Introduction to Mathematical Physics/Dual of a topological space

DefinitionEdit

Definition:

Let   be a topological vectorial space. The set   of the continuous linear form on   is a vectorial subspace of   and is called the topological dual of  .


DistributionsEdit

chapdistr

Distributions\index{distribution} allow to describe in an elegant and synthetic way lots of physical phenomena. They allow to describe charge "distributions" in electrostatic (like point charge, dipole charge). They also allow to genereralize the derivation notion to functions that are not continuous.

Definition:

L. Schwartz distributions are linear functionals continuous on  , thus the elements of the dual   of  .

Definition:

A function is called locally summable if it is integrable in Lebesgue sense over any bounded interval.

Definition:

To any locally summable function  , a distribution   defined by:

 

can be associated.

Definition:

Dirac distribution,\index{Dirac distribution} noted   is defined by:

 

Remark:

Physicist often uses the (incorrect!) integral notation:

 

to describe the action of Dirac distribution   on a function  .


Convolution of two functions   and   is the function   if exists defined by:

 

and is noted:

 

Convolution product of two distributions   and   is (if exists) a distribution noted   defined by:

 

Here are some results:

  • convolution by   is unity of convolution.
  • convolution by   is the derivation.
  • convolution by   is the derivation of order  .
  • convolution by   is the translation of  .

The notion of Fourier transform of functions can be extended to distributions. Let us first recall the definition of the Fourier transform of a function:

Definition:

Let   be a complex valuated function\index{Fourier transform} of the real variable  . Fourier transform of   is the complex valuated function  \sigma</math> defined by:

 

if it exists.

A sufficient condition for the Fourier transform to exist is that   is summable. The Fourier transform can be inverted: if

 

then

 

Here are some useful formulas:

 

 

 

 

 

 

Let us now generalize the notion of Fourier transform to distributions. The Fourier transform of a distribution can not be defined by

 

Indeed, if  , then   and the second member of previous equality does not exist.

Definition:

Space   is the space of fast decreasing functions. More precisely,   if

  • its  derivative   exists for any positive integer  .
  • for all positive or zero integers   and  ,   is bounded.

Definition:

Fourier transform of a tempered distribution   is the distribution   defined by

 

The Fourier transform of the Dirac distribution is one:

 

Distributions\index{distribution} allow to describe in an elegant and synthetic way lots of physical phenomena. They allow to describe charge "distributions" in electrostatic (like point charge, dipole charge). They also allow to genereralize the derivation notion to functions that are not continuous.

Statistical descriptionEdit

Random variablesEdit

Distribution theory generalizes the function notion to describe\index{random variable} physical objects very common in physics (point charge, discontinuity surfaces,\dots). A random variable describes also very common objects of physics. As we will see, distributions can help to describe random variables. At section secstoch, we will introduce stochastic processes which are the mathemarical characteristics of being nowhere differentiable.

Let   a tribe of parts of a set   of "results"  . An event is an element of  , that is a set of  's. A probability   is a positive measure of tribe  . The faces of a dice numbered from 0 to 6 can be considered as the results of a set  . A random variable   is an application from   into   (or  ). For instance one can associate to each result   of the de experiment a number equal to the number written on the face. This number is a random variable.

Probability densityEdit

Distribution theory provides the right framework to describe statistical "distributions". Let   be a random variable that takes values in  .

Definition:

The density probability function   is such that:

 

It satisfies:  

Example:

The density probability function of a Bernoulli process is:

 

Moments of the partition functionEdit

Often, a function   is described by its moments:

Definition:

The   moment of function   is the integral

 

Definition:

The mean of the random variable or mathematical expectation is moment  

Definition:

Variance   is the second order moment:

 

The square root of variance is called ecart-type and is noted  .

Generating functionEdit

Definition:

The generatrice function\index{generating function} of probability density   is the Fourier transform of .

Example:

For the Bernouilli distribution:

 

The property of Fourier transform:

 

implies that:

 

Sum of random variablesEdit

Theorem:

The density probability   associated to the sum   of two {\bf independent} random variables is the convolution product of the probability densities   and  .

We do not provide here a proof of this theorem, but the reader can on the following example understand how convolution appears. The probability that the sum of two random variables that can take values in   with   is   is, taking into account all the possible cases:

 

This can be used to show the probability density associated to a binomial law. Using the Fourier counterpart of previous theorem:

 

So

 

Let us state the central limit theorem.

Theorem:

The convolution product of a great number of functions tends\footnote{ The notion of limit used here is not explicited, because this result will not be further used in this book. } to a Gaussian. \index{central limit theorem}

 

Proof:

Let us give a quick and dirty proof of this theorem. Assume that   has the following Taylor expansion around zero:

 

and that the moments   with   are zero. then using the definition of moments:

 

This implies using   that:

 

A Taylor expansion yields to

 

Finally, inverting the Fourier transform: