Giuseppe Negro
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 10h comment Homeomorphism from real interval to an arc of a circle This is unnecessarily complicated IMHO. The standatd parameterization of the unit circle, which is also mentioned by the OP, is simpler. 1d comment Analytic Vectors (Nelson's Theorem) Just wanted to let you know that I stumbled on your post and I found it really useful. Thank you 1d reviewed Approve Strong convergence of Spectral Projection 1d comment How to determine standard equation of a conic from the general second degree equation? This book should be useful. I recommend it. 1d revised Local boundedness of weak solutions for a elliptic equation in divergence form some rewordings to improve clarity (hopefully) 1d comment Given two column vectors $a$ and $b$, what is the determinant of $A$ if $A=Id-ab^T$ HINT: This is a rank 1 perturbation of $I_n$, so it only has one eigenvalue besides $1$. You can easily compute such an eigenvalue and then compute the determinant as the product of all eigenvalues. 2d revised Linearspan of Gaussians dense in Schwartz space added 592 characters in body 2d comment Linearspan of Gaussians dense in Schwartz space (NOTE: your definition of $g_\epsilon$ should be $g_\epsilon(x)=\epsilon^{-n}e^{\frac{|x|^2}{2\epsilon^2}}$, don't forget the $\epsilon^{-n}$). The idea is best explained when $\phi\in L^1(\mathbb{R}^n)$, in which case the convolution is the plain vanilla one and not this annoying distributional contraption. I am adding some lines in my answer. Apr 30 comment Singular values of the differentation operator It seems to me that $M(D)$ is badly computed. Only the first column is right. In the second one, you should put the coordinates of $D(\sqrt{\frac32}x)$ in the orthogonal basis, not in the old one $1, x, x^2$. Apr 30 comment How to find the radius of convergence given recurrence relation? @Did: My bad. That is why I only commented, instead of answering. Apr 29 answered Linearspan of Gaussians dense in Schwartz space Apr 28 awarded Nice Question Apr 28 comment Equality about integral on unit sphere, involving Lebesgue measure Specifically, what you need here is the equality $$d\omega_n=(1-t^2)^{\frac{n-3}{2}}dt d\omega_{n-1},$$ where $t=\xi\cdot \eta$. Here there are some rough notes I wrote about this computation, but they are not meant to be read by anyone else than myself, so if you find them unclear please switch to Müller's books. Apr 28 comment Equality about integral on unit sphere, involving Lebesgue measure Have a look in the very first pages of Müller's "Spherical Harmonics" (or his later "Analysis of spherical symmetry in Euclidean space"), it is a classic in this kind of computations and more or less all those books on spherical-stuff have it in their bibliographies. Apr 27 comment Linear Algebra with functions I like to use this method: math.stackexchange.com/a/269694/8157 but the linked Q&A contains many others. Apr 27 comment How to find the radius of convergence given recurrence relation? I guess you should analyze $|c_n|^\frac1n$ in the limit $n\to \infty$. I suspect that $c_n$ is bounded, which would imply that the radius of convergence is at least $1$. If you want the precise radius of convergence, I would guess that it is exactly $1$, because I also suspect that the sequence does not tend to $0$ as $n\to \infty$. Apr 27 comment Show that $(C^1([a,b]), d_{\infty})$ is not complete The example is OK if $a=-1, b=1$. You should now find a way to generalize it to an arbitrary interval. HINT: An affine change of variables $x=\alpha X+\beta$ transports your example sequence $f_n(x)$ on $[-1,1]$ into a sequence on $[\beta-\alpha , \alpha+\beta]$. Apr 23 revised Is local Lipschitz continuity sufficient for an ODE to have a unique solution? deleted 1 character in body Apr 19 revised Which functions are tempered distributions? added question 3 + improved definition of polynomially growing function + retag Apr 19 comment Why is the image of the implicit function in the implicit function theorem not open? I don't know how to answer, but I would like to point you to Bruce Blackadar's manuscript: wolfweb.unr.edu/homepage/bruceb (follow the link for the manuscript on Real Analysis). The chapter on inverse/implicit function theorem is very carefully crafted and it treats many subtle details that usually go uncared for in standard textbooks.