Fejér singular integral

An integral of the form

$$\sigma_n(f,x)=\frac1\pi\int\limits_{-\pi}^\pi f(x+t)\Phi_n(t)dt,$$

where

$$\Phi_n(t)=\frac{1}{2(n+1)}\frac{\sin^2(n+1)t/2}{\sin^2t/2}$$

is the Fejér kernel. The Fejér singular integral is an integral representation of the Fejér sums (cf. Fejér sum) $\sigma_n(f,x)$.

For references see Fejér sum.