11.6.2 Mean Square Approximation
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
. This will be
shown next. It is nothing but a simple computation. Recall the Fourier coefficients
Then using this fact as needed, consider the following computation.
Then adding and subtracting 2π ∑
Therefore, to make
as small as possible for all choices of bk, one should let bk = ak, the kth Fourier coefficient.
Stated another way,
for any choice of bk. In particular,
and a simple computation of the above sort shows that also
Now it is easy to prove the following fundamental theorem.
Theorem 11.6.5 Let f ∈ R
and it is periodic of period
Proof: First assume f is continuous and 2π periodic. Then by 11.23
and the last expression converges to 0 by Theorem 11.6.4
Next suppose f ∈ R
for all x
. Then the construction used in
yields a continuous function h
which vanishes off some closed interval contained
Since h vanishes off some closed interval contained in
is extended off
periodic, it follows the resulting function, still denoted by h,
is continuous. Then using
the inequality (For a better inequality, see Problem 2
and from 11.24 and 11.25 this is no larger than
and by the first part, this last term converges to 0 as n →∞.
Therefore, since ε
this shows that for n
This proves the theorem.