Can someone explain to me about Dirac distribution?

Consider the vector space of all infinitely differentiable functions f(x) on the real line

R. There are various subspaces of interest. The space S consists of all such functions

which decrease faster than 1/(1 + |x|^n) for all n, as do all their derivatives.

Example: f(x) = Polynomial(x) e^(-x^2). However, I don't want to get too technical.

A distribution is a linear function T : S --> C.

Example: Let G be a bounded integrable function.

The corresponding distribution is T_G (f) = Int_(-oo,oo) f(x) G(x) dx

The Dirac "delta function" (which is really a distribution) is defined by the

property

Int_(-oo,oo) f(x) delta(x) dx = f(0); this is a "sifting" property

more generally, Int f(x) delta (x - a) dx = f(a)

Every function can be regarded as a (continuous) sum of weighted delta functions

g(a) delta(x-a). To solve a linear differential equation like y" + b(x) y' + c(x) y = g

it is enough to solve when the right side is delta(x-a). Along with boundary conditions,

the solution is the Green's function. Likewise in higher dimensions.

delta(x) can be regarded as a limit of ordinary functions. For example,

let g_n(x) = n/2 for |x| < 1/n, 0 elsewhere. The area under the graph of g_n is 1.

It is easy to see that for any continuous function f(x),

liim_(n--> oo) Int_(-oo,oo) f(x) g_n(x) dx = f(0)

Another way of looking at this: if G is differentiable, with derivative G' not growing rapidly,

Int (f G' dx) = - IInt (f' G dx) because f and f' --> 0 rapidly at oo.

If T is any distribution we DEFINE T' by T'(f) = - T(f') .

Let H(x) = 0 if x < 0, 1 if x >= 0. Then T_H ' is the delta distribution.

I leave it as an exercise to check this.

Intuitively, Integral _(-oo,x) delta(t) dt = 0 if x < 0, 1 if x > 0 because you pick up

a unit mass at t = 0.

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