Image of problem Problem
Consider the extended space $$Ω^x$$ which consists of all functions χ(x) such that
$$<χ|\phi>=\int_ {-\infty}^{\infty}χ^*(x)\phi(x) dx $$
is finite for all phi(x) in the nuclear space Ω. Every function in H (Hilbert Space) satisfies this condition,so H is a subspace of $$Ω^x$$. Which of the following functions belong to Ω(nuclear space), to H, and/or to $$Ω^×$$
Hint: you may want to use the fact that the fourier transform of a square-integrable function exists and is a finite function.
$$(a) \sin(x)$$ $$(b) \sin(x)/x$$ $$(c) x^2\cos(x)$$ $$(d) \frac{\log(1+|x|)}{1+|x|}$$ $$(e) \exp(−x^2)$$ $$(f) x^4e^{−|x|}$$
Comments/Questions:
Of course, I don't expect help with each one, but if someone can help with one or two that would be great. I'm having trouble with the Fourier transformation and when it applies.
I'm also having trouble with the concepts. If I were to find the fourier transform of one of the functions, what would that entail? What does that say about the function? If I were to find the fourier transform of (a)$\sin(x)$, do I do $$\sin^2(x)$$
I'm sorry, I'm terribly confused