DCM
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Member for 10 years, 1 month
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Rellich Embedding Theorem for the $2$-Sphere
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Rellich Embedding Theorem for the $2$-Sphere
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Rellich Embedding Theorem for the $2$-Sphere
Ah! Proposition 2.4 of books.google.co.uk/… uses the kind of local charts needed here (the second sentence of the proof).
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Rellich Embedding Theorem for the $2$-Sphere
This MO post seems relevant Re. the use of geodesic normal coordinates: mathoverflow.net/questions/327475/…
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Rellich Embedding Theorem for the $2$-Sphere
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Rellich Embedding Theorem for the $2$-Sphere
Re. relevance of extra structures: I don't think the Kahler structure really gives you any extra useful information.
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Projection of a function $f\in L^1(\Omega)$ onto a finite dimensional subspace
Are you primarily interested in an answer for a specific choice of $g_1,\dots,g_k$, or in full generality? I guess you'd have said if it was the former, but no harm in asking!
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Why is this test function admissible? [Paper explanation]
"Thus $\psi(u_n(t))\chi_{(0,t)}$is basically the function we use to integrate against $P(u_n), f_n$ and for that reason the wide term "test function" is adopted." - that is my understanding yes.
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Substitute Concrete Value in Conditional Expectation
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Substitute Concrete Value in Conditional Expectation
@IosifPinelis: I don't disagree with anything you've written, it just seemed to me that the OP was asking a slightly different question. It seems not given that the OP has accepted your answer :)
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Substitute Concrete Value in Conditional Expectation
I was going to use the word distribution instead of `functional' there for a second...
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Substitute Concrete Value in Conditional Expectation
I read this question as being about the functional $\mu_{X,Y,f}:\psi\longmapsto \int\int\psi(y)f(x,y)(dP^{X,Y}(x,y) - dP^X(x)dP^Y(y))$ for a fixed choice of $f$.
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Substitute Concrete Value in Conditional Expectation
What happens $P_X$ and $P_Y$ supported on finite (or compact) sets and let $f(x,y)=1$?
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Substitute Concrete Value in Conditional Expectation
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Substitute Concrete Value in Conditional Expectation
*probability, not 'probabilit'...
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Substitute Concrete Value in Conditional Expectation
The notion of a regular conditional probabilit (sometimes called a transition kernel) is usually helpful with this sort of thing. Do you have a particular set of circumstances where you'd like your $\varphi$ to exist?