All Questions
9,780 questions
1
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102
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Pertubations of self-adjoint first order operators
If we consider the self-adjoint operator $L= L_0 + h$ on the appropriate Sobolev spaces of maps from $S^1$ to $\mathbb{R}^N$, say, where $L_0$ is a first order self-adjoint operator and $h$ is a ...
- 11
5
votes
0
answers
161
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$L^p$ estimates for Ornstein-Uhlenbeck: what is known beyond hypercontractivity?
Consider an infinite-dimensional Gaussian random vector $X$, and a positive random variable $f(X) \in L^p, p > 1$. Let $f(X) \sim \sum_n f_n(X)$ be its (formal) chaos expansion. Let $(U_\rho, \rho \...
- 4,010
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0
answers
213
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Equivalent Gaussian measures
Let $\mu$ be a gaussian measure with eigenpair $\{e_k,2^{-k}\}$ and $\nu$ with eigenpair $\{ Te_k,2^{-k}\}$. Here, T is the unitary operator given by $Tx = x - 2\left\langle x,v \right\rangle v$. Let $...
- 33
9
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1
answer
450
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A question on infinite dimensional Gaussian measure and affine tranformations.
Let $\gamma_\infty$ denote the product Gaussian measure on $\mathbb{R}^\mathbb{N}$. Which $a,b \geq 0$ satisfy that for every Borel set $K\subseteq \mathbb{R}^\mathbb{N}$ of positive measure, $a K + ...
- 3,547
2
votes
1
answer
137
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Continuous depedence of the spectrum on elements
Suppose $a_n \to a$ in a unital C*-algebra $A$. If $\lambda_n \in \sigma(a_n)$ and $\lambda_n \to \lambda$, then $\lambda \in \sigma(a)$. Does the converse hold?
So if $\lambda \in \sigma(a)$, does ...
- 388
1
vote
0
answers
85
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Coersivity of a bilinear form [closed]
I need to proof the coersivity of the following bilinear form. a,b and c are scalars, u is the velocity vector field and p is the pressure. Any help is much appreciated!
$$ B(\textbf{u},\textbf{v}) = ...
- 11
2
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1
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469
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If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
- 3,245
0
votes
1
answer
169
views
Exponential Convexity Results [closed]
$\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous
and
$$\sum _{i, j=1}^n\xi_i\xi_jh(x_i+x_j)\geq 0,$$
for all $n\in\mathbb{N}$ and all ...
- 153
2
votes
1
answer
680
views
spectra of sums in (Banach) algebras
A similar question was already asked in question titled "Spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]".
Answer there led me to the following question.
If for ...
- 179
1
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0
answers
125
views
Convergence of solutions of the volterra integral equation with convergent kernels
Consider the following Volterra integral equation
$$
g(t) = \int_0^t K_n(t,s)w_n(s) ds
$$
where g(t) and K_n(t,s) are continuous and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ ...
- 111
2
votes
0
answers
126
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Nearly injective Banach spaces
There was a problem about nearly injective metric spaces posed by Aronszajn and Panitchpakdi which I actually solved in the past but it still remains open (as long as I know) for the Banach spaces--so ...
- 7,500
2
votes
0
answers
787
views
Regarding a proof in Bourbaki's Topological Vector Spaces
On Bourbaki's TVS Chapter IV pages 33-34, the last part of Proposition 2 can be formulated as follows:
Notations:
$K$ - The underlying field which is the real or complex number field;
$X$ - A ...
- 21
0
votes
1
answer
259
views
Convolution with an element in the dual space
We recall that if $f_1\in L^p(\mathbb R)$ and if $f_2\in L^q(\mathbb R)$ where
$1 \lt p \lt \infty$ and $\frac 1p+\frac1q=1$ then the function $f_1\ast f_2(x)=\int_{\mathbb R} f_1(x-y) f_2(y)dy$ is a ...
