All Questions
3,841 questions with no upvoted or accepted answers
4
votes
0
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197
views
Bailey's lemma in number theory
A pair of sequences $(α_n,β_n)$ is called a Bailey pair if they are related by
$$\beta_n=\sum_{r=0}^n\frac{\alpha_r}{(q;q)_{n-r}(aq;q)_{n+r}}$$
or equivalently
$$\alpha_n = (1-aq^{2n})\sum_{j=0}^n\...
- 41
4
votes
0
answers
70
views
Local energy estimate in a semiclassical regime
Let us consider $h_n=(2n+1)^{-1/2}\to 0$ as $n\to \infty$ be a small parameter, which we just write as $h$ for convenience, and $u_h : \mathbb{R} \to \mathbb{R}$ be functions satisfying $Pu_h=0$ (I ...
- 489
4
votes
0
answers
481
views
Generalized Jensen's inequality for positively homogeneous functions
The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...
- 519
4
votes
0
answers
246
views
A question regarding the Hahn-Banach theorem and Banach limits
Set theorists typically prove the existence of Banach limits (EBL) using the Ultrafilter Theorem or, its equivalent, the Boolean Prime Ideal Theorem (BPI). Analysts, on the other hand, typically prove ...
- 6,453
4
votes
0
answers
256
views
Finding minimum of function using its Fourier transform
Say we have a function $f:\mathbb{R}^n \to \mathbb{R}$ such that
$$
f(x) = \int_{\mathbb{R}^n} \psi(k) e^{ik \cdot x} dk
$$
where $\psi$ is the Fourier dual of $f$. Say we further know that $\psi >...
- 595
4
votes
0
answers
622
views
Simple constructive proof for the hyperplane separating theorem (HST)?
The Hyperplane Separation Theorem (HST) is usually proved through the existence of a unique minimum-norm vector in a nonempty closed convex set. I think this is an existential proof which applies to ...
- 599
4
votes
0
answers
161
views
Hodge theory in higher eigen-spaces?
Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology
$$\mathcal{H}(E) \simeq H(E).$$
A classical example with differential forms ($E = (\Omega,d)$) ...
- 5,230
4
votes
0
answers
164
views
Convergence rates for kernel empirical risk minimization, i.e empirical risk minimization (ERM) with kernel density estimation (KDE)
Let $\Theta$ be an open subset of some $\mathbb R^m$ and let $P$ be a probability distribution on $\mathbb R^d$ with density $f$ in a Sobolev space $W_p^s(\mathbb R^d)$, i.e all derivatives of $f$ ...
- 6,853
4
votes
0
answers
81
views
Does this sequence of functions converge in a distributional sense?
Let $f\in W^{1,12/5}(\mathbb{R}^3)$ (time-independent), let $K^{\epsilon}$ be a uniformly in $\epsilon$ bounded sequence in $L^{1}\cap L^{7/5}(\mathbb{R}^3)$ and let
$$\tilde{K}^{\epsilon} := K^{\...
- 469
4
votes
0
answers
120
views
Are fibers in the corona of $H^\infty$ separable?
Let $\mathcal{M}(H^\infty(\mathbb{D}))$ denote the spectrum of the Banach algebra $H^\infty$ and $\mathcal{M}_z(H^\infty(\mathbb{D}))$ the fiber over $z\in \mathbb{D}$, i.e. $\{\varphi\in \mathcal{M}:...
- 81
4
votes
0
answers
188
views
Branch cuts, inverse Fourier transform and large time asymptotics
Let the Fourier transform of $f(t)$ be defined as $F(\omega) = \int_{-\infty}^\infty dt f(t) e^{i\omega t}$ for values of $\omega$ where the integral exists. What are the precise conditions on $F(\...
- 1,150
4
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311
views
Estimates of the Frobenius norm of commutator
Let $A,B$ be two unitary matrices in $U(n)$, and $\|\cdot\|_{F}$ denote the Frobenius norm (or Hilbert Schmidt norm on the finite dimensional $M_n(\mathbb{C})$). I am looking for estimates of the ...
