 Fejér sum

With the $N$-term convention, the Fejér sum of an integrable $2\pi$-periodic function isThe Fejér kernel formula gives the finite frequency expansionEach mode is tapered linearly rather than sharply retained or discarded. The degree is at most $N-1$, and ${\sigma }_{1}f$ is the mean of $f$.

For $f(x)=\cos x$ and $N\ge 2$, the result is $(1-1/N)\cos x$. This checks the precise finite-index error in Fejér saturation on a Fourier mode. The nonnegative kernel gives Fejér summation is a uniform-norm contraction, and Fejér's theorem explains uniform convergence for continuous input. Neither property asserts that a sum of a sign-changing function is pointwise nonnegative.