Fourier Series through Complex Analysis
Scott Rome
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Continuous periodic functions are functions on S 1
Let C2π (R) denote functions f : R → R which satisfy f (x + 2π) = f (x), that is, f is 2πperiodic, henceforth periodic. To be more precise, it would be more convenient to define
the space as
C2π (R) = {f : f : [0, 2π) → R is continuous and lim f (x) = f (0)},
x→2π
where there exists a bijection to periodic continuous functions on R by identifying the
functions with their periodic extensions to all of R.
Many times, people identify C2π (R) with functions defined on the unit circle S 1 ⊂ C.
This we will show they are isomorphic as sets. That is, there exists a bijection between
them. In which case, we may think of them for our purposes as “the same thing.”
We will construct an explicit map. Consider first the map ` : [0, 2π) → S 1 defined by
θ 7→ eiθ
(1.1)
has a well defined inverse on [0, 2π) (hence the half open interval– so ` would be injective!).
For each f ∈ C2π (R), there is a corresponding map on the unit circle fs = f ◦ `−1 . Then
if we consider L : C2π (R) → C(S 1 ) given by
f 7→ fs
it is straightforward to show this is a bijection. First, we can show it is 1-1. If L(f ) = L(g)
for some functions f and g, then
(f − g) ◦ `−1 ≡ 0.
Since `−1 is a bijection, that implies f − g ≡ 0, proving the claim. Take any h ∈ C(S 1 ),
then the function h ◦ ` is a function on C2π (R) such that L(h) = h ◦ ` ◦ `−1 = h. This proves
the theorem.
Theorem 1.1. C2π (R) ∼
= C(S 1 ) as sets. That is, we may identify continuous periodic
functions on R with continuous functions on S 1 .
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The above theorem is the reason why (one version of) the Fourier transform is sometimes thought of as being a map of functions defined on the unit circle– to emphasize this
relationship.
It is also true that this can be a way to view continuous functions on R such that their
limits to ±∞ is 0 (or equal). Then using stereographic projection, you may again identify
the space of C(R) with C(S 1 ).
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Existence of Fourier series as a consequence of complex
analysis
Now that we may identify continuous periodic functions defined on R with continuous
functions on S 1 . Thus we can view a function f ∈ C2π (R) that is periodic as a function
f : S 1 ⊂ C → R. We can extend f to a harmonic function in T = {z ∈ C : |z| ≤ 1} using
the Cauchy Integral Formula. Indeed, define We can extend f to a harmonic function in
T = {z ∈ C : |z| ≤ 1} using the Cauchy Integral Formula. Indeed, define for w ∈ T ,
Z
1
f (ζ)
u(w) :=
dz.
(2.1)
2πi S 1 z − w
Notice, many books on complex analysis have the integral representation be valid only
on the interior of the region outlined by the curve. Since S 1 is sufficiently smooth and
f is continuous, it is true that the representation extends to the boundary (see Complex
Analysis by Kodaira).
Thus this function is holomorphic as a consequence of Cauchy’s Integral Theorem and
therefore has a power series expansion around 0 which is unique and at least holds up to
the boundary. Call the series
X
u(z) =
ak z k .
(2.2)
k∈Z
Utilizing that f on the real line is periodic, we can write the Fourier series for f via a
restriction to the unit circle for u:
X
f (x) = u(eix ) =
ak eikx .
(2.3)
k∈Z
We must point out that the “slick” part of this method is that we know from analyticity
that this Fourier series exists and converges. Now the battle begins over the dreaded 2π
in the exponent... Before we start, notice we can also recover the Fourier coefficients if we
recall that eix is an orthnormal set with respect to the the L2 inner product, using (2.3),
Z
−ikx
ak = (f, e
)L2 ([0,2π)) =
f (x)e−ikx dx.
(2.4)
[0,2π)
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The lack of 2π comes from the choice of normalization. Regardless, we have just derived
the Fourier series (and proven it converges trivially via complex analysis!).
Because of this formulation, many of the properties of the Fourier series come for free
from complex analysis as well. Indeed, you gain all the theorems about analytic functions.
In particular, you may do termwise differentiation and integration, and you know that the
uniform limit of functions with Fourier series again have a Fourier series defined by the
limit of the coefficients.
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