1,163 reputation
613
bio website cornellmath.wordpress.com
location Ithaca, NY
age 31
visits member for 4 years, 5 months
seen Sep 27 at 22:56
I was a graduate student at Cornell University studying harmonic analysis with Camil Muscalu. Now I'm a postdoc at Washington University in St. Louis.

Aug
12
awarded  Nice Answer
Jul
31
awarded  Scholar
Jul
31
accepted Prove that …, f(x-2), f(x-1), f(x), f(x+1), f(x+2),… is algebraically linearly independent without the Fourier transform
Jul
31
comment Prove that …, f(x-2), f(x-1), f(x), f(x+1), f(x+2),… is algebraically linearly independent without the Fourier transform
I'm embarrassed I didn't make that observation myself about the finite dimensionality of $V$. It's a nice little argument. It also highlights that the $2$ in the $L^2$ condition isn't key as you wind up in a finite dimensional setting. Showing that the inner product, $\langle f(x-k_0), f(x-k)\rangle$, tends to zero is equivalent to proving the Riemann-Lebesgue lemma, but if you just approximate $f$ with a $C_c^\infty$ function, I think all you need is the dominated convergence theorem and Cauchy-Schwarz, so trigonometric polynomials never enter the picture. Thanks!
Jul
29
comment Prove that …, f(x-2), f(x-1), f(x), f(x+1), f(x+2),… is algebraically linearly independent without the Fourier transform
Sorry, maybe I'm just being dense, but why does linear dependence imply the space is finite dimensional? Certainly if you have one dependence relation, you will have infinitely many, but it doesn't seem obvious at first glance that I should get a finite dimensional space.
Jul
29
awarded  Student
Jul
29
asked Prove that …, f(x-2), f(x-1), f(x), f(x+1), f(x+2),… is algebraically linearly independent without the Fourier transform
Apr
30
awarded  Yearling
Dec
20
comment When is a collection of exponentials dense in $L^2(K), |K|<\infty$
Alexei Poltoratski gave a very nice series of lectures with a number of comments in this direction this summer at Clemson. I don't think he really discussed the R^d case, but notes similar to the lectures he gave are available here: internetanalysisseminar.gatech.edu/lectures_uncp Some of the references may point you in the right direction.
Sep
14
awarded  Informed
Jun
25
awarded  Citizen Patrol
May
25
awarded  Necromancer
Apr
30
awarded  Yearling
May
8
comment Square roots of the Laplace operator
Now that I think about it, the analytic continuation might need some additional work... there might be some poles at certain integers $\alpha$, so you would need to add an additional term to balance things out. This computation has to be written down someplace. Sorry that I don't know of a good place.
May
8
comment Square roots of the Laplace operator
Yes, much of this stuff is fairly disjointed, as I recall, which is unfortunate. Many of the basic computations are not terribly difficult (just tricky), so I think they are not written down in many places. The issue you are having in the computation in your comment comes from having to use some analytic continuation technique to extend the range of possible $\alpha$s in the inverse transform of $|\xi|^\alpha$. $|x|^{-n-1}$ is not locally integrable, as Tom pointed out, so its a bit more delicate.
May
8
answered Square roots of the Laplace operator
Apr
30
awarded  Yearling
Nov
23
comment Carleson's Theorem (on the Adeles and other exotic groups)
I don't think you can accept more than one answer, but that's ok. I just hope what I wrote helped in some way!
Oct
19
comment Can the supremum of continuous functions be discontinuous on a set of positive measure?
Hm, I suppose you could also have just made it 1 on $K_n$ and 0 on $E$ and avoided that $\sup(-f_n)$ business...
Oct
19
answered Can the supremum of continuous functions be discontinuous on a set of positive measure?