- 949
4
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answers
212
views
"Cyclic vector" of sequence of operators
I recently encountered the following somewhat random-looking problem in my research. At first I thought that should not be too hard, but now, the more I think about it, the more interesting it seems.
...
- 9,853
4
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answers
116
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$\mathcal{C}^1(\overline{\Omega})$ gradient bounds for the Dirichlet problem of the heat equation on general domains
I am studying the heat equation on a general bounded domain $\Omega \subset \mathbb{R}^+ \times \mathbb{R}^n$ with continuously differentiable Dirichlet data $\phi$ on the boundary,
$$
\left\{
\begin{...
- 213
4
votes
0
answers
160
views
An estimate for the Benjamin-Ono equation from T. Tao's well-posedness paper
In https://arxiv.org/abs/math/0307289 (eq. (8)),
for a (smooth) solution of the equation $$u_t - uu_x + Hu_{xx} = 0$$
(where $H$ denotes the Hilbert transform) the following estimate is stated (...
- 839
4
votes
0
answers
193
views
Min max of a quadratic form of plus-minus ones
Does the following limit exist?
$$
\lim_{n \to \infty}\, n^{-3/2} \min_{a_{ij}=\pm 1}\max_{x_{j}=\pm 1}\left|\sum_{1\leq i <j \leq n} a_{ij}x_{i}x_{j} \right|
$$
There is no any significant ...
- 3,946
4
votes
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answers
132
views
$L_1$-subspace of the predual of a von Neumann algebra
If $M$ is a type $II$ von Neumann algebra, then the predual has a complemented subspace isometric to $L_1(0,1)$. It follows from the existence of expectation. However, I don't know whether such a ...
- 617
4
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0
answers
184
views
Weak* HI Banach spaces
The following question is inspired by Bill's nice unpublished result that the dual of a non separable Banach space is decomposable. (See the previous posts Decomposable Banach Spaces, Hereditarily ...
- 986
4
votes
0
answers
83
views
Crossed products of A by ℤ: non-stably isomorphic examples
What are some good sources of examples (and/or the simplest example) for:
Pairs of automorphisms $\phi,\psi:A\to A$ over the same base $C^*$-algebra $A$
with non-stably isomorphic crossed products, i....
- 598
4
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0
answers
127
views
Can the injective envelope ever be injective for $*$-homomorphisms?
The answers to the question "Is the injective envelope functorial" resoundingly remind us that the injective envelope of a C$^*$-algebra really belongs in the category of completely positive ...
- 3,984
4
votes
0
answers
255
views
Pointwise convergence of kernels of Hilbert-Schmidt operators
Lately I was discussing different types of convergence for Hilbert-Schmidt operators and during that discussion we ended up talking about pointwise convergence of Fourier series. I have already asked ...
- 1,045
4
votes
0
answers
165
views
Tensor product of representations on a compact quantum group
Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz) with function algebra $(C(\mathbb{G}), \Delta)$.
Let $X \in M(B_0(H)\otimes C(\mathbb{G}))$ and $Y \in M(B_0(K)\...
- 175
4
votes
0
answers
174
views
Techniques for showing non-degeneracy results (PDE)
Motivation:
Consider the equation,
$$-\Delta u = u^p$$
in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...
- 537
4
votes
0
answers
104
views
Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$
Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials?
In this post, $\phi$ is said to be polyharmonic in $\mathbb{...
- 41
4
votes
0
answers
656
views
Eigenvalues of Matérn covariance function
Recall that Matérn covariance function $C_\nu(d)$ is defined as
$$
C_\nu(d)=\sigma^2\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{\rho}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{\rho}\right), ...
- 397
4
votes
0
answers
116
views
Improving log-Sobolev inequalities via quadratic regularisation
Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$.
For suitable functions $g \geqslant 0$, define
$$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{...
