Dirichlet kernel

The Dirichlet kernel is the trigonometric polynomialAt multiples of $2\pi$, the quotient is interpreted by its removable value $2N+1$. Summing a finite geometric series proves the formula. In the convention $\hat{f}(k)=(2\pi {)}^{-1}{\int }_{-\pi }^{\pi }f(t)e^{-ikt}dt$, the Fourier partial sum is $S_{N}f(x)=(2\pi {)}^{-1}{\int }_{-\pi }^{\pi }{D}_N(t)f(x-t)dt$.

Its normalized integral is one, but it changes sign. Its growing L1 norm explains why sharper truncation need not improve convergence for every continuous input; the positive Fejér kernel behaves differently.