Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,435 questions with no upvoted or accepted answers
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Convergence of the difference quotient of a BV function
Consider a BV function $u \in BV(\mathbb{R}^N; \mathbb{R}^N)$.
What can be said about the difference quotient
$$
\frac{u(x+\epsilon y)-u(x)}{\epsilon}
$$
regarding its convergence as $\epsilon \to 0$...
- 839
2
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0
answers
165
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Jacobian and Jacobian matrix of solutions of ODE with Sobolev vector field
Let $\Phi$ be the Lagrangian flow (defined as in page 6 of this paper) of the ODE
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{...
user124345
2
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0
answers
279
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Relationship between $p$-capacity and Riesz $s$-capacity of a set
What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set?
Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
- 839
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Reference on iterated integrals against projection valued measures
I know (to some extent) how integration over $\mathbb{R}$ of a Borel-measurable function against a projection-valued measure works. Recently while reading a paper I came across calculations in which ...
- 183
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230
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Does Every Extreme Point Maximize Some Linear Functional
Let $L^2$ be the set of all square-integrable functions $f:[0,1] \to [0,1]$ and $S \subset L^2$ be a closed and convex subset of $L^2$ containing the function that is constant and equal to zero. Are ...
- 355
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83
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Existence of a fixed point for this operator
I'm looking for some mathematical results that I might be able to apply to see if an operator I'm considering has a fixed point.
In particular consider,
$$ Ag(x) = \Big\{ \xi(x) + K(g(x))^\frac{1}{\...
- 121
2
votes
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131
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Variant of Sion's minimax theorem
Sion's minimax theorem assumes that $f:X\times Y\to\mathbb{R}$ is being minimized w.r.t. $x$ and maximized w.r.t. $y$, where at least one of $X,Y$ is compact (additional (quasi)convexity and semi-...
- 6,541
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61
views
Second derivative of a functional defined by an integral
I was reading the following example from the book Methods in Nonlinear Analysis (Zhang, Springer) on page 10: First, everything was fine:
Example 2. Let $X = C^1(\overline \Omega, \mathbb R^N)$, $Y =...
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148
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Approximation of functions in $L^p(R^d;L^\infty)$
Assume that the function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$. Can we find a class of functions $f_n\in C_b^2(R^d;L^\infty(B_R))$ such that
$$...
- 411
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answers
57
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The significant role of dual frames in the progress of Frame theory
For a given frame $\{\zeta_i\}_{i=1}^\infty$, any Bessel sequence $\{\eta_i\}_{i=1}^\infty$ satisfying in the following identity for every $\xi\in H$
$$\xi=\sum_{i=1}^\infty \langle \xi, \eta_i\...
- 21
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views
1D Schrödinger Equation with Measure-Valued Coefficients
I've been looking at one of the simplest systems I can think of: a one-dimensional infinite square well on $[0,1]$ with Hamiltonian given by the following:
$$\hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}...
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Off-diagonal estimates for Poisson kernels on manifolds
Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...
- 1,090
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example in $L^p_{s}-$Sobolev spaces
We define $L^p-$
Sobolev spaces as follows:
$$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^d): \mathcal{F}^{-1} [\langle \cdot \rangle^s \mathcal{F}(f)] \in L^p(\mathbb R^d) \}$$
where $\langle \...
- 305
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The dual of the space of continuous sections in a vector bundle
If $X$ is a compact Hausdorff space, one may view the space of complex, continuous functions on it as the space of continuous sections in the trivial Hermitian bundle $X \times \mathbb C$. By the ...
- 5,407
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Differences among various index theories in critical point theory
Index theories help characterize critical points of functionals having certain symmetries. What are the differences (regarding problems they can be applied to) between for example these ones?
the ...
- 839
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answers
255
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Sobolev Multiplication on non-compact manifold
We know that for a compact Riemannian $n$-dim manifold $(M,g)$(the boundary could be nonempty), the Sobolev Multiplication Theorem states that $L^p_k\times L^q_l⟶L^r_m$, where $1/r−m/n>1/p−k/m+1/...
- 1,915
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3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators
During my studies, I came across several different Stone spaces, e.g.:
(i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators;
...
- 207
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159
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What is the boundary of the set $\{ x : dist (x ,\partial \Omega) > \alpha \}$ for a domain $\Omega$?