- 801
4
votes
0
answers
382
views
Reference Aubin T.- Espaces de Sobolev sur les varietes Riemanniennes. Bull. Sc. Math. 100, (1976) 149-173
I have been searching without success for the reference:
Aubin T.- Espaces de Sobolev sur les varietes Riemanniennes. Bull. Sc. Math. 100, (1976) 149-173
It is cited in many related works. In ...
- 201
4
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0
answers
201
views
For what $C^*$ algebras $A$ do different types of projection equivalence agree?
For example,
For what $C^*$ algebras $A$ is unitary equivalence the same as mvn equivalence for projections.
For what $C^*$ algebras $A$ is unitary equivalence the same as homotopy equivalence for ...
- 183
4
votes
0
answers
263
views
Is there a notion of „flatness” in point-set topology?
In algebraic geometry, flat morphisms are usually associated with the intuition that they have „continuously varying fibers”. Is there a notion in topology formalizing the same intuition? Consider for ...
- 1,153
4
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0
answers
194
views
$L^\infty$ solutions for parabolic Neumann problem (heat equation)
Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs:
$$u_t - \Delta u = f$$
$$\partial_\nu u = 0$$
$$u|_{t=0} = u_0$$
where $f \in L^p(0,T;L^r(\Omega))$ and $...
- 307
4
votes
0
answers
127
views
Algebra properties regarding Gevrey spaces: closed under multiplication
In page 24 of the paper Landau Damping: Paraproducts and Gevrey Regularity, the authors claimed an algebra property of Gevrey spaces, the formula (3.14), without giving a proof. So I'm asking for a ...
- 517
4
votes
0
answers
115
views
Delta distributions that are smooth on strata of a singular manifold
This is a mild reformulation of a previous question. Let $R = C^\infty(\mathbb{R}^N)$ and let $I$ be an ideal in $R$ which cuts out an $n$-dimensional "singular $C^\infty$ manifold $X$" in $\...
- 7,952
4
votes
0
answers
194
views
What are the "local degrees of freedom" in the space of smooth functions?
Let $C^k$ be the set of $k$th-order smooth real functions $f:\mathbb{R}\to\mathbb{R}$, and $C^\infty$ the set of smooth real functions. One can specify an $f\in C^k$ by specifying all its derivatives ...
- 49
4
votes
0
answers
197
views
Approximation of a holomorphic function vanishing at a submanifold by polynomials
Suppose $M$ is a complex affine algebraic manifold in ${\mathbb C}^n$ (I mean, a set of common zeroes of a system of polynomials on ${\mathbb C}^n$, which is at the same time a smooth manifold). ...
- 7,426
4
votes
0
answers
109
views
Is the heat semigroup on a manifold the limit of the heat semigroups associated to a compact exhaustion?
Let $M$ be a paracompact Riemannian manifold, and $E \to M$ a Hermitian vector bundle endowed with a Hermitian connection $\nabla$. Write $M$ as an exhaustion $\bigcup _{j \ge 0} U_j$ with relatively ...
- 5,407
4
votes
0
answers
210
views
A more general version of the Fejér-Riesz theorem
A classical result, known as the Fejér-Riesz theorem, states that any Laurent polynomial $p(z)=\sum_{|k|\leq N} c_kz^k$
(the coefficients $c_k$ are complex numbers) which is nonnegative on the torus $\...
4
votes
0
answers
111
views
What is the native Hilbert space associated with the kernel $\frac{\sum \min{(x_i,y_i)}}{\sum \max{(x_i,y_i)}}$?
In this answer on MSE it is shown that the function
$$ K:(\mathbb{R}^{>0})^n\times (\mathbb{R}^{>0})^n\rightarrow\mathbb{R}\,\quad K(x,y)=\frac{\sum_{i=1}^n\min{(x_i,y_i)}}{\sum_{i=1}^n\max{(x_i,...
- 316
4
votes
0
answers
202
views
Double commutant of compact operators
So my question is straightforward. Let $\mathfrak{X}$ be a (complex, if necessary) Banach space and $K\colon\mathfrak{X}\to\mathfrak{X}$ a nonzero compact operator. Denote by $\mathcal{C}(K)$ the ...
