Fejér's theorem
In mathematics, Fejér's theorem, named after Hungarian mathematician Lipót Fejér, states that if f:R → C is a continuous function with period 2π, then the sequence of Cesàro means of the sequence of partial sums of the Fourier series of f converges uniformly to f on .
Explicitly,
where
and
with Fn being the nth order Fejér kernel.
A more general form of the theorem applies to functions which are not necessarily continuous. Suppose that f is in L1. If the left and right limits f of f exist at x0, or if both limits are infinite of the same sign, then
Existence or divergence to infinity of the Cesàro mean is also implied. By a theorem of Marcel Riesz, Fejér's theorem holds precisely as stated if the mean σn is replaced with mean of the Fourier series.
