The partial sums of the Fourier series of f do a better job approximating f in the mean
square sense than any other linear combination of the functions, eikθ for
|k|
≤ n. This will be
shown next. It is nothing but a simple computation. Recall the Fourier coefficients
are
∫ π
ak = 1-- f (θ)e−ikθ
2π −π
Then using this fact as needed, consider the following computation.
∫ | |2
π || ∑n ikθ||
−π ||f (θ)− bke || dθ
k= −n
∫ π ( n ) ( n )
= f (θ) − ∑ bkeikθ f-(θ)− ∑ ale−ilθ dθ
−π k=− n l=−n
Since h vanishes off some closed interval contained in
(− π,π)
, if h is extended off
[− π,π]
to
be 2π periodic, it follows the resulting function, still denoted by h, is continuous. Then using
the inequality (For a better inequality, see Problem 2.)
( )
(a+ b +c)2 ≤ 4 a2 + b2 + c2
∫ ∫
π |f − S f|2dx = π (|f − h|+ |h − S h|+ |S h− S f|)2dx
−π n − π n n n