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Questions tagged [fa.functional-analysis]

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

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Decomposition of a contractive representation into an orthogonal sum for the $n$-dimensional case. Has this been done yet?

I know that it has been done for the two-dimensional case. Marek Kosiek showed it in his work "Decomposition of operator representations of the algebra $R(K_1 \times K_2)$" and "...
S-F's user avatar
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Locally compact groupoid with range map restricted to isotropy groupoid is open

Suppose the action groupoid 𝐺=𝐻⋉𝑋, where 𝐻 is a locally compact group and 𝑋 a locally compact space is such that isotropy subgroups of H are isomorphic to each other. Can this be an example of a ...
K N SRIDHARAN NAMBOODIRI's user avatar
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On constructing the canonical boundary operator for a given differential operator

Given an $n\times n$ matrix $$X=\begin{pmatrix} x_{11} & x_{12} & \cdots & x_{1n} \\ x_{21} & x_{22} & \cdots & x_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n1}...
Ryan Hendricks's user avatar
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Orthogonalization of symmetric non-degenerate bilinear forms

It is well-known that given a field $k$ with characteristic different from $2$, every symmetric non-degenerate bilinear form $B$ over a finite-dimensional space can be orthogonalized. This means that ...
Luiz Felipe Garcia's user avatar
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What is the maximum of $ \frac{\sin(n(x+a))}{\sin(x+a)} + \frac{\sin(n(x-a))}{\sin(x-a)}$?

I have asked this here. Due to inactivity and no satisfying answers, I am asking here. Hope that's okay. We know the global maxima of the function $\frac{\sin(nx)}{\sin(x)}$ is $n$ (thanks to this ...
RajaKrishnappa's user avatar
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Find the maximum of an expression under the logconcave assumption

Let $F(v)$ be a cdf over $\left[0,v_{max}\right]$, $1-F(v)$ is logconcave. The corresponding density function is $f(v)$. Let $p^m$ solve $1-F(v)-f(v)v=0$ (it is a FOC of a profit maximization problem)....
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Measurable selection for the mean value theorem

When we use the mean value theorem we come across the problem of measurability of the argument. The problem is somehow like that: Let $f:\Omega\times [0,1]\to\mathbb{R}$ be a Caratheodory function (i....
Bogdan's user avatar
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Is the Bures metric equivalent to the Euclidean one?

Let $K=\mathbb R$ (reall numbers) or $K=\mathbb C$ (complex numbers). Define $\mathcal M_n$ to be the space of $n\times n$ matrices $A=(a_{i,j})_{1\le i,j\le n}$, with $a_{i,j}\in K$. Let $\|\cdot\|$ ...
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convolution of the fundamental solution with the homogeneous solution

I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero? Let $U$ and $E$ ...
Alucard-o Ming's user avatar
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Heine-Borel property for (probability) measures on $\mathcal{S}'$?

For Euclidean space $\mathbb{R}^n$, it is a well-known fact that the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is a Montel space. Moreover, the space of tempered distributions $\mathcal{S}'(\mathbb{R}...
Isaac's user avatar
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Comonotone solution for Optimal Transport problems with supermodular surplus

In Alfred Galichon's book Optimal Transport Methods in Economics the foollowing result is stated for OT problems on the real line. Theorem 4.3.(i) Assume that $\Phi$ is supermodular. Then the primal ...
Francesco Bilotta's user avatar
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Locally uniformly convexity in kernels (generalized definition of graphon) with cut norm

Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions $W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert ...
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Mean value property for fractional laplacian

I just started reading about fractional Laplacian. I am curious on the following questions Does fractional laplacian i.e., $(-\Delta)^su=0$ in $\mathbb{R}^n$ this equation satisfies any mean value ...
Emmie's user avatar
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Domain of a Jacobi operator with unbounded coefficients

Is it possible to describe the domain of a Jacobi operator explicitly? Let $J$ be the linear operator acting on a real sequence $(u_{n})_{n\in\mathbb{N}}$ by $$ J(u_{n}) = a_{n+1} u_{n+1} + a_{n} u_{n-...
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Does the sequence of bounded symmetric square integrable holomorphic functions have a convergent subsequence?

