Fejér kernel

The Fejér kernel F_N(x) is the Cesàro mean of the Dirichlet kernels, defined as F_N(x) = (1/N) Σ_{n=0}^{N-1} D_n(x) = (1/N) (sin(Nx/2) / sin(x/2))². Unlike the Dirichlet kernel, the Fejér kernel is non-negative, which guarantees that the Cesàro means of a Fourier series converge uniformly for continuous functions — a result known as Fejér's theorem. The Fejér kernel thus provides a more robust summability method than the raw partial sums, trading the sharpness of Dirichlet convergence for the reliability of averaged convergence. It exemplifies a general principle in analysis: when a natural approximation fails, averaging often succeeds. The Fejér kernel also appears in the study of Convergence of Fourier series and has applications in approximation theory and signal smoothing.