All Questions
10,198 questions
2
votes
0
answers
30
views
Dual of homogeneous Triebel-Lizorkin
Let $ p, q \in (1,\infty)$ and consider the homogeneous Triebel- Lizorkin space $\dot{F}^{s}_{p,q}$ to be the space of all tempered distributions (modulo polynomials) with
$$
[f]^{p}_{\dot{F}^{s}_{p,q}...
- 31
0
votes
0
answers
81
views
Measurable Extension
Let $(\Omega, \mathcal{F})$ be a measurable space and $X$ some metric space (probably Polish) with the Borel $\sigma$-algebra and a function $f: \Omega \times X \to \mathbb{R}$. Usually, functions ...
- 136
0
votes
0
answers
34
views
Support of a function acting on an algebra?
Quick: for a measurable function $f$ its support on Euclidean space is clearly just the subset where $f$ does not vanish.
Now, let’s have $f$ acting on an finite Lie algebra, f.e. $\mathfrak{gl}$ as $...
2
votes
0
answers
67
views
Regularity and decay of Fourier-like series on a manifold
Let $D$ be a first-order self-adjoint elliptic differential operator acting on sections of a vector bundle $S$ over a closed manifold $M$. Then it is well-known that the various eigenspaces $E_\lambda$...
- 1,903
1
vote
0
answers
53
views
Reference for Density question
Let $ B $ be a reflexive, separable Banach space and $ p \in (1,\infty)$. Then denote by $L^{p}(B)$ the space of all functions $$ f : \mathbb{R}^{n} \to B $$ with
$$
\int_{\mathbb{R}^{n}} \vert f \...
- 31
2
votes
1
answer
142
views
Estimating a solution to an Euler-type ODE
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer and $a$ be a real number.
Let $u(r)$ be a function on $[1,\infty)$ ...
- 969
0
votes
1
answer
100
views
Projection on a countable union of linear subspace
For any natural number $n$, $V_n$ denotes a closed linear subspace of a $L_2(m)$ space, which is an Hilbert Space, where $m$ denotes a finite measure. Moreover $(V_n)$ is increasing, that is $V_n$ is ...
- 31
1
vote
0
answers
78
views
Trace theorem for $L^2([0,1]; H^k(S^2))$
Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer.
Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
- 969
0
votes
0
answers
22
views
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 $...
- 1,334
0
votes
2
answers
167
views
Is a signed measure $\mu$ on $\mathbb{R}^d$ characterized by the transform $\mathcal{L}_\mu (\lambda ):=\int e^{\langle \lambda,x\rangle }\mu (dx)$?
In the book "Probability Theory" by Achim Klenke there's the following theorem: a finite measure $\mu$ on $[0,\infty )$ is characterized by its Laplace transform $\mathcal{L}_\mu(\lambda):=\...
- 15
2
votes
0
answers
75
views
Regularity of solutions to an elliptic boundary value problem
Let $M = [1,\infty)\times S^2$. For an integer $k \geq 2$ and number $\tau<0$, define the space $L^2_{\tau}([1,\infty);H^k(S^2))$ to be all $H^k(S^2)$-valued functions $u$ on $[1,\infty)$ with $\...
- 969
8
votes
1
answer
537
views
Reference request: Expository paper on the use of functional analysis in differential and integral equations
Some textbooks on functional analysis do not hint that a major raison d'être of the subject is its use in the study of differential and integral equations. The reader could go all the way through ...
1
vote
0
answers
56
views
Finding thin plate spline subjected to boundary conditions
I am trying to formulate a problem as a PDE. What I want to know is if my formulation is correct, if it admits solution and what am I missing.
This question is related to : Thin-Plate-Spline ...
- 115
6
votes
1
answer
528
views
A functional equation
I am working on some physics problem and got stuck with the following equation: Let $a$ be a very small positive number. Is there a bounded function $F$, $0 \leq F \leq 1$, such that for all $x \in \...
- 63
0
votes
0
answers
44
views
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
...
- 101
4
votes
0
answers
158
views
Measurability of $L^{p}(L^{q})$ integrable functions
Let $ F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ be a function with the property that
$
\int_{\mathbb{R}^n} \big[ \int_0^\infty |F(x,r) |^q \, dr \big]^{p/q} \, dx < \infty
$
In addition we ...
- 41
0
votes
0
answers
40
views
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
$$
\tag{(A)...
- 5,405
0
votes
0
answers
53
views
Vectors of complex exponentials span $\mathbf{C}^N$
Let $Q = [0,1]\times [0,1]$. Let $1\leq k < \infty$ and $\{(x_l,\xi_l)\}_{l=1}^{k}\subseteq Q$ be such that $(x_i,\xi_i)\ne (x_j, \xi_j)$ for $i\ne j$. Additionally, for $1\leq l \leq k$, let $n_l\...
