Questions tagged [fa.functional-analysis]

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

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Is $\phi(t)=\|P(w+td)-w\|_X/t$ nonincreasing if $X$ is "only" a uniformly smooth and uniformly convex reflexive Banach space?

For a Hilbert space $X$ it is known that the function $\phi(t)=\frac{1}{t}\|P(w+td)-w\|_X$ with $t>0$ is nonincreasing. Here, $P:X\to C$ denotes the projection operator and $w \in C, d \in X$ are ...
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Explicit estimates on summability kernels

A "summability kernel" is a sequence of functions $k_n:[0,1)\to \mathbb C$ such that $$ \int_0^1 k_n(t) \mathrm d t =1,$$ $$ \int_0^1 |k_n(t)| \mathrm d t =O(1),$$ with an implied constant ...
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7 votes
1 answer
200 views

Square-root lattices: where do they appear?

As an experimental physicist working on crystallography I'm often dealing with the reconstruction of an object from intensity data that emerge from an imaging device. In mathematics the problem is ...
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A statement on completeness of complex exponentials

I'm currently reading a paper by Olevskii on almost integer translates: https://www.sciencedirect.com/science/article/pii/S0764444297878731 In this paper the author considers for a given sequence $\{ \...
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On invariant subspaces of a nonautonomous heat semigroup

Consider the one dimensional heat equation with a spacetime dependent smooth conductivity coefficient $\sigma(t,x)>0$ on $[0,T)\times (0,1)$ for some $T>0$ subject to source terms $F$, that is ...
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The inversion of the Laplacian transform Pazy's Book "semigroups of linear operators and applications to Partial differential equations"

This question has been posted on Math Stack Exchange but no reply, and so I have to put it here. My question is: In Pazy's Book page 26, the author gives a proof of Lemma 7.1, the lemma 7.1 says that: ...
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41 views

Elliptic partial differential equations with Robin boundary condition and domain of fractional power of Robin Laplacian operator

This question has been posted on Mathematics Stack Exchange but got no response, and so I put it here. When I read the paper "On the attractor for a semilinear wave equation with critical ...
2 votes
1 answer
120 views

Function monotony between [0,T] and $L^2$

Let $\Omega\subseteq\mathbb{R}^N$ be a bounded and smooth domain. If $z:[0,T]\to L^2(\Omega)$ is a function in $H^1([0,T],L^2(\Omega))$ with the property that $z'(t)(x):=z'(t,x)>0$ a.e. on $\Omega$ ...
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On Riesz decomposition of Volterra operator

Let $T:L^2((0,1)) \to L^2((0,1))$ be the Volterra operator defined by $$ Tf(x) = \int_0^x f(t)\,dt.$$ Given any $\lambda\neq 0$ it is well known that there exists some positive integer $n=n(\lambda)$ ...
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3 votes
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145 views

Boundedness of an extension operator

Let $d \ge 2$ be a positive integer. For $x=(x_1,\dotsc,x_{d-1},x_d)$, we write $x'=(x_1,\dotsc,x_{d-1})$. Let $\mathbb{H}^d=\{x=(x',x_d) \mid x_d>0\}$ denote the $d$-dimensional upper half-space. ...
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For a vector field $f$ and a measure $\mu$, does conservativity $\mu$-almost everywhere have a sense?

$\DeclareMathOperator\Jac{Jac}$ Let $\Omega$ an open star shaped subset of $\mathbb{R}^d$ and $f : \Omega \rightarrow \mathbb{R}^d$ a differentiable vector field. For $x \in \mathbb{R}^d$, we denote ...
6 votes
1 answer
508 views

Spectrum of the complex harmonic oscilllator

Let $$ H_\lambda=-\frac{d^2}{dx^2}+\lambda^2 x^2,\quad\lambda>0. $$ It is known that the spectrum of $H_\lambda$ is the set $\{(2n-1)\lambda,n\in \Bbb N^*\}$. Now put $$ (U_\mu \phi)(x)= e^{\mu\...
5 votes
1 answer
102 views

