Hilbert's Spaces

The goal of this section is to find a class of functions wich Fourier Series makes sense. The idea that guide us is to search an equivalence between a functions space and some succesions space. Then this will allow us to make and equivalence between Fourier coefficinets and functions represented by them.

Here after we have the Fourier series in complex format, that is we take:

\(S_{N}(x)=\sum_{n=-N}^{N} f_{n}e^{2\pi nx} \)

This can be done because:

\(cosnx= \displaystyle \frac{ e^{inx}+e^{-inx}}{2} \)
\(sinnx= \displaystyle \frac{ e^{inx}-e^{-inx}}{2i} \)

By substitution these formulas in:

\(f(x)=\frac{A_{0}}{2}+\sum_{n=1}^{\infty}(A_{n}cosnx+B_{n}sinnx) \)      

we obtain the Forier Series in complex format.

We will also consider the space of \( 2\pi \)-Periodics-with-square-integrable functions, that is the set:

\( L^2(T)=\{f: f(x)=f(x+2 \pi) : \int_{-\pi}^{\pi}|f(x)|^2<\infty \} \)

This is an important example of Hilbert space , where besides two functions are equivalent if they are different in a zer-measure set, we will explain it at length below.

Extended theory

We call:

\( \mathbb{I} ^2(-\pi, \pi)=\{f: \int_{-\pi}^{\pi}|f(x)|^2<\infty \} \)

Now we define over this set following equivalence relationship.

\(f, g \in \mathbb{I} ^2(-\pi, \pi), f R g \Leftrightarrow f=g\; (a.e) \)

where a. e. means almost everywhere or equivalently, the set where functions are different has measure zezo.

To equivalence classes set given by:

\( \mathbb{I} ^2(-\pi, \pi)/R \)

We simply call it as:

\( L ^2(-\pi, \pi) \)

Within this space we can define a scalar product as follows:

\( <f,g >=\int_{-\pi}^{\pi}f(x)\overline{g(x)}dx \)(1)

It is possible to show that this functions space\( L ^2(-\pi, \pi) \) wndowed with the inner product (1) is actually a de hecho un Hilbert space. Besides in it we can define a norm:

\(||f(x)||=\int_{-\pi}^{\pi}|f(x)|^2 dx \)

We consider now the succesions space of square-summable, that is the set:

\( l^2(Z)=\{(a_{n}) _{n=-\infty}^{\infty} : \sum_{n=\infty}^{\infty}(a_n)^2 < \infty \} \)

We can define, ss we have done in the functions space a scalar product and a norm as follows:

\( < (a_{n}, b_{n}) >= \sum_{n=-\infty}^{\infty}(a_n)(b_n) \)

\( ||(a_{n})||^2 = \sum_{n=-\infty}^{\infty} |a_n|^2 \)

This space \( l^2(Z) \), is also a Hilbert space.

Following theorem guarantees the existence of an Orthonormal system complete for all Hilbert space Wich means that all element in a Hilbert Space H, can be expressed as a infinite linear combination of elements of the space . Other way to say the same is that the sub-space spanned by the succesion is dense in H.

Time to note that we are working with infinite-dimensions vector-space: The basis here has infinite elements.

Every Hilbert space has one complete Orthonormal system .

Now given a Hilbert space H, and an Orthonormal system {un} in H, we can build the Fourier transformation as follows:

\( T:H \rightarrow l^2(Z) \)
\( \: \: \: \: \; x \rightarrow T(x)=\{ <x,u_{n} > \} _{n=0} ^{\infty} \)

This application is well defined because of the Bessel inequality

\( \sum_{n=-\infty}^{\infty} <x, u_{n} > \leqslant || x ||^2 \)

Theorem (Complete Orthonormal system characterization)

Let's H Hilbert space and lets {un} Orthonormal system within H. Then following statements are equivalent:

1) \( \{u_{n} \} \) is a complete orthonormal complete.

2) The space spanned by \( \{u_{n} \} \), is dense in H.

3) Bessel inequality becomes the Plancherel identity

\( \sum_{n=-\infty}^{\infty} <x, u_{n} > = || x ||^2 \)

4) Fourier transform is actually an isometry.

Note: Last condition is the called Riesz-Fischer Theorem.


Every Hilbert spaceis isomorphic and isometric to a space of type L 2
So, every Hilbert space H, can be consider as one L 2, wich is isomorphic and isometric to a succesions space called l2(Z). By kowing this, we can proceed now in the other way around and think that if Fourier coefficents were a complete Orthonormal system in l2(Z), then we should be able to identify such space with L 2 functions space.


Functions succession given by:

\(u_{n}(x)=e^{2\pi nx} \)

Makes up a complete Orthonormal system

This is the way to build the equivalence between the sets L2 and l2(Z). By using the Fourier Coefficients, with the map:

\( T:H \rightarrow l^2(Z) \)
\( \: \: \: \: \; f \rightarrow \hat{f}=\{f_n \)

Where \( f_n = < f,e^{2 \pi nx} > =\int_{0}^{1}f(x)e^{-2\pi nx}dx \)

is isomorphic and isometric because of the Plancherel Identity:

\(||f||=|| \hat{f}=\sum_{n=- \infty}^{ \infty } f_n \)

And the Perseval identity holds as well:

\( <f,g >=< \hat{f},\hat{g} >=\sum_{n=- \infty}^{ \infty } f_n g_n \)

In particular if {an} ∈ l2(Z), then partial sums given by

\( S_{n}(x)= \sum_{n=-N}^{N } a_{n} e^{2 \pi nx} \)

Conforms a convergent succession to a function, f ∈ L2

This is the way to identify a function f(x) with it Fourier series:

\( f(x)= \sum_{n=- \infty}^{ \infty } f_{n} e^{2 \pi nx} \)

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