Let $\Omega$ is a bounded open domain in $\mathbb R ^n$, and $\alpha \geq 0$ a real number, and consider the set $ E_\alpha = \{ x \in \Omega : \text{dist}(x , \partial \Omega) > \alpha\} $, which ...
- 747
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On the infimium of a functional
Let $(M^n,g)$ be a closed Riemannian manifold. Define
$$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0<f\in C^{\infty}(M),\; \int_Mf^2\;d\nu=1\},$$
where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\...
- 169
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150
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Limit circle/point of an ODE with finite eigenvalues
Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+...
- 155
2
votes
0
answers
64
views
Integral equation with kernel defined in a rectangle
Let us consider $$f(x) + \lambda \int_0^4 {K(s,x)f(s)ds=0} ,{\text{ x}} \in {\text{(0}}{\text{,1)}}$$
Observe that the kernel is not defined on a square.
My question: Can I apply the classical ...
- 617
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0
answers
164
views
Convergence of Eigenvalues and Eigenvectors for Uniformly Form-Bounded Operators
Suppose that $A$ is an operator on a dense domain $D(A)\subset L^2$ with compact resolvent, and with quadratic form $q(f,g):=\langle f,Ag\rangle$.
Let $(r_n)_{n\in\mathbb N}$ be a sequence of ...
- 891
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answers
266
views
Optimal transport between Gaussian mixtures and their centers
I have a question about bounding the Wasserstein loss between a continuous Gaussian mixture and a discrete uniform distribution of its centers. In particular, let $P=\frac 1 k \sum_{i=1}^k \mathcal{N}(...
- 161
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306
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Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum
I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...
- 7,609
2
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answers
100
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k-rank numerical range of an operator
Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...
- 231
2
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answers
97
views
Is $\ell_2(A)$ a Hilbert C$^*$-module with Opial property?
If $A=Mat_{n\times n}(\mathbb{C}) $, Is $\ell_2(A)$ a Hilbert $A$-module with Opial property?
Opial property: If ($w-\lim x_n=0 $) then $
(\liminf \lVert x_n\rVert<\liminf \lVert x_n-y \...
- 327
2
votes
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answers
125
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Inclusion of convex polytopes and embedding from $\ell_2$ to $\ell_\infty$
I would like to dig deeper into the problem posted Probability that a convex shape contains the unit ball:
If you pick n points uniformly at random from the surface of a d
dimensional sphere of ...
- 133
2
votes
0
answers
31
views
Funk transform of density supported on an embedded curve
A Funk transform is a certain invertible linear transformation on the space of square-integrable functions on $S^2$. I think its domain can be extended to include densities supported on embedded ...
- 305
2
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0
answers
327
views
Dual of the space of affine functions
Let $M^+(D)$ be the space of all positive measures on a closed convex subset $D$ of a locally convex topological vector space $E$. Two measure $\mu, \nu\in M^+(D)$ one can define a partial ordering $\...
- 521
2
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answers
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views
Integral with product of two infinite sums
I am looking for references and results on integrals with product of two infinite sums:
$$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$
Above integral is ...
- 1,199
2
votes
0
answers
42
views
Dual representation of problems involving $f$-divergences
Studying some problems arising in decision-making under model uncertainty, I'm led to consider the following problems.
Let $\mathbb E_P$ and $\mathbb V_P$ denote the expectation and variance ...
- 6,853
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votes
0
answers
124
views
Is every nonexpansive retract of a Hilbert space closed and convex?
Given a closed and convex subset $C\subset H$ of a Hilbert space $H$, the metric projection is a nonexpansive retraction of $H$ onto $C$. This implies that every closed and convex subset of a Hilbert ...
- 799
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votes
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views
Volume of critical points decreases under symmetric decreasing rearrangement
In the lecture note http://www.math.utoronto.ca/almut/rearrange.pdf, it was stated that the volume of the set of critical points decreases under symmetric decreasing rearrangement. It seems so obvious ...
- 21
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answers
51
views
What restrictions on the form of an integral equation have a unique solution f=0?
We're stuck on the following question in a problem relevant to a physics paper on AdS/CFT that we are working on. Given a Fredholm equation of second kind with the form $f$+$\int_D K f\,dx = 0$, where ...
- 31
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votes
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answers
217
views
Confusion over dentability and denting points
I'm trying to learn about dentable sets and denting points in Banach spaces, but I'm running into some trouble. Part of the issue is that I've attacked this topic in a really sloppy, disorganised ...