Properties of non-integer powers of the Hodge Laplacian

Consider a complete smooth Riemannian manifold $(M,g)$. I think that it is not difficult to prove that the $k$ Hodge Laplacian is essentially selfadjoint in the relevant $L^2$ space of $k$ forms, ...
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Estimation of a scalar product under the action of an isomorphism

Introduce the following assumptions and definitions: $\Omega\subset \mathbb{R}^3$ a bounded Lipshitz domain, $(L^2(\Omega))^3$ the space of vector square integrable functions, $M$ a $3\times 3$ real ...
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1 answer
279 views

Bounded operator on $L^2(\Bbb R^2)$

Let $\lambda\in \{z\in\Bbb C\mid \text{Re}(z)>0\quad \text{and}\quad \text{Im}(z)>0 \}$. Consider the operator $T_\lambda: L^2(\Bbb C)\to L^2(\Bbb C)$: $$ f\mapsto T_\lambda(f)(z)=\int_{\Bbb C}f(...
4 votes
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81 views

Weighted logarithmic Sobolev inequality

$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that $$ \Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu $$ where the entropy $$ \Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...
5 votes
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105 views

Proof of density of smooth functions in $H(\operatorname{curl})$ with $L^2$ tangential trace

I am trying to understand the proof of the following statement that is presented in “Finite Element Methods for Maxwell's Equations” by Peter Monk and Yangwen Zhang. The original source of the proof ...
2 votes
1 answer
92 views

Example of a compact operator that is not uniformly continuous

I want to find a Banach space $E$ and a compact operator $K:[0,1]\times E \rightarrow E$ (that is, $K$ maps every bounded sequence onto a sequence that converges up to a subsequence) satisfying the ...
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Concentration compactness lemma and the best Sobolev constant

It is well known that the best Sobolev constant can be achieved on $\mathbf{R}^n$. More precisely, we have the following theorem (A): Let $\frac{1}{q}=\frac{1}{2}-\frac{1}{n}$, $$S=\inf\limits_{{u\in ...
8 votes
1 answer
219 views

Why operator systems?

A $\mathrm{C}^*$-algebra $\mathcal{A}\subset B(\mathsf{H})$ is a norm-closed, self-adjoint subalgebra of bounded operators on a Hilbert space. If we then take a unital self-adjoint (possibly closed) ...
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Boundary conditions for first-order nonlinear system of PDEs

Consider the following system of PDEs for the dependent variables $x=x(u,v)$ and $y=y(u,v)$, \begin{align} E(u,v)\:x_v^2-2F(u,v)\: x_vx_u+G\:x_u^2&=\Delta^2\\ E(u,v)\:y_v^2-2F(u,v)\: y_vy_u+G\:y_u^...
4 votes
1 answer
138 views

The complement of $L_1(0,1)$ in $L_1(0,1)^{**}$

Let $\mu$ be a finite measure, like the Lebesgue measure in $(0,1)$. It is well-known that $L_1(\mu)$ and its second dual $L_1(\mu)^{**}$ are Banach lattices, $L_1(\mu)$ is a projection band in $L_1(\...
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1 vote
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Approximation in Bochner spaces

Is there any result like the Bramble-Hilbert lemma for Bochner spaces? More specifically: let $H$ a (e.g.) Hilbert space and $I$ a compact interval, $L \in L(H^k(I,H), Y)$ for $Y$ a normed space. If $...
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Spectral measure for a finite set of mutually commuting normal operators

The following question is from Exercise $\S 11.11$ in A Course in Operator Theory written by John B. Conway: Suppose $\{N_1, \cdots, N_p\}$ is a finite set of mutually commuting normal operators in $...
2 votes
1 answer
76 views

Representing solutions of $-\Delta u+au=f$ when $a\leq 0$

Let $\Omega=[0,1]\times [0,1]$ be the square. We say a function $f\in H^1(\Omega)$ is periodic on $\Omega$ if $f(x,0)=f(x,1)$ and $f(0,y)=f(1,y)$ (in the sense of traces of course). Now consider the ...
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How to prove an equality involving Laguerre polynomials

