kernel approaches it's value at zero. In other words, I am looking to understand the big oh term below
$$\int\_{-\pi}^\pi f(t)F\_N(t) dt - f(0) = O(1/N^?)$$
where
$$F\_N(t) = \frac{1}{2\pi}\cdot\frac{\sin^2(Nt/2)}{N\sin^2(t/2)} = \sum\_{n = -N+1}^{N-1}\left(1-\frac{|n|}{N}\right)\cos(nt)$$
is the $N$th Fejer Kernel and a delta-sequence. I can't find this result anywhere. Thank you in advance.



Rate of convergence of Fejer kernel to the Dirac delta function
For any $\delta,x$ such that $0<\delta\leq |x|<\pi$ one has $|F\_N(x)|\leq[2\pi(N+1)\sin^2(\delta/2)]^{-1}$, so the error in the delta-function approximation is of order $1/N$.


Two explicit examples, if $f(t)=\cos t$ one has
$$\int\_{-\pi}^\pi f(t)F\_N(t) dt =1-\frac{1}{N}+{\cal O}(N^{-2}).$$
and if $f(t)=\sin^2(t/2)$ one has
$$\int\_{-\pi}^\pi f(t)F\_N(t) dt =\frac{1}{2N}.$$


