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 functionst they represent.
Here we will take Fourier series in complex form, ie the limit of the partial sums on
\(S_{N}(x)=\sum_{n=-N}^{N} f_{n}e^{2\pi nx} \)
This is possible because
\(cosnx= \displaystyle \frac{ e^{inx}+e^{-inx}}{2} \)
\(sinnx= \displaystyle \frac{ e^{inx}-e^{-inx}}{2i} \)
Now just replace in
\(f(x)=\frac{A_{0}}{2}+\sum_{n=1}^{\infty}(A_{n}cosnx+B_{n}sinnx) \)
so, we get the complex format of the Fourier Series
And consider all functions Periodic square-integrable, ie the set
\( L^2(T)=\{f:periodicas: \int_{-\pi}^{\pi}|f(x)|^2<\infty \} \)
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.
Hilbert Spaces
Basic concepts and Principles
Extended Theory
Lets
\( \mathbb{I} ^2(-\pi, \pi)=\{f: \int_{-\pi}^{\pi}|f(x)|^2<\infty \} \)
Now we make the following equivalence relationship
\(f, g \in \mathbb{I} ^2(-\pi, \pi), f R g \Leftrightarrow f=g\; (a.e) \)
a. e. minds almost everywhere, in the sense of the set of points where the functions are different has measure zero.
To equivalence classes set
\( \mathbb{I} ^2(-\pi, \pi)/R \)
\( L ^2(-\pi, \pi) \)
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
\( <f,g >=\int_{-\pi}^{\pi}f(x)\overline{g(x)}dx \)(1)
and so, obtain the following norm
\(||f(x)||=\int_{-\pi}^{\pi}|f(x)|^2 dx \)
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
\( l^2(Z)=\{(a_{n}) _{n=-\infty}^{\infty} : \sum_{n=\infty}^{\infty}(a_n)^2 < \infty \} \)
Now, we can define the inner product and norm as follows
\( < (a_{n}, b_{n}) >= \sum_{n=-\infty}^{\infty}(a_n)(b_n) \)
\( ||(a_{n}||^2 = \sum_{n=-\infty}^{\infty} |a_n|^2 \)
The following Theorem, shows us that all Hilbert space contains at least a Hilbert Space has got a Orthonormal System
Now, given a Hilbert Space, H and an Orthonormal System {un} of H, we can define the Fourier Transform as follows
\( T:H \rightarrow l^2(Z) \)
\( \: \: \: \: \; \: \; x \rightarrow T(x)=\{ <x,u_{n} > \} _{n=0} ^{\infty} \)
The application is well defined by the Bessel inequality
\( \sum_{n=-\infty}^{\infty} <x, u_{n} > \leqslant || x ||^2 \)
Note: The last of these conditions is also called The Riesz-Fischer Theorem
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
In this way you can make a correspondence through Fourier Coefficients between the sets
\( T: L^2(T)=H \rightarrow l^2(Z) \)
\( \: \: \: \: \; \: \; f \rightarrow \hat{f}=\{f_n \)
\( f_n = < f,e^{2 \pi nx} > =\int_{0}^{1}f(x)e^{-2\pi nx}dx \).
As this application is isomorphic and isometric, we have the Plancherel identity
\(||f||=|| \hat{f}=\sum_{n=- \infty}^{ \infty } f_n \)
And Perseval identity also holds
\( <f,g >=< \hat{f},\hat{g} >=\sum_{n=- \infty}^{ \infty } f_n g_n \)
In particular, if {an} ∈ l2(Z) then the partial sums
\( S_{n}(x)= \sum_{n=-N}^{N } a_{n} e^{2 \pi nx} \)
Forms a sequence converging to some function, f ∈ L2
\( \mathbb{I} ^2(-\pi, \pi)=\{f: \int_{-\pi}^{\pi}|f(x)|^2<\infty \} \)
Now we make the following equivalence relationship
\(f, g \in \mathbb{I} ^2(-\pi, \pi), f R g \Leftrightarrow f=g\; (a.e) \)
a. e. minds almost everywhere, in the sense of the set of points where the functions are different has measure zero.