Assume $\mu<0$. Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$ and $f\in L^2(\Bbb C)$. How to prove that $$\sum^\infty_{k=0}\int_{\Bbb C} f(z)\frac{1}{2(2k+1)-\mu}L^0_k(\lvert z\rvert^2)...
1 vote
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N-wave solution of conservation law $u_t + (u - u^2)_x = 0$

How can we compute the "N-wave" source-solution of the conservation law $$u_t + (u - u^2)_x = 0, $$ that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...
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2 votes
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Commutative Banach algebras with zero-dimensional maximal ideal space and disjoint supports of Gelfand transforms

Let $A$ be a commutative semi-simple unital Banach algebra and let $\Delta$ be the maximal ideal space of $A$. Denote by $\widehat{\cdot}\colon A\to C(\Delta)$ the Gelfand transform. If $\Delta$ is ...
2 votes
1 answer
114 views

$\Psi$ in finite Wiener–Itô Chaos implies existence of continuous representative on neighborhood of Cameron–Martin space?

Let $(\Theta, H, \mu)$ be an abstract Wiener space, i.e. let $(\Theta, \lVert \cdot \rVert_{\Theta})$ be a separable Banach space, let $(H, \langle \cdot, \cdot \rangle_{H})$ be a separable Hilbert ...
6 votes
1 answer
319 views

Harmonic analysis for a beginner

I am currently dealing with discrete Fourier transform and correlation technique to construct the spectrum of a broad band signal. It's already known that if I have enough observations of the signal, ...
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Minimal value of condition number of positive-definite matrices, w.r.t $\ell_1$ norm

Let $n \ge 1$ be an integer and let $p \in [1,2]$. For any $n \times n$ positive-definite matrix $M$, let $\kappa_p(M)$ be the condition number of $M$ w.r.t to the $\ell_p$-norm on $\mathbb R^n$, i.e \...
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13 votes
1 answer
305 views

Is there a reflexive Banach space whose ball is not the convex hull of its extreme points?

Let $X$ be a reflexive Banach space. Then the convex hull of the extreme points of the unit ball is weakly dense by the Krein-Milman theorem and Kakutani's theorem. My question is, if there is an ...
13 votes
1 answer
414 views

Does locally nilpotent imply nilpotent for continuous self-maps of intervals?

Let $f\in C([0,1],[0,1])$ be such that: $$\forall x\in [0,1], \; \exists k\in \mathbb N, \; f^{\circ k}(x)=0.$$ Is it true that $f$ is nilpotent (i.e., that there is some $k$ such that $f^{\circ k}=0$)...
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2 votes
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Why is the essential numerical range defined as $W_e(T) = \bigcap_{K\in \mathcal K(H)} \overline{W(T+K)}$?

I have been introduced to the following definition of the essential numerical range of a bounded, linear operator on a separable, infinite-dimensional Hilbert space: $$W_e(T) := \bigcap_{K\in \mathcal ...
2 votes
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55 views

Is it possible to extend Borel's lemma to the case of functional derivatives?

Let us think of a collection of tempered distributions $\{ T(x_1, \cdots, x_n)\}_{n=0}^\infty$. Here I will specifically set $x_i \in \mathbb{R}^4$ since I am considering quantum field theory and ...
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Necessary and sufficient conditions for $\mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}) \ge 0$ for all $t$

Let $A$ and $B$ be positive-definite matrices of the same size. For any $t \ge 0$, define $$ u(t) := \mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}). $$ Question. What are necessary and ...
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2 votes
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Anisotropic Calderon-Zygmund decomposition

I am looking for the following version of Calderon-Zygmund decomposition, consider an function $f \in L^1(R^{d+1})$ and cylinders of the form $Q_{R,R^p}$ for some fixed $p \in (0,\infty)$, The ...
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4 votes
1 answer
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Chain rule in Sobolev space

In the theory of Sobolev space, we have the following chain rule: For a uniformly Lipschitz function $F : \mathbf{R}\to \mathbf{R}$ such that $F(0)=0$, and $u\in W^{1,1}(\mathbf{R}^n)$, then we have ...
3 votes
1 answer
323 views

Equivalence between two fractional Sobolev spaces

For $s \in (0,1)$, we consider the spectral fractional Laplacian \begin{align} (-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k \end{align} where \begin{align*} \begin{cases} ...
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5 votes
1 answer
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Gaussian-to-Gaussian transformations are affine a.e.?