- 171
0
votes
1
answer
114
views
Ball in separable Banach space has positive Gaussian measure
I have (presumably non-degenerate) Gaussian $\mu$ over separable Banach space $X$. I would like to prove that for any ball of radius $r$ centered at $x$, $\mu(B_r(x))$. I know how to prove this in ...
- 379
1
vote
1
answer
127
views
approximating differentiable functions with double trigonometric polynomials
Let $Q = [0,1]^2$. For sake of notation, let
$$
f^{(i,j)}(x,\xi) = \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x,\xi).
$$
Fix some non-negative integer $k$. Moreover let $f\in C^k(Q)$ if
$$
\|...
- 171
6
votes
0
answers
293
views
Looking for Mackey's PhD thesis, "The subspaces of the conjugate of an abstract linear space"
I'm looking for a copy of George Mackey's PhD thesis, The subspaces of the conjugate of an abstract linear space (Harvard Univ., 1942), but am currently struggling to find one online, with the only ...
- 11.8k
2
votes
0
answers
132
views
Convergence of integral operators' inverses
Let $K$ be a positive definite integral operator with continuous kernel $K(x,y)$ defined by
$$
Kf(x) = \int_0^1 K(x,y) f(y) \, dy.
$$
Let $K_n$ denote the matrix $K(x_i, x_j)$ with $x_i = i /n$ and ...
- 620
3
votes
2
answers
250
views
Existence of a positive measurable set with disjoint preimage under iterated transformation
Let $(X,\mathcal B,\mu)$ be a atomless probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left({x\in X: T^n(x)=x}\right)=0$ for every $n\ge 1$. Let $A\in \mathcal ...
- 105
2
votes
0
answers
151
views
Does the complex interpolation space $(L^1(\mathbb{R}),W^{2,1}(\mathbb{R}))_{\frac{1}{2}}$ continuously embed into $L^\infty(\mathbb{R})$?
The complex interpolation space between $(L^p(\mathbb{R}),W^{2,p}(\mathbb{R}))_\theta$ with interpolation parameter $\theta=\frac{1}{2}$ is known to be $W^{1,p}(\mathbb{R})$ for $1<p<\infty$. As ...
- 989
0
votes
1
answer
65
views
Strict positive definite function gradient tuple
I have a (Gaussian) random function (aka "stochastic process" or "random field") $(f(t))_{t\in \mathbb{R}^d}$. I now want to consider the vector valued random function $g=(f, \...
- 385
2
votes
1
answer
197
views
Prékopa-Leindler style inequality?
Does anyone know a simple proof of the following Prékopa-Leindler style inequality:
If we have $f_1,f_2,g_1,g_2$ strictly positive functions on $\mathbb{R}$ such that, for any $x_1,x_2 \in \mathbb{R}$,...
- 125
1
vote
1
answer
121
views
An asymptotic integral with complex phase
Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds
$$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
- 4,145
0
votes
1
answer
53
views
Rate of convergence of the minimum point over a product space
Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that
$f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$.
$f(\theta, \epsilon) > 0$ for ...
- 434
-1
votes
1
answer
139
views
$L^1$ convergence
Setting
For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R})...
- 125
3
votes
0
answers
105
views
Maximal-type inequality for a Borel probability measure supported on a subset of $L^2(\mathbb{R}^d)$
Let $\mu$ be a Borel probability measure on $L^2(\mathbb{R}^d)$ for $d\ge 1$ which is moreover supported on the unit sphere
$$S=\{\phi\in L^2(\mathbb{R}^d): \| \phi\|_{ L^2(\mathbb{R}^d)}=1\}.$$
Let ...
- 850
0
votes
1
answer
77
views
Decay rate of minimum point over a product space
Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that
$f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$.
$f(\theta, \epsilon) > 0$ for ...
- 434
3
votes
0
answers
73
views
About the J-method of interpolation
In the classic text of J.Bergh and J.Lofstrom, Interpolation Spaces, the $J$-method of real interpolation defines a functor in the following way: For $0<\theta<1$ and $1\leq q\leq \infty$ we ...
2
votes
1
answer
79
views
Hausdorff-Lipschitz continuity of cone correspondence
Let $\mathbb{R}_+$ denote the strictly positive real numbers, let $\mathcal{X} \subset \mathbb{R}^n$ and $\mathcal{P} \subset \mathbb{R}^m$ be compact and convex subsets, let
\begin{equation}
f: \...