- 415
4
votes
2
answers
581
views
An analogue of Hilbert-Schmidt theorem for multilinear forms
Let $H$ be a (the) real separable Hilbert space. The Hilbert--Schmidt theorem says that a compact self-adjoint operator $A$ has an eigenfunction expansion. Instead of operator, we can think of a ...
- 1,533
1
vote
1
answer
154
views
Is the speed of a curve in $ \ell^\infty $ zero a.e. if the derivative of each component is zero a.e.?
Let $ A $ be an $ \mathcal{H}^1$-measurable subset of $ \mathbb{R} $ and $ \gamma \colon A \subseteq \mathbb{R} \to \ell^\infty $ be a Lipschitz mapping with the Lipschitz constant $ L $. Also, assume ...
- 520
5
votes
2
answers
958
views
L1 distance from a trigonometric susbspace
How to check, whether the $L^{1}$ distance between a finite exponential sum $S_{F}(x)=\sum\limits_{n\in F} \exp(inx)$ and the $L^{1}$-closure of subspace $\mathrm{span}\left(\exp(inx): n\in \mathbb{Z}\...
- 696
4
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0
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107
views
Is Wiener's Tauberian theorem true in Wiener space?
Let $\gamma$ be the standard product Gaussian measure in $\mathbb{R}^\infty$, and let $\mu$ be a finite variation measure, not necessarily positive, such that $\mu \ll \gamma$.
Is the following true?
...
- 4,010
4
votes
0
answers
454
views
Binomial Expectation of Convex Function
Suppose $x$ has a binomial distribution with chance $\alpha$ drawn $k$ times, and let $f(x)$ be a positive convex real valued function. I would like to evaluate
$$\frac{\partial}{\partial \alpha} \...
- 41
1
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1
answer
160
views
Contractively complemented subspaces without contractively complemented complement
Can someone give me an example of a Banach space $X$ and contractive projection $P\in\mathcal{B}(X)$ such that $\ker P$ is not a range of any contractive projection $Q\in\mathcal{B}(X)$?
- 1,697
2
votes
1
answer
202
views
tensor product with projective topology
There are two Banach spaces $X,Y$. These spaces have unconditional Schauder bases $\{e_i\}$ and $\{f_i\}$ respectively.
Is this right that $e_i\otimes f_j$ is the unconditional Schauder basis in $X\...
- 21
5
votes
0
answers
1k
views
Generalized Stone Weierstrass theorem
Given a smooth function $f$ on some compact $K$ in the euclidean space $\mathbb{R}^d$, does exist a sequence of polynomial functions $p_n$ such that $p_n$ and all of its derivatives converge uniformly ...
- 367
4
votes
1
answer
985
views
weak convergence in infinite dimensional spaces
Weak convergence can be tricky when dealing with infinite dimensional spaces. For example, the usual Levy's continuity theorem does not extend readily to separable Banach spaces.
Consider a (...
- 2,133
2
votes
0
answers
185
views
Showing $\langle \frac{\partial b(v)}{\partial t}, v \rangle_{H^{-1}(\Omega), H^1(\Omega)} = \frac{d}{dt}\int_{\Omega}\Psi^*(b(v))$
Let $b$ be continuous and increasing with $b(0) = 0$. Define $\Psi(t) = \int_0^t b(s)\;ds$ and $\Psi^*(s) = \sup_{r \in \mathbb{R}} (sr-\Psi(r))$.
(Note $\Psi^*(b(s)) + \Psi(s) = sb(s)$).
Let $v \...
- 173
3
votes
1
answer
328
views
Laplacian on coset spaces
Edited after @J. Martel's comment: Let us consider the sphere $S^n$ (embedded in $\mathbb{R}^{n+1}$). We know that if $X_i$ represent the vector fields on $S^2$ giving the rotation about the $x_i$-...
- 33
0
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0
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114
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Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space
Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map $...
- 5,529
1
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1
answer
686
views
analytic continuation of a Laplace transform from a countably infinite set of points?
Let $f(\lambda)=\int_0^\infty e^{-\lambda s} F(ds)$, where $F$ is the distribution of a positive random variable. Suppose I know the value of $f(n)$ for $n=0,1,2,\cdots$. Is this enough to uniquely ...
