Fejér's theorem is a result in the theory of Fourier series, specifically concerning the convergence of the Fourier series of a periodic function. It states that if \( f \) is a piecewise continuous function on the interval \([-L, L]\), then the sequence of partial sums of its Fourier series converges uniformly to the average of the left-hand and right-hand limits of \( f \) at each point.
Articles by others on the same topic
The Cesàro means of the Fourier series of a continuous periodic function converge uniformly to that function. The Fejér kernel is nonnegative with total mass one after normalization, and its mass outside every fixed neighborhood of zero tends to zero. Uniform continuity controls the near contribution, while the vanishing tail controls the rest. For an function the same positive-kernel argument gives an contraction, which is useful even when pointwise behavior is unavailable.