Most general form of nascent delta function

[mmzaj]

Dear all

I'm wondering if you can help me find the most general formula of all nascent delta functions. all i have found a somewhat random forms . I'm looking for a general elegant formula that all the forms can be derived from .

thanks in advance .

 

[arildno]

The delta function isn't really a function at all, it is a distribution/linear functional.
You might read the following thread:
https://www.physicsforums.com/showthread.php?t=73447

 

[mmzaj]

thanks a lot , really informative , but here is the thing , I'm looking for a function ( or a functional ) g such that :

lim x ->0 g (x,y) = delta(y).

well , obviously any nascent delta function satisfies the condition , but the problem is that I am also interested in g(x,y) every where , not only when x -> 0 , and there is no constrains on g(x,y) other than it equals delta when x->0 . hence , the program is to find the most general form of the nascent delta functions , and construct a linear combination of them . i tried Fourier series and transform to check if there are common properties of the nascent delta functions in the frequency domain - as we electrical engineers like to call it - but in vain . here is the dilemma . can you help me here .

thanks in advance .

 

[mmzaj]

now , after a long struggle i found that what I'm looking for isn't any thing but a linear combination of probability distributions ( such as the normal distribution ) G (x,y) such that lim x->0 G(x,y) = delta (y) . but again , there is no information about the number of distributions that satisfy that condition . the normal distribution does , cauchy distribution does , but are there other distributions like that ?

 

[Hurkyl]

The Dirac delta is not special; it's just another point in the space of distributions. There are exactly as many things that converge to the zero distribution as there are that converge to the Dirac delta distribution.

There really isn't a classification for things that converge to the Dirac delta other than "it converges to the Dirac delta".