Let $f$ be a bounded holomorphic function on $\mathbb D^2$ and $s : \mathbb C^2 \longrightarrow \mathbb C^2$ be the symmetrization map given by $s(z) = (z_1 + z_2, z_1 z_2),$ for $z = (z_1, z_2) \in \...
Anacardium's user avatar
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Existence of sequence of regular projections

Reading the book :Krasnosel'skii, M.A.; Pustylnik, E.I.; Sobolevskii, P.E.; Zabreiko, P.P. (1976), Integral Operators in Spaces of Summable Functions, Leyden: Noordhoff International Publishing, 520 p....
Guillermo García Sáez's user avatar
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Operator globally hypoelliptic

An operateor $T$ is globally hypoelliptic if : $u\in S'(\Bbb R^n),Tu\in S(\Bbb R^n)$ imply $u\in S(\Bbb R^n)$. My question why if $u\in L^2(\Bbb R^n): Tu =\lambda u$. Then $u\in S(\Bbb R^n)$. where $\...
zoran  Vicovic's user avatar
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Can $L^p([0,1])$ be built up from countably infnite copies of $l^p({F})$ , where $F$ is a finite set or $\mathbb{N}$?

I know that $L^p([0,1])$ is not isometrically isomorphic to $l^p(\mathbb{N})$ when $p\neq 2$? But, there is an isometric copy of $l^p(\mathbb{N})$ inside $L^p([0,1])$. My question is that whether $L^...
John Depp's user avatar
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A Poincaré inequality holds for $p>2$ but fails for $p\leqslant 2$

I am confused with the following example taken from page 6 of Sobolev Met Poincaré, by Hajłasz and Koskela (MR1683160, Zbl 0954.46022). Let $(X,d,\mu)$ be a metric measure space and let $\Omega\subset ...
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Equality between operators, on dense subspace, from a quadratic form point of view

Let $L \ge 1$ and consider a finite box $\Omega = [0,L]^{d} \subset \mathbb{R}^{d}$. The set of functions: $$\psi_{p}(x) = \frac{1}{L^{d/2}}e^{i\langle p,x\rangle} \quad p\in \frac{2\pi}{L}\mathbb{Z}^{...
MathMath's user avatar
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Has anyone seen such a function/quantity?

I am dealing with a problem wherein I encounter the following quantity- $$ Q_{d, \epsilon}(t_0) = \sup_{t' \notin B(t_0, \epsilon)} \inf_{t \in B(t_0, \epsilon)} \frac{d(t') - d(t)}{t'-t}. $$ Here,...
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Sequential compactness via Arzela-Ascoli theorem for uniform state spaces

Let $X$ be a uniform topological space and $C([0,1],X)$ the space of continuous functions from [0,1] to $X$. Assume that for subsets of $X$ sequential compactness and compactness are equivalent. Let $(...
PDEprobabilist's user avatar
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Weakly compact set

I want to show that if the set $$ \big\{u \in L^{q}([0, n] ; X): u(t) \in \phi(t, x(t)), t \in[0, n]\big\} $$ is weakly compact, then the set $$ \mathcal{S}_{\phi}(x)=\Big\{u\in L_{loc}^{q}(\mathbb{R}...
Mathlover's user avatar
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Time regularity vs space regularity for parabolic PDE

Suppose that there exist separable Hilbert spaces $V, H, X$ such that $V\hookrightarrow H\hookrightarrow X\hookrightarrow V'\,$ continuously, where $V'$ denotes the dual of the Hilbert space $V$. Let ...
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A Lipschitz function induced by the infimum of the length of curves

Recently I have read a paper, Quasiconformal Images of Hölder Domains, written by S. M. Buckley in 2004, published by Annales Academiæ Scientiarum Fennicæ Mathematica. I am confused about page 33 of ...
Javier's user avatar
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Equality of two measures on functional spaces