- 1,453
4
votes
0
answers
104
views
Which linear forms are linear combinations of point evaluations?
Let $f_1,\ldots,f_r\in\mathbb{C}[x,y]$ and consider the subalgebras $A_1,\ldots,A_r$ of $\mathbb{C}[x,y]$ that are generated by $f_1,\ldots,f_r$, i.e., $A_i=\mathbb{C}[f_i]$. Using some dimension ...
- 3,031
4
votes
0
answers
149
views
Cyclic vectors for the translation operator
Let $b\in \mathbb{R}\neq 0$, and consider the translation operators:
$$
\begin{align}
T_b:C(\mathbb{R}) & \rightarrow C(\mathbb{R})\\
f &\mapsto f(\cdot + b).
\end{align}
$$
*Are there known ...
- 5,405
4
votes
0
answers
143
views
If theorem valid for compactly supported distribution then is it also valid for tempered distribution?
I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution.
For instance,
Theorem: Any $A \in \Psi^{m}$ ...
- 309
4
votes
0
answers
145
views
Hamel basis with all coordinate functionals discontinuous
If $X$ is an infinite dimensional (separable) Banach space, can one find a Hamel basis $(x_\alpha)_{\alpha\in\Lambda}$ such that all coordinate functionals $x_\alpha^*(x_\beta)=\delta_{\alpha,\beta}$ ...
- 1,361
4
votes
0
answers
135
views
Zygmund class, Schwartz class and Littlewood-Paley projection operators
I'm studying Littlewood-Paley theory in harmonic analysis, where I encountered the following problem related to the Zygmund class of functions:
Consider the Zygmund class of functions defined as ...
- 349
4
votes
0
answers
146
views
Fourier transform without characters (Eigenfunctions of an operator)
Let's consider a very simple problem in quantum mechanics:
We have, in $\mathbb R,$ a potential barrier of the form
$$
V(x) = V_0 \mathbf 1_{[-a,a]}(x),
$$
where $\mathbf 1_{[-a,a]}$ denotes the ...
- 1,755
4
votes
0
answers
310
views
Sobolev spaces and spectral theorem
Consider a generalised harmonic oscillator of the form
$$
A(x,D_x) = \langle x \rangle (1-\Delta_x)\langle x \rangle, \quad x \in \mathbb{R}^n,
$$
where $\langle x \rangle = (1+|x|^2)^{1/2}.$ The ...
4
votes
0
answers
123
views
Restricting a function defined on an étale groupoid to an isotropy group
Let $\mathcal G$ be an étale groupoid, let $x$ be a point in the unit space of $\mathcal G$, and let $\mathcal G(x)$
be the isotropy group of $x$.
If $f$ is a continuous, complex valued, compactly ...
- 2,263
4
votes
0
answers
548
views
Understanding vector-valued analytic functions vs holomorphic functional calculus
Let $A$ be a unital Banach algebra over complex numbers and call elements of $A$ "vectors". Let $\Omega$ be an open set in $\mathbb{C}$ and $H(\Omega)$ the space of analytic functions on $\...
- 83
4
votes
0
answers
140
views
A convex function is "usually" subdifferentiable
Let $X$ be a locally convex topological vector space, and let $f:X\to\mathbb R\cup\{\infty\}$ be a proper, convex, lower semicontinuous function, whose effective domain $D:=f^{-1}(\mathbb R)$ is ...
- 537
4
votes
0
answers
231
views
Spectral theorem for unbounded operators
Part of the Spectral theorem for unbounded operators states that if $A$ is a self adjoint unbounded operator and $B$ is a bounded operator such that $BA$ is contained in $AB$, then $B$ commutes with ...
- 771
4
votes
0
answers
95
views
When the Jacobian of unstable measure converges
Let $T:X \to X$ be a hyperbolic map on the compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product ...
- 1,043