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1d
comment Spectral theory of $\frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2}$ and $4 \frac{\partial ^2}{\partial z\partial \bar{z}}$
The spectral theories of two things differing by a positive real constant are the same. Do you mean to be asking a different question?
2d
comment Eigenvalues of the imaginary part of the Symplectic action on Siegel upper half plane
It is a bit difficult for me to understand the genuine question here. There is the old identity $\det \Im \gamma z=|\det (cz+d)|^{-2}\cdot \det \Im z$ which applies as well in the Siegel modular case as in the elliptic modular case. If the question is about better algorithms, I have no idea.
Apr
22
comment Error estimate in the spectral theorem of compact operators on a Hilbert space
Partial sums of general compact operators' spectral decomposition do not converge in $L^2$. That latter convergence is exactly the property of being Hilbert-Schmidt, and most compact operators (self-adjoint or not) are not Hilbert-Schmidt. (Also, H-S ops need not have positive semi-definite kernels...) And, even then, the partial sums of H-S kernels admit no better estimate for convergence than would something in $\ell^2$ admit estimates for its convergence. That is, arbitrarily bad-but-convergent possibilities do occur.
Apr
22
comment Error estimate in the spectral theorem of compact operators on a Hilbert space
Hilbert-Schmidt operators are a proper sub-class of compact operators, namely, exactly the class of compact operators whose naturally-plausible kernel does converge in $L^2$. "Reproducing kernels" are kernels giving the identity map on some function-space, and possessing some pointwise or uniformly locally pointwise convergence ($L^2$ convergence for the identity map could happen only on a finite-dimensional space...)
Apr
22
comment Order of metaplectic operator
No, the metaplectic group is a two-fold cover...
Apr
21
answered Dual operators between Hilbert spaces: with or without Riesz representation
Apr
21
answered Error estimate in the spectral theorem of compact operators on a Hilbert space
Apr
21
answered Examples of categorical adjunctions in analysis, Lie theory, and differential geometry?
Apr
21
comment Proof of a fact about traces
Also, btw, closing with "kind wishes" does not disturb me in the least!
Apr
19
comment Harmonic functions in tempered distribution sense
The observation that $\hat{u}$ "has singular support" can be made more precise, of course: for a tempered distribution $u$ annihilated by $\Delta, $, $|x|^2\cdot \hat{u}=0$, which implies that the support of $\hat{u}$ is contained in the zero-set of $|x|^2$, namely, $\{0\}$. By classification of distributions supported on a point, $\hat{u}$ is a linear combination of $\delta$ and derivatives, so $u$ is a (harmonic) polynomial, which is not $L^2$ unless it's $0$.
Apr
10
comment Some question on haar measure for sumsets of closed subsets of profinite groups
As suggested by @Venkataramana, I significantly edited the question in a direction that makes it ask the only question I could see that might make sense. And then the answer below is a good answer.
Apr
10
revised Some question on haar measure for sumsets of closed subsets of profinite groups
Cleaned up question to directly ask the only thing that made sense.
Apr
9
comment Maass form properties and their fourier coefficients
Perhaps it is under-appreciated that the fact that those particular Bessel functions are the only ones that appear (as opposed to another solution of the same second-order ODE) is due to the asymptotics at infinity, which is a singular point, etc. Understanding those asymptotics in that case is nearly 100 years old by now, but in higher rank (e.g., $SL_3(\mathbb Z)$...), this is Casselman's subrepresentation theory, etc., which is pretty serious business.
Apr
9
comment Maass form properties and their fourier coefficients
One perhaps-important quibble is that the Fourier coefficients are not "strictly" multiplicative, but only "weakly": $a_{mn}=a_m\cdot a_n$ generally only when $m,n$ are coprime. This is considerably different from the strict multiplicativity of Dirichlet characters. The Hecke operators give the second-order recursion for prime-power coefficients $a_{p^n}$, in contrast to the in-effect first-order recursion for Dirichlet characters.
Apr
8
comment commutators in upper triangular matrices
Need $p>2$, to start.
Apr
4
revised Examples of high level math that can be motivated to laypeople
added 680 characters in body
Apr
4
answered Examples of high level math that can be motivated to laypeople
Apr
4
awarded  Yearling
Mar
31
comment Quantum Mechanics derivation of Wallis' Formula?
Yes, Liouville's theorem is useful, not static.
Mar
31
comment Quantum Mechanics derivation of Wallis' Formula?
Euler's product formula for $\sin^2$ does not need the general apparatus of Weierstrass or Hadamard products: in brief, first use Liouville's theorem to prove the partial fraction expansion of $1/\sin^2 x$, second regroup and observe that this is the derivative of $\cot x$, third, observe that this is the logarithmic derivative of $\sin x$. E.g., as math.umn.edu/~garrett/m/complex/08a_product_sine.pdf