- 351
2
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1
answer
372
views
Complementable subspaces of $(c_{00}(S),\Vert\cdot\Vert_1)$
Let $\ell_{1,0}(S)=(c_{0,0}(S),\Vert\cdot\Vert_1)$ be a space of functions on a set $S$ with finite support, endowed with $\ell_1$ norm. Could you answer the at least one of the following questions
...
- 1,697
3
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0
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236
views
Is every covariance operator the covariance of a measure?
Let $X$ be a topological linear space over $\mathbb R$ which is complete and Hausdorff with a dual space that separates points. Let $k : X^* \to X$ be an arbitrary covariance operator. i.e., any ...
- 8,512
3
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0
answers
189
views
Can we obtain topology results using analysis in metric measures spaces?
Let $M$ be a smooth compact manifold. It is known that a lower bound on the Ricci curvature is equivalent to the convexity of the entropy on $\mathcal{P}^2(M)$ (Von Rennesse and Sturm '05), but I don'...
- 215
0
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1
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901
views
Schwartz space inequality
Let $g$ be a function in the Schwartz space $\mathscr S (\mathbb R)$. Show that for any $l \ge 0$, we have $\sup_x |x|^l |g(x-y)|\le A_l (1+|y|)^l$ by considering separately the cases $|x|\le 2|y|$ ...
2
votes
3
answers
1k
views
Baire category theorem
Let's call the following conditions (1): $X$ is a complete metric space with metric $d$, $X = \cup_{n=1}^\infty A_n$. Let $\bar{A}$ denote the closure of $A$.
Let's call the following statement (2): ...
- 498
3
votes
2
answers
1k
views
Sequences of linear combinations of measures
Let $X$ be a Polish space. Let $J\in\mathbb{N}$.
Let $\lbrace a^n_1\rbrace_n,\dots,\lbrace a^n_J\rbrace_n$ be $J$ sequences of reals.
Let $\lbrace \mu^n_1\rbrace_n,\dots,\lbrace \mu^n_J\rbrace_n$ be ...
user12713
0
votes
1
answer
321
views
Increasing regularity for $L^2$ function
Suppose that we have a function $u$ on $\mathbb{R}^2$ such that $r\frac{\partial}{\partial\theta}u \in L^2(\mathbb{R}^2)$. Suppose it is also given that $u$ lies in some fractional Sobolev space $H^s(\...
- 1
9
votes
2
answers
477
views
An extension of Gaussian Isoperimetry
The Gaussian isoperimetric inequality (Tsirelson,Sudakov, Borell) states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian ...
- 949
3
votes
0
answers
74
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Semi-continuity of the dimension of the null space
Suppose $T_n : X \rightarrow X$ is a sequence of Fredholm operators on a Banach space such that $T_k \rightarrow T$ strongly (in the induced operator norm). If $N_k$ and $N$ denote the dimensions of ...
- 537
2
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0
answers
79
views
Point Spectrum of a Second Order System of Differential Equations
Consider the following operator acting on $H^1(\mathbb{R})$
$$
\mathcal{L} \left(\begin{array}{c} \phi \\ \psi \end{array}\right) = -\left(\begin{array}{c} \phi \\ \psi \end{array}\right)'' + V(x) \...
- 516
-1
votes
1
answer
77
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Parseval frame, convergence of $\sum_{k=0}^\infty \left\|g_k\right\|$ [closed]
Let $\mu$ be a Borel probability measure on $[0, 1)$, and $\{g_k\}_{k=0}^\infty$ be a Parseval frame for $L^2(\mu)$. Does
$$\sum_{k=0}^\infty \left\|g_k\right\|$$
converges?
- 297
-4
votes
1
answer
200
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How I can choose $(t_1,t_2,...,t_{r}) \in (0,1)^{r}$ such that $f^{(k)}\left(1-2\prod_{j=1}^{k}t_{j}\right)=0$?