It is well known that if $\mu$ and $\nu$ are two measures on the space $C^0([0,1],\mathbb{R}^n)$ of continuous mappings from $[0,1]$ to $\mathbb{R^n}$ endowed with its Borel $\sigma$-algebra satisfy $$...
Anico's user avatar
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Hilbert spaces that include algebraic polynomials

This question is motivated by a phrase I found in several books/papers about approximation theory, for example, M.J.D.Powell's Approximation Theory and Methods: ''Let $\mathcal{H}$ be a Hilbert space ...
FDK's user avatar
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How to prove that $f(x) := |x|^{\frac{\lambda - n}{p}}(1 - \psi(x))$ satisfies a specific property related to its limit at the origin

Disclaimer. I have asked this question a month ago on MSE (click here to access the original post) and even bountied it. I got an answer on MSE, but unfortunately I don't feel like it has enough ...
Rodrigo's user avatar
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Projection onto Shift Invariant Subspaces of $H^2$

Every shift invariant subspace of the Hardy space $H^2(\mathbb{D})$ is either $\{0\}$ or is of the form $\varphi H^2$ for some inner function $\varphi$. I know that if $\varphi(0) \neq 0$, then the ...
bm3253's user avatar
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Show that functional $J$ satisfies Palais-Smale condition iff every $P-S$ sequence is bounded

Let $H$ be a Hilbert space and let $K: H \rightarrow \mathbb{R}$ be $C^1$ and such that $\nabla K: H \rightarrow H$ sends bounded sets into precompact sets. Consider the functional $J: H \rightarrow \...
YuerWu's user avatar
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Sufficient condition for interpolation

If we have a couple of two compatible banach spaces (in this sense) $(X,Y)$ and a sequence of Banach spaces $\{Z\}_{\theta\in[0,1]}$ which are intermediate between $X$ and $Y$ satisfying: $Z_0=X$, $...
mejopa's user avatar
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Maximal function on mixed $L^{p}$

Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is $$ \Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
User091099's user avatar
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Dual of closure

Currently I'm studying about abstract interpolation theory for my research. One of the basic ways to construct new interpolation spaces, given an interpolation space $E$ with respect to a compatible ...
Guillermo García Sáez's user avatar
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The size of super level sets and the symmetry on a sphere

Let $u$ be a smooth function defined on the sphere $\mathbb{S}^2$, and let $R \in \mathrm{SO}(3)$ be a three-dimensional rotation. Define $$ S_R = \{x \in \mathbb{S}^2 : u(x) \neq u(Rx)\}. $$ Suppose ...
MathLearner's user avatar
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Measurable Extension

Let $(\Omega, \mathcal{F})$ be a measurable space and $X$ some metric space (probably Polish) with the Borel $\sigma$-algebra and a function $f: \Omega \times X \to \mathbb{R}$. Usually, functions ...
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Support of a function acting on an algebra?

Quick: for a measurable function $f$ its support on Euclidean space is clearly just the subset where $f$ does not vanish. Now, let’s have $f$ acting on an finite Lie algebra, f.e. $\mathfrak{gl}$ as $...
relativeentropy's user avatar
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Approximation of Lipschitz functions by convex combinaison of Lipschitz functions depending on projections

Let $K\subset \mathbb R^2$ be compact. For any $c>0$, denote by ${\rm Lip}_c(K)$ the collection of Lipschitz functions $f:K\to\mathbb R$ whose Lipschitz constant is less than or equal to $c$. Set $...
GJC20's user avatar
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Integral of gradient of a function times a vector fields, null whatever the function, implies null divergence and tangential limits conditions

I'm reposting this question from math.stackexchange, as I haven't got answers so far. At the beginning of "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de ...
Oersted's user avatar
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Iterating partially-unconstrained optimization with projection

