Dear elmolla,
thanks for your post and for your PM (i did not see you post untill today)
first ... do not worry, i think you are really good at maths
YEP, my mistake, i mix up the kroneker and the delta function:|, thanks for your clarification :idea:
BUT I have, again
, some questions :
1. You said before that you calculated the transform from the variable x to variable Ω and from variable y to variable Φ. i.e. the transform kernel is exp[-j(Ωx+Φy)], not h.
Thus, this means that Ω and Φ are the new set of dimensions, or domain, after the transform. So why you wrote '' F(α(x,y))=lim[N->∞]((1/N)∑δ(Ω-xh,Φ-yh²)) ''? You have to take off the ''x'' and ''y''.
But it's not a problem, we will continue and suppose that F(α(x,y))(Ω,Φ)=lim[N->∞]((1/N)∑δ(Ω-h,Φ-h²))
2. When you calculated the invers transform you have to but exp[j(Ωx+Φy)], i mean instead
lim[N->∞]((1/N)∑(from h=0 to ∞)∫∫δ(Ω-xh,Φ-yh²)dΩdΦ)
you have to write
lim[N->∞]((1/N)∑(from h=0 to ∞)∫∫δ(Ω-h,Φ-h²) exp[j(Ωx+Φy)] dΩdΦ)
3. And my final question (or remark) :
You are right when you said that ∞²>∞ because ∞²/∞=∞
But,
i think that (perheps i'm wrong) you are not alloweded to write this :
lim[N->∞]((1/N)∑(from h=0 to ∞)∫∫δ(Ω-xh,Φ-yh²)dΩdΦ)
=lim[N->∞]((1/N)∑(from h=0 to ∞)∫δ(Φ-yh²)dΦ)
Because even if
lim[N->∞] N²/N=∞ (it means that ∞²>∞)
we can not state (I think) ... that
lim[N->∞, J->∞] N²/J=∞ because J and N is not the same variable, for exemple you can take N=d and J=exp(d) ... then :
lim(N->∞) <=> lim(d->∞)
and
lim(J->∞) <=> lim(d->∞)
Thus,
lim[N->∞, J->∞] N²/J <=> lim[d->∞] N²/J
BUT
lim[d->∞] N²/J=0 Then even if [N->∞, J->∞] we can have lim[N->∞, J->∞] N²/J=0
I mean that you can say that ∞²>∞ if you have the same variable, otherwise it's not true ...
Thus, i think that you can not write :
lim[N->∞]((1/N)∑(from h=0 to ∞)∫∫δ(Ω-xh,Φ-yh²)dΩdΦ)
=lim[N->∞]((1/N)∑(from h=0 to ∞)∫δ(Φ-yh²)dΦ)
because we do not know how Φ, Ω and N will vary
Once again, I not really good on maths, but i was really surpise but your development (YOU ARE SMART...)...
Thanks I really appreciate your help...
Nab ...