# Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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### Poisson summation formula for infinite dimensional spaces

Let $M$ be an orientable, compact smooth manifold with and $H^{-1}(M)$ be the dual space of $$H^{1}(M)=\{f:\int |f|^2+(\nabla f)^2<\infty\}$$ I know it is well known that (see Julien Dubedat, page ...
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### Kolmogorov entropy of a subset of $L^1$

How can we estimate the Kolmogorov $\epsilon$-entropy $$H_\epsilon (A,L^1(\mathbb R))$$ where $A = \{f:\mathbb R \to [0,K] \text{ s.t.$f \in L^1$and has total variation$TV(f) \le M$}\}$?
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### Condition on kernel convolution operator

I am studying O'Neil's convolution inequality. Let $\Phi_1$ and $\Phi_2$ be $N$-functions, with $$\Phi_i(2t)\approx \Phi_i(t), \quad i=1,2$$ with $t\gg 1$ and let $k \in M_+(\mathbf R^n)$ is the ...
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### Nonseparable Hilbert spaces

Being nonseparable Banach space is in fact nothing special: one meets the first examples in the standard functional analysis course, when one learns about $\ell^p$ or $L^p[0,1]$ spaces-these spaces ...
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### Weak convergence in a product space

Given a function $f: Y\longrightarrow Y$ ($Y$ is a Banach space). Assume that $f$ satisfies: If $y_n \rightharpoonup y$, then $f(y_n)\rightharpoonup f(y) \text{ in } Y$; $f$ is weakly compact; ...
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### Question/References on the Skorokhod M1 topology

Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
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### The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$

The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$ is defined by: $$V_k f(x)=\int_{\mathbb{R}^d}f(y)d\mu_x(y),$$ where $d\mu_x$ is a probability measure on $\mathbb{R}^d$ with support in the ...
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### A strong duality for convex functional optimization that admits Lipschitz continuity constraints?

Problem Statement I am looking for formal proof---hopefully textbook material---of two items: an analogue to Slater's condition [1] that obtains strong duality for optimization of convex functionals; ...
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### Isolated points of the spectra of self-adjoint operators on Hilbert spaces

Let $T$ be a (everywhere defined) self-adjoint operator on a complex Hilbert space $\mathcal{H}$. I am interested in results that give (non-trivial, possibly mild) sufficient conditions on $T$ to ...
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### How can we deal with singular integral equation that is a sum of the Cauchy kernel and Hilbert kernel

I am a theoretical physicist dealing with Matrix Models. In this context, one often encounters singular integral equations whose solution gives the eigenvalue density in the so-called large $N$ limit. ...
let Y be a contact (3) manifold and X be its symplectization. let's say the Reeb dynamics is at least Morse Bott. let $u: \Sigma \rightarrow X$ be a $J$ holomorphic curve. I know the usual model for a ...