Let $f:H\to \mathbb{R}$ be a strictly convex Fréchet differentiable, coercive function on a separable Hilbert space $H$ and let $C_1,C_2\subseteq H$ be closed and convex. I want to optimize $$ \tag{(A)...
ABIM's user avatar
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Vectors of complex exponentials span $\mathbf{C}^N$

Let $Q = [0,1]\times [0,1]$. Let $1\leq k < \infty$ and $\{(x_l,\xi_l)\}_{l=1}^{k}\subseteq Q$ be such that $(x_i,\xi_i)\ne (x_j, \xi_j)$ for $i\ne j$. Additionally, for $1\leq l \leq k$, let $n_l\...
Doofenshmert's user avatar
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Projection measure and an integral formula for Lipschitz functions

Let $n\geq m\geq 0$ be integers and put $k=n-m$. Let $A\subset\mathbb{R}^n$ be Borel measurable, we define the projection measure of $A$ as $$\mu_k(A):=\underset{P_1, \ldots, P_r}{\sup_{A=A_1\sqcup\...
Alexander's user avatar
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Notation $\le_{a,b,n,\ldots}$ in Analysis

In modern Analysis, especially Functional Analysis, one proves, or one uses inequalities of the form $$F(X)\le_{a,\ldots,n}G(X).$$ The meaning of the subscripts in the inequality sign means that there ...
Denis Serre's user avatar
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Criteria giving sufficient conditions for a Borel measure to have compact support

I am interested in criteria that guarantee that a Borel probability measure has compact support. I outline two below and I am hoping to gather more as answers (if they exist). The first sufficient ...
Dispersion's user avatar
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Johnson-Lindenstrauss type result for matrix factorization

The type of result I want is: given matrix $A\in \mathbb{R}^{m\times n}$ and error tolerance $\epsilon$, what is a lower bound on $k$ such that $\|A - UV\|_{??}\le \epsilon$, where $U \in\mathbb{R}^{m\...
optimal_transport_fan's user avatar
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When does the Hermite series converge pointwise and when is it uniformly bounded?

Let $\gamma$ denote the standard Gaussian measure on the real line, and consider $f \in L^2(\gamma)$. Since the Hermite polynomials $\{H_n\}_{n \geq 0}$ are a complete orthonormal system, we may ...
Drew Brady's user avatar
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Generator of an analytic semigroup

Perhaps I have a naive question. My question is as follows: When we consider a Cauchy proposition of the following form: $$ \begin{cases} x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\ x(0)=...
Mathlover's user avatar
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Are there probability densities $\rho, f_n$ such that $\lim_n \frac{[\rho * f_n]_\alpha}{\|\rho * f_n\|_\infty} = \infty$?

We fix $\alpha \in (0, 1)$. Let $[f]_\alpha$ be the best $\alpha$-Hölder constant of $f: \mathbb R^d \to \mathbb R^k \otimes \mathbb R^m$, i.e., $[f]_\alpha := \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x-...
Akira's user avatar
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A problem about how to understand the existence of derivative of level set in Mountain-pass theorem

I'm confused about the Mountain pass theorem in Lemma4 of here. Background : $$ \begin{gathered} I_\lambda(u)=\frac{1}{2} \int_M\left|\Delta_g^{\frac{m}{2}} u\right|^2 d \mu_g-\frac{\lambda}{2 m} \log ...
Elio Li's user avatar
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Characterization for the multipliers of Schwartz space

Is the following true? A function $m:\mathbb R^n\to\mathbb C$ is a Schwartz multiplier (i.e. $[f\mapsto mf]:S(\mathbb R^n)\to S(\mathbb R^n)$ is bounded linear) iff the following: For every $\alpha$ ...
Liding Yao's user avatar
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When is a symmetric block Toeplitz matrix invertible?

Let $$ Q = \begin{bmatrix} Q_0 & Q_1 & Q_2 & \cdots\\ Q_{-1} & Q_{0} & Q_1 & \cdots\\ Q_{-2} & Q_{-1} & Q_0 & \cdots\\ \vdots & \vdots & \vdots & \ddots ...
Benjamin Tennyson's user avatar

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