Fejér's theorem


Fejér's theorem

In mathematics, Fejér's theorem, named for Hungarian mathematician Lipót Fejér, states that if "f":R → C is a continuous function with period 2π, then the sequence (σ"n") of Cesàro means of the sequence ("s""n") of partial sums of the Fourier series of "f" converges uniformly to "f" on [-π,π] .

Explicitly,:s_n(x)=sum_{k=-n}^nc_ke^{ikx},where:c_n=frac{1}{2pi}int_{-pi}^pi f(t)e^{-int}dt,and:sigma_n(x)=frac{1}{n}sum_{k=0}^{n-1}s_k(x)=frac{1}{2pi}int_{-pi}^pi f(x-t)F_n(t)dt,with "F""n" being the "n"th order Fejér kernel.

A more general form of the theorem applies to functions which are not necessarily continuous harv|Zygmund|1968|loc=Theorem III.3.4. Suppose that "f" is in "L"1(-π,π). If the left and right limits "f"("x"0±0) of "f"("x") exist at "x"0, or if both limits are infinite of the same sign, then

:sigma_n(x_0) o frac{1}{2}left(f(x_0+0)+f(x_0-0) ight).

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 (C, 1) mean σ"n" is replaced with (C, α) mean of the Fourier series harv|Zygmund|1968|loc=Theorem III.5.1.

References

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