Fourier Analysis is the decomposition of any square-integrable functions into an infinite series of trigonometric functions. Here we show that the trigonometric polynomial functions are dense in the space of periodic continuous functions, and thus can be used as good approximations. This implies that any repeating pattern-as-function, however complex, as long as having continuity, has a series of simple trigonometric functions that will approximate it within a given error-tolerance.
We begin thus with some definitions:
- A subset
, the space of continuous real functions from any metric space
, is a function algebra if it is closed under addition, scalar multiplication, and function multiplication, i.e.
.
vanishes at point
if
.
Note:Non-Vanishing in
, i.e.
.
separates points if
. \
Note:Separates
a countable collection of
are dense in
- a trigonometric polynomial is defined by:
Note: the function algebra of all trigonometric polynomials,separates points of
and vanishes nowhere
The Stone-Weierstrass Theorem: If
is a compact metric space and
is a function algebra in
that vanishes nowhere and separates points, then
is dense in 
i.e. 
Note: We can see immediately that
is dense in the periodic functions on 
Lemma 1: If
vanishes nowhere and separates points then
with specified values at any pair of distinct points, i.e. 
since
vanishes nowhere, there must at least a function for each point not equal to 0 at that point, i.e.
. Since
is an algebra,
and is non-zero at either point, i.e.
. Since
separates points, let
be a separator of these two points, i.e.
. Thus, there is a linear combination of
and
with the specified values:
Letting , so with
, i.e. there is a linear solution, implying that
is the specified function with
.
Lemma 2: The closure of a function algebra in
is a function algebra
it follows that
, which is thus closed under addition, function multiplication, and scalar multiplication, and so a function algebra.
Proof of the Stone-Weierstrass Theorem: 
Prove: by approximating the absolute value function on the interval
by a polynomial.
From the Weierstrass Approximation Theorem: since is a continuous function, on
polynomial
that approximates it, in that
The closures of a function algebra is closed under absolute value, minimum, and maximum operations.
Let, which has a zero constant term, yet still approximates
since
from Weierstrass Approximation Theorem so
. Writing
and
as an approximation of
, by Lemma 2
is an algebra so
since
has no constant term and can thus be generated by the 3 closed function algebraic operations. With
by the sequence of
approximations with
. Since
and by repetition, so are the max \& min for a finite number of functions.
separates
: Due to compactness of M,
for which the linear combination of the product of their separating and non-vanishing functions (
) are dense in
, by Lemma 1,
.
(i) Fixingand letting
vary, since
are continuous (i.e.
), so is
, and thus because
neighborhood around q such that for
. Since
is compact and
, a finite subcover. Defining
, since
for some
for some
.
(ii) Now repeating the above for variable,
is continuous with
neighborhood around p such that for
, a finite subcover. Setting
and since
for some
for some
.
Fromsince
for some i and
, therefore
. Thus, from
.
A corollary of the Stone-Weierstrass Theorem: Any
-periodic continuous function of
can be uniformly approximated by a trigonometric polynomial
\[T(x)=a_0 + \sum_{k=1}^{n}a_k \cos kx + \sum_{k=1}^{n}a_k \sin kx]\ parameterizes the unit circle
by
. Since
is compact, a
-periodic continuous function of
is equivalent to a continuous function on
. The trigonometric functions
are an algebra. If
, so
vanishes nowhere. And,
so it separates points. Thus, by the Stone-Weierstrass Theorem, the trigonometric polynomials
are dense in
.
This result may be extended to any p-periodic function by changing the trigonometric polynomial class
\[T_p(x)=a_0 + \sum_{k=1}^{n}a_k \cos \frac{k}{r}x + \sum_{k=1}^{n}a_k \sin \frac{k}{r}x, \ r=\frac{p}{2\pi}]\
Fourier Analysis
The Trigonometric Functions are an orthonormal basis to
and so their linear combinations with
Fourier coefficients are dense, with the coefficients going to zero.
References
- Pugh, Real Mathematical Analysis, pp. 234-239.
- Elias M. Stein & Rami Shakarchi, Real Analysis: Measure Theory, Integration, \& Hilbert Spaces. Princeton Lectures in Analysis, Vol. III, pp. 160,170,171.