- 55
2
votes
0
answers
69
views
Schauder Frames in nuclear vector spaces
In recent years, the definition of frame has been extended to locally convex topological vector spaces (lcs) (1). In particular, let $X$ be a lcs and $X'$ its dual. A sequence $\big((x_n,y_n)\big)_{n\...
- 655
11
votes
1
answer
676
views
Entropy arguments used by Jean Bourgain
My question comes from understanding a probabilistic inequality in Bourgain's paper on Erdős simiarilty problem: Construction of sets of positive measure not containing an affine image of a given ...
- 196
0
votes
0
answers
57
views
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
$$\mu_k(A):=\underset{P_1, \ldots, P_r}{\sup_{A=A_1\sqcup\...
- 75
2
votes
1
answer
126
views
Subspaces of $C_0$ on which $p$-norm are equivalent?
I have a question concerning the generalization of the following fact.
Let $E = C^0([0,1],\mathbb{R})$ endowed with the $\|.\|_\infty$ norm. One can show that if $F$ is a subspace of $E$ for which ...
- 125
0
votes
1
answer
102
views
On weighted Fourier transforms
Suppose that $f\in L^{\infty}((0,1))$ and that there exists $c_1,c_2>0$ such that
$$ \left|\int_0^1 e^{i \xi x} e^{-|\xi|^{-1}x}f(x)\,dx \right| \leq c_1 e^{-c_2|\xi|} \quad \forall\, |\xi|>1.$$
...
- 4,145
3
votes
0
answers
128
views
Image of trace operator on $W^{2,1}(\mathbb{R}^2)$
It is known that for a domain $\Omega\subset \mathbb{R}^2$ with $C^1$ boundary $\partial\Omega$, that the trace operator is bounded and surjective from $T:W^{1,1}(\Omega)\to L^1(\partial\Omega)$.
For ...
- 989
0
votes
1
answer
127
views
Continuous extensions of tangent vector fields
Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
- 434
0
votes
1
answer
80
views
Continuous modification of tangent vector fields
Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
- 434
3
votes
1
answer
249
views
Is Toeplitz operator on the Bergman space bounded iff its symbol is bounded?
It is fairly well known that if $T_\varphi$ is a Toeplitz operator on the Hardy-Hilbert space, then $\lVert T_\varphi \rVert = \lVert \varphi \rVert _{\infty}$.
Now, if $\varphi \in L^\infty (\mathbb ...
- 151
6
votes
2
answers
449
views
Asymptotic behavior of the "Cauchy square" series
$\renewcommand{\ge}{\geqslant}\renewcommand{\le}{\leqslant}$
$\newcommand{\pa}[1]{\left( #1 \right)}$
Let us take $\alpha > 0$, $x_1 := \alpha$ and for any $n \ge \mathbb{N}$,
\begin{align*}
\boxed{...
- 61
2
votes
0
answers
138
views
Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?
The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm
$$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$
The space $L^2([a,b]\times S^2)$ ...
- 969
1
vote
0
answers
92
views
Multilinear non-commutative Khintchine inequality
Let $g_1,\ldots,g_k$ be independent standard Gaussians and for each index $(i_1,\ldots,i_k)\in [n]^k$ let $A_{i_1,\ldots,i_k}$ be a $d\times d$ symmetric matrix.
Question: Is there a known bound for ...
- 156
4
votes
0
answers
262
views
Spectrum of ring in algebraic geometry vs spectrum of Banach algebra
For a commutative unital Banach algebra $A,$ and $x\in A,$ we have $\lambda \in \sigma_A(x)$ if and only if $\phi(x) = \lambda$ for some algebra homomorphism $\phi:A \to \mathbb C.$ The set of all ...
- 1,755
1
vote
1
answer
112
views
A bilinear estimate with a simple one-dimensional oscillatory integral kernel
Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$.
I am trying to show that
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
- 852
0
votes
0
answers
149
views
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 ...
- 52.3k
2
votes
0
answers
28
views
Monotonicity of the rank of finitely generated closed sublattices
Let $E$ be a Banach lattice. We consider finitely generated elements in the class of all closed sublattices of $E$. Let $F$ be a closed sublattice $E$. Then a subset $G \subseteq F$ is called ...
- 467
2
votes
0
answers
78
views
What is known about $\operatorname{gap}(A) = \|A\| - r(A)$ for bounded operators on Hilbert spaces?
The gap of a bounded linear operator on a Hilbert space is defined as $$\operatorname{gap}(A) := \|A\| - r(A),$$
where $r(A)$ denotes the spectral radius of $A$. A natural question to ask is - for ...
- 165