12.6. MEAN SQUARE APPROXIMATION 287

Note how these kernels are nonnegative, unlike the Dirichlet kernels. Also there isa large bump in the center which gets increasingly large as n gets larger. The fact thesekernels are nonnegative is what is responsible for the superior ability of the Fejer means toapproximate a continuous function.

Theorem 12.5.3 Let f be a continuous and 2π periodic function. Then

limn→∞∥ f −σn+1 f∥0 = 0.

Proof: Let ε > 0 be given. Then by part 2. of Lemma 12.5.2,

| f (x)−σn+1 f (x)|=∣∣∣∣∫ π

−π

f (x)Fn+1 (y)dy−∫

π

−π

Fn+1 (y) f (x− y)dy∣∣∣∣

=

∣∣∣∣∫ π

−π

( f (x)− f (x− y))Fn+1 (y)dy∣∣∣∣≤ ∫ π

−π

| f (x)− f (x− y)|Fn+1 (y)dy

=∫

δ

−δ

| f (x)− f (x− y)|Fn+1 (y)dy+∫

π

δ

| f (x)− f (x− y)|Fn+1 (y)dy

+∫ −δ

−π

| f (x)− f (x− y)|Fn+1 (y)dy

Since Fn+1 is even and | f | is continuous and periodic, hence bounded by some constant Mthe above is dominated by

≤∫

δ

−δ

| f (x)− f (x− y)|Fn+1 (y)dy+4M∫

π

δ

Fn+1 (y)dy

Now choose δ such that for all x, it follows that if |y| < δ then | f (x)− f (x− y)| < ε/2.This can be done because f is uniformly continuous on [−π,π] by Theorem 6.7.2 on Page112. Since it is periodic, it must also be uniformly continuous on R. (why?) Therefore, forthis δ , this has shown that for all x, | f (x)−σn+1 f (x)| ≤ ε/2+4M

∫π

δFn+1 (y)dy and now

by Lemma 12.5.2 it follows

|| f −σn+1 f ||0 ≤ ε/2+8Mπ

(n+1)sin2(

δ

2

)4π

< ε

provided n is large enough.

12.6 Mean Square ApproximationThe partial sums of the Fourier series of f do a better job approximating f in the meansquare sense than any other linear combination of the functions, eikθ for |k| ≤ n. This willbe shown next. It is nothing but a simple computation. Recall the Fourier coefficients are

ak =1

2π

∫π

−π

f (θ)e−ikθ dθ

Also recall that ∫π

−π

eikθ e−ilθ dθ =

{2π if l = k0 if l ̸= k