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Apr
6
revised Question about Singer's theorem
edited body
Apr
6
asked Question about Singer's theorem
Apr
5
comment Which are the conditions for a Lorentz space $L^{p,q}$ to be order-continuous?
Thanks for the answer! Do you now any book/notes where this kind of questions about Lorentz spaces are treated in detail?
Apr
1
asked Which are the conditions for a Lorentz space $L^{p,q}$ to be order-continuous?
Mar
8
accepted A question involving Fourier transform
Mar
8
comment A question involving Fourier transform
Very good answer, Priyatham. Thanks a lot!
Mar
8
revised A question involving Fourier transform
added 111 characters in body
Mar
8
comment A question involving Fourier transform
If we take $\infty$ instead of $x$, taking Fourier tranforms, we get that $T(xe^{\frac{-x^2}{2}})=T(e^{-x}\ast g)=T(e^{-x})\cdot T(g)$. Using the fact that $T(xe^{-x})(\xi)=\sqrt{2\pi}i(-\xi) e^{\frac{-\xi^2}{2}}$ and $T(e^{-x})(\xi)=\frac{1}{1+i\xi}$ we arrive to $T(g)(\xi)=\sqrt{2\pi}i(-\xi)e^\frac{-\xi^2}{2}(1+i\xi)$. With Fourier inversion, we would get $g$, but i'm having problems with it...
Mar
8
comment A question involving Fourier transform
I also thought that. In the notes is written this way. With $\infty$, would it make more sense I think. Looks like some convolution
Mar
8
comment A question involving Fourier transform
We work with this definition of Fourier transform: $T(f)(\xi)=\int_{\mathbb{R}}f(x)e^{-i\xi x}\;dx$,
Mar
8
comment A question involving Fourier transform
Thanks for the answer, Priyatham. But are the hypothesis of differentiation under integral sign verifyed? And do you have any idea on how could this be solved using Fourier transform techiniques?
Mar
8
comment A question involving Fourier transform
I know its involved because the problem comes from some class notes about Fourier transform.
Mar
8
asked A question involving Fourier transform
Mar
4
comment A property of the Fourier transform
To apply BigM's argument, the function $f(t)g(x)e^{-ixt}$ should be integrable. Is that obvious?
Mar
4
asked A property of the Fourier transform
Mar
1
accepted About the range of an operator and its adjoint
Feb
27
comment About the range of an operator and its adjoint
I don't get this...could you detail it a bit? How this leads to $T^*$ having closed range?
Feb
27
asked About the range of an operator and its adjoint
Feb
24
accepted About dependence on initial conditions
Feb
24
comment About dependence on initial conditions
Very clear and detailed explanation, froggie! Thanks a lot!