Table of Contents

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

*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)*Fixing and 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 .

From since 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.