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Jan
6
awarded  Informed
Jan
3
revised Relationship between decay of Fourier transform and smoothness in $L^2$
added 86 characters in body
Jan
3
accepted Relationship between decay of Fourier transform and smoothness in $L^2$
Jan
3
comment Relationship between decay of Fourier transform and smoothness in $L^2$
actually can you elaborate on the proof that $\xi \hat{f} \in L^1$?
Jan
3
comment Relationship between decay of Fourier transform and smoothness in $L^2$
Incidentally, integrability lets me complete my method as well via Fubini theorem. Thanks!
Jan
3
comment Relationship between decay of Fourier transform and smoothness in $L^2$
Also, the integral is not a standard integral, but it is an $L^2$ limit of functions of $y$
Jan
3
comment Relationship between decay of Fourier transform and smoothness in $L^2$
How do you show that $L \xi \hat{f}(\xi)$ is integrable?
Jan
3
revised Relationship between decay of Fourier transform and smoothness in $L^2$
added 403 characters in body
Jan
3
comment Relationship between decay of Fourier transform and smoothness in $L^2$
Using which dominating function?
Jan
3
asked Relationship between decay of Fourier transform and smoothness in $L^2$
Dec
29
accepted Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$
Dec
23
comment Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$
Since we know G(x,y) to be doubly periodic on the plane, we may as well flip your triangle on the line x=y. Then we will have deduced the function for the region y <= x. I think this finishes the question. I have also found the answer using my book's method and I added it as an edit.
Dec
23
revised Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$
added 501 characters in body
Dec
23
comment Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$
Can you possibly touch up your answer to deduce the final expression for G(x,y) as given in my text? Then I will accept your answer
Dec
23
comment Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$
But actually this answer is looking very good!
Dec
23
comment Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$
I think you must also be more careful about where the expression for $g(z)$ is valid, since it is periodic it cannot be equal to the parabola everywhere.
Dec
22
comment Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$
Oh! $\cos(a)\cos(b) = (\cos(a+b) + \cos(a - b)) / 2$ I think you have a sign messed up in the very beginning
Dec
22
comment Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$
To clarify, I the reason I said that I don't think it's correct is because my book gives $\frac{x^2 - x + y^2 - y - |x-y|}{2} + 1/3$ to be the final answer for G(x,y)
Dec
22
comment Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$
Isn't your expression for $g'(z)$ the expansion of a sawtooth wave? And therefore $g''(z)$ should indeed be a constant as you say. I think the entire computation for $g''(z)$ can be avoided if we accept the formula given here mathworld.wolfram.com/FourierSeriesSawtoothWave.html which shows the slope of the wave would be 1/2, using L=1 and then restricting to [0,1]
Dec
22
comment Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$
This is a nice attempt but I don't think that this is the correct answer