 Hilbert Spaces

Basic concepts and Principles

The purpose of this section is to find the class of functions for which the Fourier series makes sense. For this, we seek a correspondence between function spaces and sequence spaces. This will entail an equivalence between the Fourier coefficients and the functions that they represent.
Here we will take Fourier series in complex form, ie the limit of the partial sums on

SN (x) = ∑ N-N f-n e2πinx

This is possible because

cos nx = (einx + e-inx ) / 2
sin nx = (einx - e-inx ) / 2i

Now just replace in

f(x) = A0/2 + ∑ n=1 ( An cos nx + Bn sen nx)      (1)

so, we get the complex format of the Fourier Series
And consider all functions Periodic square-integrable, ie the set

L2 (T) = { f functions: ∫π |f(x)|2 < ∞ }

This is an important example of Hilbert Space , is the space of square integrable functions, which also we will consider two functions as the same if they differ only in a set of measure zero.

Hilber't spaces are like vectorial spaces seen in the Linear Algebra section, but with the difference of the infinite dimension at Hilbert spaces case.

Extended Theory

Lets

£2 (-π, π) = { f functions: ∫π |f(x)|2 < ∞ }

Now we make the following equivalence relation

f, g ∈ £2 (-π, π) , f R g <=> f(x) = g(x) a.e.

a. e. minds almost everywhere, in the sense that the set of points where the functions are different has measure zero.
To equivalence classes set

£2 (-π, π) / R

we call      L2 (-π, π)

Note that this whole argument is just to say two functions equivalents if it are equal except on a set of measure zero.

On this space we make the scalar product

<f, g> = ∫πf(x) g(x) dx     (1)

and so, obtain the following norm

|| f ||; = (∫π | f(x) | 2 dx) 1/2

It can be shown that this space of square integrable functions endowed with the inner product (1) is a Hilbert Space .
We consider now the space of square summable sequences, ie we consider the set

l 2 (Z) = { ( an )n=-∞ : ∑n=-∞ ( an ) 2 < ∞ }

Now, we can define the inner product and norm as follows

< ( an ), ( bn )> = ∑n=-∞ an bn

|| (an) || = ∑n=-∞ (an) 2

The following Theorem, shows us that all Hilbert space contains at least a Hilbert Space has got a Orthonormal System

Theorem
All Hilbert Space has a Orthonormal System Complete.

Now, given a Hilbert Space, H and an Orthonormal System {un} of H, we can define the Fourier Transform as follows

T: H → l 2(Z)
x   → T(x) = { <x, un> }n=-∞

The application is well defined by the Bessel inequality

(∑n=-∞ < x, un > )1/2 ≤ || x ||

Theorem (Complete orthonormal systems characterization)

Let H be a Hilbert Space and {an} an Orthonormal System .

Them are Equivalent

1) {un} Is a complete Orthonormal System

2) The subspace spanned by {un} , ie, the set of finite linear combinations of elements {un} is dense in H.

3) The dBessel inequality becomes the Plancherel Identity

∑n=-∞ < x, un > 2 = || x ||2

4) Fourier Transformation is an Isometry

Note: The last of these conditions is also called The Riesz-Fischer Theorem

Corolary
All Hilbert Space is isomorphic and isometric to a space of kind L 2

Thus, any Hilbert space, H is essentially an L 2 which is isomorphic and isometric to a sequence space called here l 2 (Z).

Knowing this last and returning at contrary sense, we can say: if the Fourier coefficients are a complete orthonormal system in l 2 (Z), then it is possible to identify l 2 (Z) with functions space L 2.

Note: We say again: In resume we are identifying the sucesions space l 2 (Z) with the functions space L 2

Theorem
The functions sequence
un(x) = e2πinx
Is a Complete Orthonormal System

In this way you can make a correspondence through Fourier Coefficients between the sets

T: L 2(T) → l 2(Z)
f → f^ = { fn }

Where fn = < f, e2πinx > = ∫π f(x) e-2πinx dx.

As this application is isomorphic and isometric, we have the Plancherel identity || f || = || f^ || = ∑n=-∞ fn

And Perseval identity also holds

<f, g> = <f^, g^> = ∑n=-∞ fn gn

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

SN (x) = ∑ N-N fn e2πinx

Forms a sequence converging to some function, f ∈ L2

Thus we can identify f(x) with its Fourier series

f(x) = ∑n=-∞ fn e2πinx

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