L2 and dirac delta function as a tempered distribution

[Vijay Aghera]

Hi i'm unsure about the following problem: if f(x)=u(x)*exp{-x*alpha} alpha>0 where u(x)is the step function then does f prime (1st derivative wrt x) belong to lebesgue space with p=2 on the real line.

f prime is a linear combination of a piecewise function and a tempered distribution. The latter is because the derivative of the step function is the dirac delta function. From wikipedia i know all square integrable functions are tempered distributions but i am wondering whether the converse is true. Any suggestions would be terrific. thanks.

[W^3]

In article
<14980286.1228141148...@nitrogen.mathforum.org>,
Vijay Aghera <vijay...@hotmail.com> wrote:

No, the delta function is not given by any L^2 function: Suppose to
the contrary delta = f in L^2. Choose any smooth function g with g(0)
= 1 having compact support. Set g_n(x) = g(nx). Then delta(g_n) = 1
for all n, but int_R f*g_n -> 0.

[Vijay Aghera]

thanks very much for your reply it was helpful.