I'm reading through my professor's class notes for Fourier Analysis and there is a remark that I can't verify.
He notes that:
$[\sigma_n (f)](x) - f(x) = \frac1{2\pi}\int_{-\pi}^\pi [f(x+t) - f(x)]K_n(t)\;\mathrm dt.$
Note that here $K_n$ is the Fejer kernel of $f$, and the factor of $2\pi$ comes from the fact that we scaled the $L_1$ norm by this factor which I have noticed is not always done.
I tried using the fact that $\sigma_n (f)$ is equal to $f*K_n$, which let me obtain:
$[\sigma_n (f)](x) - f(x) = \frac1{2\pi}\int_{-\pi}^\pi f(x+t)K_n (t)\;\mathrm dt - f(x)$
I can't see how to get the $K_{n}$ inside the integral, let alone attached to the factor of $K_{n}$. Thanks for the patience, I just am learning how to use this site. :)
Note that the definition of $\sigma_n (f)$ is $\frac1{n}\sum\limits_{j=0}^{n-1}S_j (f)$, where $S_j (f) = \sum\limits_{m=-k}^k\widehat{f}(m) e^{imt}$ is the $j$-th partial sum of the Fourier series of $f$.