Want to learn more about Delta Function and Generalized Function

[chingkui]

I am reading a electrical engineering book about digital signal processing, in the process, those Fourier transform and discrete time Fourier transform of constant and the exponential function lead to the delta function. I understand how to manipulate them formally, but I have serious trouble with them in terms of their mathematical foundation, such as why the DTFT "exist" (which definitely doesn't converge in the usual sense) and result in the delta function. I was told I need the theory of generalized function, which I have no background. I want to find some books or online lecture notes or resources to study them myself. Can anyone point me to the easiest way to study them rigorously without too big a detour to other area of mathematics?
I have undergraduate analysis as well as some measure theory background. Please let me know what else I need to understand generalized functions, and what books would be best for this purpose. Thank you.

 

[George Jones]

A truly outstanding (but still rigorous) introduction to distributions is chapter 9, Generalized functions, from the book Fourier Analysis and its Applications by Gerald Folland.

Regards,
George

 

[arildno]

Hi, chingkui!
In: https://www.physicsforums.com/showthread.php?t=73447, I've commented on how we can make sense of the Dirac delta function, in particular how we can construct an integral representation of what the Dirac function(al) "does to" a function.
The treatment is fairly rigorous, but it does not explicitly define generalized functions/distributions or delve too much into these topics.

It should, however, be sufficient in showing that there is some sense in the Dirac delta function after all..

 

[kleinwolf]

Dirac delta is a functional (function acting on function and giving the argument of the entry function as result), but FT is an operator, function->function. I remember that sometimes there are funny things with FT, for example radial part of Coulomb potential FT :
$$|FT[1/r]=\int_0^\infty \frac{e^{ikr}}{r}r^2dr=\frac{r}{ik}e^{ikr}|_{r=0}^\infty-\frac{1}{ik}\int_0^\infty e^{ikr}$$
by parts...the first term is infinite, and the second not defined.
However if you take a Yukawa potential and redo it :
$$FT[\frac{e^{-\alpha r}}{r}]=\int_0^\infty\frac{e^{(ik-\alpha)}r}{r}r^2dr=\frac{r}{ik-\alpha}e^{(ik-\alpha)r}|_{r=0}^\infty-\frac{1}{ik-\alpha}\int_0^\infty e^{ik-\alpha}rdr\rightarrow_{\alpha\rightarrow 0}\frac{-1}{k^2}$$
Hence $$lim_{\alpha->0}FT[\frac{e^{-\alpha r}}{r}]\neq FT[\frac{1}{r}]$$...we could believe FT is hence not continuous...but it is because the Coulomb potential is not a square inegrable (L^2)...