Timeline for Inverse transform of $\int K(s,t) \rho(s) dS $

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Mar 30, 2022 at 13:46	comment	added	Dirk		A compact operator on a Hilbert space has a singular value decomposition which can be obtained by application of the spectral theorem for compact and self-adjoint operators to the operator $F^*F$. This may be found in books on functional analysis (while most treat the spectral theorem, not all treat the singular value decomposition). A domain in R^d is just an open and connected subset of R^d. It may be that everything is the same if $V$ is a surface, but I haven't thought about it.
Mar 30, 2022 at 10:11	comment	added	MikeTeX		Note: I'm not sure what you mean by "domain in $\mathbb R^d$". In my setting, $V$ is a surface in $\mathbb R^d$ or the union of disjoint surfaces. I meant a differentiable manifold of dim d-1, or the union of such manifolds. To fix my ideas, I was considering objects like spheres, ellipsoids etc. in $\mathbb R^3$.
Mar 30, 2022 at 10:02	comment	added	MikeTeX		Thank you for answering me. My knowledge is limited in this domain, and I lost you when you said "Consequently, the singular values of F go to zero and hence, no continuous inverse exist". Could you elaborate a bit in your answer? Regarding my motivating pseudo-demonstration, I am of course aware that it is not viable, even as an "approach" to this problem; it was just "kind of" informal approach.
Mar 30, 2022 at 9:24	history	answered	Dirk	CC BY-SA 4.0