Let $\mathcal{G}_n = \{ N(\mu,\Sigma) ; \mu \in \mathbb{R}^n, \Sigma > 0\}$ be the collection of Gaussian distributions on $\mathbb{R}^n$ with full support. If $f : \mathbb{R}^n \to \mathbb{R}^k$ ...
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6 votes
0 answers
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What is the closed cone generated by constant and coordinate functions and closed under taking $f\mapsto\max(f,0)$?

Let $C$ be the smallest closed convex cone of functions from $\mathbb{R}^n$ to $\mathbb{R}$ that contains all constant functions, all coordinate functions, and such that $\max(f,0)\in C$ whenever $f\...
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1 vote
1 answer
215 views

Why we have $f=0$

Define the Fourier transform for a suitable function $f\in L^1(\Bbb R)$ by $\widehat{f}(\xi)=\int_{\Bbb R}f(x)e^{-ix\xi} dx$. Assume the condition $$\int_{\Bbb R}\int_{\Bbb R}|\widehat{f}(\xi)f(x)|^...
3 votes
0 answers
90 views

Ergodic diffeomorphisms of the circle

From the paper Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889. the following result is known: Let $(E,\Sigma, \mu)$ be a ...
5 votes
1 answer
161 views

Isomorphic embedding of $l^n_{\infty}$ into $l_1^m$?

Given $n$, is there a $C(n)$-isomorphic embedding of $l^n_{\infty}$ into $l^m_1$ for sufficiently large $m$ and $C(n)<<\log(n)$? For $n=2$ this can done with $m=2$. There are some results about $...
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2 votes
1 answer
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On the equality $\{f\in C(X), f|_A=0\}+\{f\in C(X), f|_B=0\}=\{f\in C(X), f|_{A\cap B}=0\}$

Let $X$ be a Tychonoff space, let $A,B\subset X$ be closed. Let $J_A$ be the set of all continuous on $X$ real-valued functions which vanish on $A$. For which $X$'s is it true that $J_A+J_B=J_{A\cap ...
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0 votes
0 answers
89 views

Almost everywhere convergent Fourier series

Apparently there is a deep theorem stating: Let $f:\mathbb{R}\to \mathbb{C}$ be a function satisfying $f(x)=f(x+2\pi)$ and $\int_0^{2\pi}|f(x)|^2dx<\infty$. Then the Fourier series of $f$ converges ...
4 votes
0 answers
63 views

Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$

For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that $$ |f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
2 votes
0 answers
45 views

Can we decompose an increasing net of functions into two increasing nets with prescribed supports?

Let $K$ be a compact Hausdorff space and let $U,V\subset K$ be open. Let $\left(f_{i}\right)_{i\in I}$ be an increasing net of continuous non-negative functions such that $f_{i}\le 1$ and $f_{i}$ ...
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1 vote
1 answer
100 views

Cameron-Martin space of product space

Suppose you have Banach spaces $\mathcal B_\alpha$ where $\alpha$ is in some index set $I$. Let $\mu_\alpha$ be Gaussian measures on $\mathcal B_\alpha$ with Cameron-Martin spaces $\mathcal H_{\mu_\...
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50 views

Continuity of "inversion operator" between function spaces

Question: When is the operation of inversion continuous as a map between spaces of invertible functions? Let $\mathcal{F}$ be a function space such that $f\in\mathcal{F}\implies$ $f$ is invertible and ...