Let $f:\mathbb{R} \to \mathbb{R}$ be a real analytic function. Assume that $f$ has simple trivial zeros at each nonpositive integer. Then, all the $k$-th derivatives $f^{(k)}$ of $f$ have necessarily ...
- 1,197
0
votes
1
answer
319
views
continuty of volume of a convex set in Rn [closed]
Let O(X) be the metric space of all compact subsets of a compact set X in Rn and let L be an element of O(X). Let vol(L) be the volume of L. How do we prove that vol(L) is a continuous function on O(X)...
- 1
2
votes
1
answer
384
views
Properties of the weak-$*$ topology
Let $X$ be a topological affine space over a complete base field $\mathbb S := \mathbb C$, $\mathbb R$ or $\mathbb Q_p$. Let $X^*$ be the dual space of continuous affine functionals equipped with the ...
3
votes
0
answers
274
views
Fourier transform and support of a distribution
Let $T \in \mathscr D'(\mathbb R^n)$ be a distribution, such that its Fourier transform $\widehat T$ is a real analytic function on $\mathbb R^n$ but it can't be continued to an entire function on $\...
- 1,329
4
votes
1
answer
367
views
Manifold structure for the set of solutions to a first order elliptic system?
Consider a bounded domain $S\subset R^2$ and an elliptic system of two first order PDEs, namely a generalization of the Cauchy-Riemann system allowing nonconstant coefficients and lower order terms. ...
- 41
3
votes
0
answers
186
views
Smooth function over a manifold into an algebra
Let $M$ be a manifold and $A$ a $*$-algebra. Does is hold that
$$C^{\infty}(M,A) \cong C^{\infty}(M) \otimes A$$
where the RHS means that you take smooth functions which map into $A$. If this holds, ...
- 31
2
votes
1
answer
400
views
Existence of a measure under certain condition
Hi everyone,
my problem seems quite simple: I have a set $\Gamma$ along with a nice $\sigma$-algebra $\mathscr{B}$. Then I have a vector space of bounded measurable functions $A \subset \mathscr{B}_{\...
3
votes
0
answers
113
views
Stationarity of Brownian motion with drift
Suppose the following SDE for $X_t$ is well-posed:
$$dX_t = \sqrt{2}\, dB_t - \nabla\Phi(X_t)\,dt.$$
For what $\Phi\in C^1(R^d)$ will $X$ have stationary distribution $u_{\infty}$? For what $\Phi$ ...
- 165
7
votes
0
answers
293
views
Complex interpolation of a Banach space and its antidual when the space has a basis
Given a complex Banach space $X$ and its antidual $\hat X^*$, it is possible in some cases to apply the complex interpolation method, and get as $(X,\hat X^*)_{1/2}$ a Hilbert space. See [F. Watbled, ...
- 4,461
0
votes
1
answer
426
views
Lower bounds for partial sums of multiplicative functions
The motivation for this enquiry is to understand something about the impact of multiplicativity for $f:\mathbb{N}\rightarrow\mathbb{C}$ on the conditional convergence of Dirichlet series
$$F(s)=\...
- 2,480
3
votes
0
answers
74
views
Trace space of $\{ t^su \in L^2(0,\infty;X) \mid t^su_t \in L^2(0,\infty;Y)\}$ for $s \in (-\frac 12, \frac 12)$
Let $s \in (-\frac 12,\frac 12)$ and let $X=D(\Lambda)$ be a Hilbert space with $\Lambda$ the infinitesimal generator of a bounded semigroup of class $C^0$ in $Y$ (which is another Hilbert space), and ...
- 251
1
vote
1
answer
908
views
About the exponential bounds for modified Bessel function
Dear colleagues,
I want to know if there are some results on the bounds of modified Bessel functions $I_\alpha(x)$ and $K_\alpha(x)$? Especially, I need the exponential bounds for them, that is to ...
- 35
1
vote
0
answers
147
views
Bounding Rayleigh quotient for stochastic matrix
Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...
- 121