To equivalence classes set
\( \mathbb{I} ^2(-\pi, \pi)/R \)
\( L ^2(-\pi, \pi) \)
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
\( <f,g >=\int_{-\pi}^{\pi}f(x)\overline{g(x)}dx \)(1)
and so, obtain the following norm
\(||f(x)||=\int_{-\pi}^{\pi}|f(x)|^2 dx \)
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
\( l^2(Z)=\{(a_{n}) _{n=-\infty}^{\infty} : \sum_{n=\infty}^{\infty}(a_n)^2 < \infty \} \)
Now, we can define the inner product and norm as follows
\( < (a_{n}, b_{n}) >= \sum_{n=-\infty}^{\infty}(a_n)(b_n) \)
\( ||(a_{n}||^2 = \sum_{n=-\infty}^{\infty} |a_n|^2 \)
The following Theorem, shows us that all Hilbert space contains at least a Hilbert Space has got a Orthonormal System
Now, given a Hilbert Space, H and an Orthonormal System {un} of H, we can define the Fourier Transform as follows
\( T:H \rightarrow l^2(Z) \)
\( \: \: \: \: \; \: \; x \rightarrow T(x)=\{ <x,u_{n} > \} _{n=0} ^{\infty} \)
The application is well defined by the Bessel inequality
\( \sum_{n=-\infty}^{\infty} <x, u_{n} > \leqslant || x ||^2 \)
Theorem (Complete orthonormal systems characterization)
Let H be a Hilbert Space and {an} an Orthonormal System .
Them are Equivalent
1) \( \{u_{n} \} \) Is a complete Orthonormal System
2) The subspace spanned by \( \{u_{n} \} \), ie, the set of finite linear combinations of elements\( \{u_{n} \} \) is dense in H.
3) The dBessel inequality becomes the Plancherel Identity
\( \sum_{n=-\infty}^{\infty} <x, u_{n} > = || x ||^2 \)
4) Fourier Transformation is an Isometry
Let H be a Hilbert Space and {an} an Orthonormal System .
Them are Equivalent
1) \( \{u_{n} \} \) Is a complete Orthonormal System
2) The subspace spanned by \( \{u_{n} \} \), ie, the set of finite linear combinations of elements\( \{u_{n} \} \) is dense in H.
3) The dBessel inequality becomes the Plancherel Identity
\( \sum_{n=-\infty}^{\infty} <x, u_{n} > = || x ||^2 \)
4) Fourier Transformation is an Isometry
Note: The last of these conditions is also called The Riesz-Fischer Theorem
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
In this way you can make a correspondence through Fourier Coefficients between the sets
\( T: L^2(T)=H \rightarrow l^2(Z) \)
\( \: \: \: \: \; \: \; f \rightarrow \hat{f}=\{f_n \)
\( f_n = < f,e^{2 \pi nx} > =\int_{0}^{1}f(x)e^{-2\pi nx}dx \).
As this application is isomorphic and isometric, we have the Plancherel identity
\(||f||=|| \hat{f}=\sum_{n=- \infty}^{ \infty } f_n \)
And Perseval identity also holds
\( <f,g >=< \hat{f},\hat{g} >=\sum_{n=- \infty}^{ \infty } f_n g_n \)
In particular, if {an} ∈ l2(Z) then the partial sums
\( S_{n}(x)= \sum_{n=-N}^{N } a_{n} e^{2 \pi nx} \)
Forms a sequence converging to some function, f ∈ L2
Thus we can identify f(x) with its Fourier series
\( f(x)= \sum_{n= \infty}^{ \infty } f_{n} e^{2 \pi nx} \)
\( f(x)= \sum_{n= \infty}^{ \infty } f_{n} e^{2 \pi nx} \)
