Sampling in a weighted Sobolev space

We show that functions f in some weighted Sobolev space are completely determined by time-frequency samples ${\{f(t_n)\}}_{n\in \mathbb{Z}}\cup {\{\stackrel{?}{f}({\lambda }_k)\}}_{k\in \mathbb{Z}}$ along appropriate slowly increasing sequences ${\{t_n\}}_{n\in \mathbb{Z}}$ and ${\{{\lambda }_n\}}_{n\in \mathbb{Z}}$ tending to ±∞ as $n\to \pm \infty$.     

Approximation of maximum of Gaussian random fields

This contribution is concerned with Gumbel limiting results for supremum ${\mathbb{M}}_{\mathit{n}}={\sup }_{\mathit{t}\in [\mathbf{0},{\mathit{T}}_{\mathit{n}}]}?|X_{\mathit{n}}(\mathit{t})|$ with $X_{\mathit{n}},\mathit{n}\in {\mathbb{N}}^2$ centered Gaussian random fields with continuous trajectories. We show first the convergence of a related point process to a Poisson point process thereby extending previous results obtained in [8] for Gaussian processes. Furthermore, we derive Gumbel limit results for ${\mathbb{M}}_{\mathit{n}}$ as $\mathit{n}\to \infty$ and show a second-order approximation for $\mathbb{E}{\{{\mathbb{M}}_{\mathit{n}}^p\}}^{1/p}$ for any $p\ge 1$.     

Superconcentration,and randomized Dvoretzky's theorem for spaces with 1-unconditional bases

Let n be a sufficiently large natural number and let B be an origin-symmetric convex body in ${\mathbb{R}}^n$in the ?-position, and such that the space $({\mathbb{R}}^n,{6?6}_B)$ admits a 1-unconditional basis. Then for any $\epsilon \in (0,1/2]$, and for random $c\epsilon \log ?n/\log ?\frac{1}{\epsilon }$-dimensional subspace E distributed according to the rotation-invariant (Haar) measure, the section $B\cap E$ is $(1+\epsilon )$-Euclidean with probability close to one. This shows that the “worst-case” dependence on ε in the randomized Dvoretzky theorem in the ?-position is significantly better than in John's position. It is a previously unexplored feature, which has strong connections with the concept of superconcentration introduced by S. Chatterjee. In fact, our main result follows from the next theorem: Let B be as before and assume additionally that B has a smooth boundary and ${\mathbb{E}}_{{\gamma }_n}{6?6}_B\le n^c{\mathbb{E}}_{{\gamma }_n}{6{\mathrm{grad}}_B(?)6}_2$ for a small universal constant $c>0$, where ${\mathrm{grad}}_B(?)$ is the gradient of ${6?6}_B$ and ${\gamma }_n$ is the standard Gaussian measure in ${\mathbb{R}}^n$. Then for any $p\in [1,c\log ?n]$ the p-th power of the norm ${6?6}_B^p$ is $\frac{C}{\log ?n}$-superconcentrated in the Gauss space.     

A note on the unique continuation property for Zakharov–Kuznetsov equation

We prove that if a sufficiently smooth solution to the initial value problem associated with the Zakharov–Kuznetsov equation$u_t+(u_{\mathit{xx}}+u_{\mathit{yy}}{)}_x+{\mathit{uu}}_x=0,(x,y)\in {\mathbb{R}}^2,t\in \mathbb{R}$is supported compactly in a nontrivial time interval then it vanishes identically.     

Gabor systems and almost periodic functions

Inspired by results of Kim and Ron, given a Gabor frame in $L^2(\mathbb{R})$, we determine a non-countable generalized frame for the non-separable space ${\mathit{AP}}_2(\mathbb{R})$ of the Besicovic almost periodic functions. Gabor type frames for suitable separable subspaces of ${\mathit{AP}}_2(\mathbb{R})$ are constructed. We show furthermore that Bessel-type estimates hold for the AP norm with respect to a countable Gabor system using suitable almost periodic norms of sequences.     

The mod 2 cohomology of the mapping class group of the Klein bottle with marked points

The purpose of this article is to compute the mod 2 cohomology of ${\mathrm{\Gamma }}^q(\mathbb{K})$, the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg–MacLane spaces $X_q=K({\mathrm{\Gamma }}^q(\mathbb{K}),1)$ and fiber bundles $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q\to X_q\to B({\mathbb{Z}}_2\times O(2))$, where $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q$ denotes the configuration space of unordered q-tuples of distinct points in $\mathbb{K}$ and $B({\mathbb{Z}}_2\times O(2))$ is the classifying space of the group ${\mathbb{Z}}_2\times O(2)$. Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses.