Wave packet analysis of Schrödinger equations in analytic function spaces

We consider a class of linear Schrödinger equations in R^d${\mathbb{R}}^d$, with analytic symbols. We prove a global-in-time integral representation for the corresponding propagator as a generalized Gabor multiplier with a window analytic and decaying exponentially at infinity, which is transported by the Hamiltonian flow. We then provide three applications of the above result: the exponential sparsity in phase space of the corresponding propagator with respect to Gabor wave packets, a wave packet characterization of Fourier integral operators with analytic phases and symbols, and the propagation of analytic singularities.     

Inequalities with exponential weights

Let R=(-∞,∞)$\mathbb{R}=(-\infty ,\infty )$ and let Q∈C²:R→R⁺=[0,∞)$Q\in C^2:\mathbb{R}\to {\mathbb{R}}^{+}=[0,\infty )$ be an even function. Then in this paper we consider the infinite–finite range inequality, an estimate for the Christoffel function, and the Markov–Bernstein inequality with the exponential weights _wρ(x)=|x|^ρe^-Q(x),x∈R$w_{\rho }(x)=|x{|}^{\rho }{\mathrm{e}}^{-Q(x)},x\in \mathbb{R}$.     

The Morse–Sard theorem for Clarke critical values

The Morse–Sard theorem states that the set of critical values of a C^k$C^k$ smooth function defined on a Euclidean space R^d${\mathbb{R}}^d$ has Lebesgue measure zero, provided k≥d$k\ge d$. This result is hereby extended for (generalized) critical values of continuous selections over a compactly indexed countable family of C^k$C^k$ functions: it is shown that these functions are Lipschitz continuous and the set of their Clarke critical values is null.     

Some results on monotone metric spaces

We present some new results on monotone metric spaces. We prove that every bounded 1-monotone metric space in R^d${\mathbb{R}}^d$ has a finite 1-dimensional Hausdorff measure. As a consequence we obtain that each continuous bounded curve in R^d${\mathbb{R}}^d$ has a finite length if and only if it can be written as a finite sum of 1-monotone continuous bounded curves. Next we construct a continuous function f such that M   has a zero Lebesgue measure provided the graph(f|M)$\mathrm{graph}(f|M)$ is a monotone set in the plane. We finally construct a differentiable function with a monotone graph and unbounded variation.     

Geometric construction of quintic parametric B-splines

The aim of this paper is to present a new class of B-spline-like functions with tension properties. The main feature of these basis functions consists in possessing C³$C^3$ or even C⁴$C^4$ continuity and, at the same time, being endowed by shape parameters that can be easily handled. Therefore they constitute a useful tool for the construction of curves satisfying some prescribed shape constraints. The construction is based on a geometric approach which uses parametric curves with piecewise quintic components.     

Convergence to Lévy stable processes under some weak dependence conditions

For a strictly stationary sequence of random vectors in R^d${\mathbb{R}}^d$ we study convergence of partial sum processes to a Lévy stable process in the Skorohod space with _J1$J_1$-topology. We identify necessary and sufficient conditions for such convergence and provide sufficient conditions when the stationary sequence is strongly mixing.     

New results for best approximation on Banach lattices

The purpose of this paper is to introduce and to discuss the concept of approximation preserving operators on Banach lattices with a strong unit. We show that every lattice isomorphism is an approximation preserving operator. Also we give a necessary and sufficient condition for uniqueness of the best approximation by closed normal subsets of X⁺$X^{+}$, and show that this condition is characterized by some special operators.     

On removable sets for convex functions

In the present article we provide a sufficient condition for a closed set F∈R^d$F\in {\mathbb{R}}^d$ to have the following property which we call c  -removability: Whenever a continuous function f:R^d→R$f:{\mathbb{R}}^d\to \mathbb{R}$ is locally convex on the complement of F  , it is convex on the whole R^d${\mathbb{R}}^d$. We also prove that no generalized rectangle of positive Lebesgue measure in R²${\mathbb{R}}^2$ is c-removable. Our results also answer the following question asked in an article by Jacek Tabor and Józef Tabor (2010) [5]: Assume the closed set F⊂R^d$F\subset {\mathbb{R}}^d$ is such that any locally convex function defined on R^d?F${\mathbb{R}}^d?F$ has a unique convex extension on R^d${\mathbb{R}}^d$. Is F   necessarily intervally thin (a notion of smallness of sets defined by their “essential transparency” in every direction)? We prove the answer is negative by finding a counterexample in R²${\mathbb{R}}^2$.     

On sums of real-valued functions with extremely thick graphs

We consider some properties of those functions acting from the real line R$\mathbf{R}$ into itself, whose graphs are extremely thick subsets of the Euclidean plane R²${\mathbf{R}}^2$. The structure of sums of such functions is studied and the obtained results are applied to certain measure extension problems.     

An invariance principle for stationary random fields under Hannan’s condition

We establish an invariance principle for a general class of stationary random fields indexed by Z^d${\mathbb{Z}}^d$, under Hannan’s condition generalized to Z^d${\mathbb{Z}}^d$. To do so we first establish a uniform integrability result for stationary orthomartingales, and second we establish a coboundary decomposition for certain stationary random fields. At last, we obtain an invariance principle by developing an orthomartingale approximation. Our invariance principle improves known results in the literature, and particularly we require only finite second moment.     

Linearly repetitive Delone sets are rectifiable

We show that every linearly repetitive Delone set in the Euclidean d  -space R^d${\mathbb{R}}^d$, with d?2$d?2$, is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice Z^d${\mathbb{Z}}^d$. In the particular case when the Delone set X   in R^d${\mathbb{R}}^d$ comes from a primitive substitution tiling of R^d${\mathbb{R}}^d$, we give a condition on the eigenvalues of the substitution matrix which ensures the existence of a homeomorphism with bounded displacement from X   to the lattice βZ^d$\beta {\mathbb{Z}}^d$ for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.     

Asymptotic profiles for wave equations with strong damping

We consider the Cauchy problem in Rⁿ${\mathbf{R}}^n$ for strongly damped wave equations. We derive asymptotic profiles of these solutions with weighted L^1,1(Rⁿ)$L^{1,1}({\mathbf{R}}^n)$ data by using a method introduced in [9] and/or [10].