Timeline for Stationary phase in spherical integral
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 25, 2020 at 16:43 | comment | added | Patch | Thanks, again!! | |
| Mar 25, 2020 at 16:37 | vote | accept | Patch | ||
| Mar 25, 2020 at 9:53 | comment | added | Bazin | @Patch (1) $J$ is indeed vector-valued, but you can handle each coordinate separately. (2) You start with $x$ such that $\vert x\vert \lambda\ge 1$; then you look at the stationary phase method with an integral near a given point $y_0$ of the sphere. You check if you have a stationary point: if not you get a better estimate with $O((\vert x\vert \lambda)^{-N})$ for all $N$. If you land on a stationary point, you check the Hessian. Note that the choice of $y_0$ in the answer above is $e_n$. | |
| Mar 25, 2020 at 6:21 | comment | added | Patch | 2) Assuming we have clarified the issue with $\mathbf{J}(x,\lambda)$ above, lets call $J_k(x, \lambda)$ the $k$-th component of the vector. Then I can see how stationary phase around $(x,z) = (0,0)$ gives us the $(|x_n| \lambda)^{(n-1)/2}$ behavior, but this would only be asymptotic behavior in each coordinate. Moreover, every $J_k$ would satisfy the same big-oh bounds involving the absolute value $|x_n|$ only; so how do I put these all together to get a uniform bound involving the norm, $|x|$? | |
| Mar 25, 2020 at 6:21 | comment | added | Patch | 1) By bringing the integral inside of the dot product, you are effectively making $\mathbf{J}(x,\lambda)$ a vector-valued function, correct? But isn't your cutoff function, $a(z)$ a scalar-valued function? Should I just be interpreting $a(z)$ as one scalar component of $\mathbf{J}(x,\lambda)$? If not, then wouldn't this seem to contradict the definition for $\mathbf{J}(x,\lambda)$ you had earlier? | |
| Mar 25, 2020 at 6:20 | comment | added | Patch | I'm still reading through this carefully, but thank you! I have a couple small clarification questions, though, so if you don't mind I'll just enumerate them here in a series of separate comments: | |
| Mar 24, 2020 at 22:30 | history | answered | Bazin | CC BY-SA 4.0 |