- 191
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Sobolev extension with boundary condition
Let $\Omega$ be a Lipschitz bounded domain of $\mathbb{R}^n$, divided in two Lipschitz subdomains $\Omega_1$ and $\Omega_2$ such that $\Omega_1 \cap \Omega_2 = \emptyset$. We define the following ...
- 143
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votes
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333
views
The determinant curvature
Let $(M,g)$ be a riemannian manifold and $R(X,Y)$ the riemannian curvature as a two form with values in the endomorphisms of the tangent bundle. I define:
$$
D_g(X,Y)=det(R)(X,Y)
$$
with $det$ the ...
- 187
2
votes
1
answer
486
views
Common eigenvector of commuting unbounded operators
Let $T$, $S$ be two self-adjoint linear operators on a Hilbert space $\mathcal{H}$ with pure point spectrum.
Then $T$ and $S$ commute if and only if they have a complete set of common eigenvectors.
...
- 21
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votes
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answers
40
views
Invariance of simple functions
Let $(A(s),D(A(s)))_{s\in\mathbb{R}}$ be a family of unbounded operators on a Banach space $X$ and $g:\mathbb{R}\rightarrow X$ be a simple function, i.e.,
\begin{align*}
g=\sum_{i=1}^n{x_i\textbf{1}_{...
2
votes
0
answers
133
views
Convex hull of piece-wise linear functions
Let $K>1$ be a positive integer. Consider a function class $$\mathcal{F}_K:=\Big\{\max_{1\leq k\leq K} a_k^\top x + b_k:\ ||a_k||\leq 1, |b_k|\leq 1, \forall 1\leq k\leq K \Big\}$$ on some compact ...
- 422
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331
views
Sobolev embeddings for vector-valued functions
I would like to know if there is a simple extension of the standard Sobolev embeddings for functions taking values in another Euclidean space.
In particular, let $\Omega \subset \mathbb{R}^n$ be a ...
2
votes
0
answers
315
views
Support of a microlocal defect measure
I'm trying to complete the proof of the Theorem 6.1 in the notes https://www.math.u-psud.fr/~nb/articles/coursX.pdf, which ensures, under certain conditions, that the support of the microlocal defect ...
- 509
2
votes
0
answers
92
views
For what functions does Nash inequality becomes equality?
For what functions does Nash inequality becomes an equality? Also any comment on the regularity of these functions (weak solutions to equality)?
Also same question about Poincare inequality.
- 698
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votes
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answers
190
views
Absence of fixed points
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$
where $x_0$ is an arbitrary but fixed ...
2
votes
0
answers
672
views
Intersection of Sobolev space with the space of continuous functions
While doing some problems, I came across the space $H=H^1(\Omega) \cap C(\Omega)$, where $\Omega$ is subset of $\mathbb{R^n}$. So far, by definition of these subspaces, We know that none of these are ...
- 141
2
votes
0
answers
148
views
Theory of distributions on various domains
The prototypical example of a distribution is the Dirac delta function, defined as a linear functional taking a well behaved test function $\phi:\mathbb{R} \to \mathbb{R}$ and returning its value at ...
- 29
2
votes
0
answers
98
views
Smooth functions with values in bornological vector space
Let $U$ be an open set in $\mathbb{R}^n$ (or more generally, a manifold) and let $V$ be a separated bornological vector space. Do we have
$$C^\infty(U, V) \cong C^\infty(U) \,\hat{\otimes}\, V,$$
as ...
- 9,853
2
votes
0
answers
257
views
The nonlinear operator defined as the commutator of a matrix and a nonlinear operator
In my studies of applied analysis and applied linear algebra, this interesting problem and concept came up:
Let us consider the space of all $ m \times n $ real matrices, and define a scalar ...
- 620
2
votes
0
answers
422
views
Hölder-Zygmund spaces of negative order
In the equation (1.1.17) in Proposition 1.1.6 (ii) in Alazard and Delort's Sobolev Estimates for Two Dimensional Water Waves, there appears a norm named $C^{-1}$, but in Chapter 6 (Appendix) of the ...
- 5,169
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votes
0
answers
152
views
Relationship between eigenvalues of compact operators $A$ and $(A+A^*)/2$
A result from 'Topics in Matrix Analysis' by Horn & Johnson
(3.3.33) is the following: For $A\in \mathbb{M}_n$, $\sum_{i=1}^k Re \lambda_i(A) \leq \sum_{i=1}^k Re \lambda_i \big(\frac{A+A^*}{2}\...
- 21