bro visit this site and you will find answer:

http://en.wikipedia.org/wiki/Colombeau_algebra

Thanks Arnab.

Is this the original definition of Colombeau?

Otherwise, who and when this definition was given?

I suggest not following  Schwartz development of distributions. Instead use infinitesimal analysis, and then see how  the original definition of Colombeau looks like!

In *R, let dx be an infinitesimal. Then define:

f(x):= 1/dx  if x\in (-dx/2, dx/2) and zero otherwise. This satisfies all the properties of generalized function without the complexities of  Schwartz development!

.

Dear Costas,

I suppose you are speaking of Robinson non standard analysis.

The definition you have given is really very simple

but I have some problems.

Your  δ  is not smooth so we have problems

in defining the derivative.

you have given not the definition of  δ  but a definition of  δ

because there are infinite  dx.

You can also change the interval of definition to  (0,dx)

and the generalized funxction do not change.

We need the Axiom of Choice for non standard analysis

so is not constructive.

In any case your suggestion may be very useful.

Dear Giancarlo,

The derivative is a generalised function. It is the same with heavyside functions. Classically this derivative may not exist, but in nonstandard mathematics there is no problem. Using Robinson infinitesimals you can discretise almost everything. The wiener measure can be approached by a suitable random walk. 

One can develop a more sophisticated approach to generalised functions using infinitesimals. The example I geve you is really simple!   

In the following article:

G.Meloni-E.Munarini, Purely algebraic teories for elementary operational calculus and generalized functions

we show that distributions are simply definable as algebraic universal solution, in the context of sheaves, to the problem of making arbitrary continuous functions infinitely differentiable in a manner compatible with the old continuous derivative.

So, having such a beautiful purely algebraic definition for the important case of the distributions, we wonder if the new generalized functions of Colombeau, or something similar, can be treated in the same way.