12.4. WAYS OF APPROXIMATING FUNCTIONS 285

12.4 Ways of Approximating FunctionsGiven above is a theorem about Fourier series converging pointwise to a periodic functionor more generally to the mid point of the jump of the function. Notice that some sort ofsmoothness of the function approximated was required, the Dini condition. It can be shownthat if this sort of thing is not present, the Fourier series of a continuous periodic functionmay fail to converge to it in a very spectacular manner. In fact, Fourier series don’t dovery well at converging pointwise. However, there is another way of converging at whichFourier series cannot be beat. It is mean square convergence.

Definition 12.4.1 Let f be a function defined on an interval, [a,b] . Then a se-quence, {gn} of functions is said to converge uniformly to f on [a,b] if

limn→∞

sup{| f (x)−gn (x)| : x ∈ [a,b]}= 0.

The expression sup{| f (x)−gn (x)| : x ∈ [a,b]} is sometimes written3 as ∥ f −gn∥0 . Moregenerally, if f is a function,

∥ f∥0 ≡ sup{| f (x)| : x ∈ [a,b]}

The sequence is said to converge mean square to f if

limn→∞∥ f −gn∥2 ≡ lim

n→∞

(∫ b

a| f −gn|2 dx

)1/2

= 0

12.5 Uniform Approximation with Trig. PolynomialsIt turns out that if you don’t insist the ak be the Fourier coefficients, then every continuous2π periodic function θ → f (θ) can be approximated uniformly with a Trig. polynomialof the form pn (θ)≡ ∑

nk=−n akeikθ . This means that for all ε > 0 there exists a pn (θ) such

that ∥ f − pn∥0 < ε . Here ∥ f∥0 ≡max{| f (x)| : x ∈ R} .

Definition 12.5.1 Recall the nth partial sum of the Fourier series Sn f (x) is givenby

Sn f (x) =∫

π

−π

Dn (x− y) f (y)dy =∫

π

−π

Dn (t) f (x− t)dt

where Dn (t) is the Dirichlet kernel, Dn (t) = (2π)−1 sin(n+ 12 )t

sin( t2 )

The nth Fejer mean, σn f (x)

is the average of the first n of the Sn f (x). Thus

σn+1 f (x)≡ 1n+1

n

∑k=0

Sk f (x) =∫

π

−π

(1

n+1

n

∑k=0

Dk (t)

)f (x− t)dt

The Fejer kernel is Fn+1 (t)≡ 1n+1 ∑

nk=0 Dk (t) .

As was the case with the Dirichlet kernel, the Fejer kernel has some properties.

3There is absolutely no consistency in this notation. It is often the case that ||·||0 is what is referred to in thisdefinition as ||·||2 . Also ||·||0 here is sometimes referred to as ||·||

∞. Sometimes ||·||2 referrs to a norm which

involves derivatives of the function.