Can anyone provide a proof for

  • Thread starter tpm
  • Start date
  • #1
tpm
72
0

Main Question or Discussion Point

Can anyone give a proof of why the product of 2 distributions can't be defined ?? :Confused:

In fact i believe (at least it should be) that if [tex] f_{n} (x) [/tex] and [tex] g_{n} (x) [/tex] are a succesion of function for [tex] n \rightarrow \infty [/tex] then the product of the 2 sucessions should be equal to the product of the 2 distributions..

hence [tex] f_{n} (x) \rightarrow d(x) [/tex] and [tex] g_{n} (x) \rightarrow e(x) [/tex] where d(x) and e(x) are 2 distributions then :

[tex] f_{n} (x) g_{n} (x) \rightarrow d(x)e(x) [/tex] ???

I have read about 'MOllifiers' and several methods for generalizing the distribution theory to include product of distributions, also couldn't the product be always defined as a 'sum' (in fact the sum of 2 distributions is defined) since:

[tex] a X b = a+a+a+a+a+a+a+a+..... [/tex] (the sum has 'b' terms)

or [tex] log(a X b )=log(a) +log(b) [/tex] :Grumpy:
 

Answers and Replies

  • #2
191
0
Can anyone give a proof of why the product of 2 distributions can't be defined ?? :Confused:
It can be defined, but only in certain special cases and in a very special way.

Take the delta distribution defined by [itex]\delta[f]:=f(0)[/itex], where [itex]f[/itex] is in Schwartz Space/ a test function. How would you define [itex]\delta^2[/itex]?
 
  • #3
tpm
72
0
you can define [tex] \delta (x) \delta (x) = \delta ^{2} (x) [/tex] in the form.

[tex] \delta ^{2} (x) \sim \frac{ sin ^{2} (Nx)}{\pi ^{2} x^{2}} [/tex]

as N-->oo (N big) , do i get the 'Field medal' for it ?? :Bigrin:
 
  • #4
1,631
4
or [tex] log(a X b )=log(a) +log(b) [/tex] :Grumpy:
let [tex] log(a )=c'[/tex], and [tex] log( b )=c"[/tex], let's say that both logs have a base "d"( i do not know how to write it )

then by definition we have from the first

d^c'=a, and d^c"=b

if we multiply side by side we get

ab=(d^c')(d^c")=d^(c'+c") so agani by definition we have
log_d(ab)=c'+c", , i guess you can see the rest?
 
Last edited:
  • #5
Stingray
Science Advisor
671
1
[tex] f_{n} (x) g_{n} (x) \rightarrow d(x)e(x) [/tex] ???
This doesn't work. Say that two sequences [itex]f_n(x)[/itex] and [itex]g_n(x)[/itex] both converge to the Dirac distribution. Depending on what these functions are, it's possible to obtain [itex]f_n(x) g_n(x) \rightarrow 0[/itex], among many other results. The limit of the product depends on more than the limits of the individual sequences. It's not unique.

[tex] a X b = a+a+a+a+a+a+a+a+..... [/tex] (the sum has 'b' terms)
That only makes sense if b is an integer. But multiplication of distributions by integers is already defined (as is multiplication by arbitrary complex numbers).
 

Related Threads on Can anyone provide a proof for

Replies
2
Views
1K
  • Last Post
Replies
10
Views
2K
Replies
6
Views
2K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
20
Views
8K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
22
Views
3K
  • Last Post
Replies
8
Views
750
  • Last Post
Replies
12
Views
2K
Replies
7
Views
1K
Top