# A uniform Riemann sum approximation of the integral of the Fejer kernels

Let $F_N(t)$ denote the Fejer kernel $$F_N(t):={1\over N+1}{\sin^2\big({(N+1)}{t\over2}\big)\over \sin^2\big( {t\over2}\big)}\ .$$ Consider Riemann sums approximation for $\int_{-\pi}^\pi F_N(t) dt$ relative to uniform subdivisions with mesh ${1\over n}$:

$$2\pi=\int_{-\pi}^\pi F_N(t)dt\ ={1\over n}\sum_{-\pi n< j <\pi n} F_N\big( {j\over n}\big) +\ r(N,n)\ .$$

I would like to obtain an estimate of the remainder term $r(N,n)$ for $n$ and $N$ large, of the following form: assuming $n=K N^2+o(N^2)$ as $N\to+\infty$, is it true that $$r(N,n)={c\over N}+o\big({1\over N}\big),$$ and, most important, I would like to compute the constant $c$