5,665 reputation
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visits member for 5 years
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Oct
18
awarded  Yearling
Oct
13
answered $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma
Jul
6
answered Proof of the Friedlander–Iwaniec theorem
Jul
2
awarded  Curious
May
5
comment Distribution of $a^2+\alpha b^2$
This isn't an answer, but there is a theorem of Atkin which is in a similar spirit: These exists a sequence of integers such that the n-th term is $n^2 + O(log(n))$ and whose sumset has positive density in the integers. ams.org/mathscinet-getitem?mr=202687
Apr
29
answered A question on Cramer's theorem
Apr
9
comment Fourier series of functions on compact groups
@Anton, If you interpret the question like that it isn't very interesting. There are certainly $L^2$ functions on the circle without absolutely convergent Fourier series. You could alternately ask for unconditional pointwise convergence (that is a.e. pointwise convergence for every ordering) but this is false for every complete infinite orthonormal system (such as Fourier series on the cirlce, again).
Apr
9
comment Fourier series of functions on compact groups
How are you ordering the summation? Note that one must specify an ordering before it makes sense to talk about pointwise convergence.
Apr
3
awarded  Nice Answer
Feb
20
comment Objections to and arguments for the simplicity of all Riemann zeros
Micah, can you give a reference for this? Is the same known for $\psi$?
Feb
20
awarded  Nice Answer
Feb
19
answered Carleson-Hunt inequality: changing order of summation
Feb
18
revised Finite field Szemeredi-Trotter theorem with unequal number of points and lines
added 651 characters in body
Feb
18
answered Finite field Szemeredi-Trotter theorem with unequal number of points and lines
Feb
14
revised Result of Beurling concerning absolute convergence of Fourier series of |f|
added 21 characters in body
Feb
14
answered Result of Beurling concerning absolute convergence of Fourier series of |f|
Feb
10
answered When is the bound in Riesz-Thorin Interpolation Theorem attained?
Feb
10
awarded  Fanatic
Jan
30
answered Number of Totally Isotropic Subspaces
Jan
15
comment Carleson's Theorem on Manifolds
Note that the answer will depend on how you order the eigenfunctions. Ordering by eigenvalue is perhaps the most natural choice. In the case of the the $2$-d torus this gives spherical summation for $2$ dimensional Fourier series. Determining if pointwise convergence holds in this setting is a longstanding open problem.