Periodic convolution operator
= Periodic convolution operator
{title2=$Af(x)=\int_0^{2\pi}K(x-x')f(x')dx'$}
A <periodic convolution operator> with continuous periodic kernel is compact on $L^2[0,2\pi]$, since its kernel is square integrable. In the normalized <Fourier series> basis it is diagonal, with multipliers $c_n=\int_0^{2\pi}K(x)e^{-inx}dx$. These multipliers include the full integral rather than its normalized <Fourier coefficient>.