Fejér's theorem - Alchetron, The Free Social Encyclopedia

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 ) = ∑ k = − n n c k e i k x ,

where

c k = 1 2 π ∫ − π π f ( t ) e − i k t d t ,

and

σ n ( x ) = 1 n ∑ k = 0 n − 1 s k ( x ) = 1 2 π ∫ − π π f ( x − t ) F n ( t ) d t ,

with F_n being the nth order Fejér kernel.

A more general form of the theorem applies to functions which are not necessarily continuous (Zygmund 1968, Theorem III.3.4). Suppose that f is in L¹(-π,π). If the left and right limits f(x₀±0) of f(x) exist at x₀, or if both limits are infinite of the same sign, then

σ n ( x 0 ) → 1 2 ( f ( x 0 + 0 ) + f ( x 0 − 0 ) ) .

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 (Zygmund 1968, Theorem III.5.1).

References

Fejér's theorem Wikipedia

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