Non-symmetric convex domains have no basis of exponentials

Maximal surface area of a convex set in $\mathbb{R}^n$ with respect to exponential rotation invariant measures

Let $p$ be a positive number. Consider probability measure $γ_p$ with density $φ_p(y)=c_{n,p}e^{-\frac{|y|^p}{p}}$. We show that the maximal surface area of a convex body in $\mathbb{R}^n$ with respect to $γ_p$ is asymptotically equal to $C_p n^{\frac{3}{4}-\frac{1}{p}}$, where constant $C_p$ depends on $p$ only. This is a generalization of Ball's and Nazarov's bounds, which were given for the case of the standard Gaussian measure $γ_2$.

Exponential convexifying of polynomials

Let $X\subset\mathbb{R}^n$ be a convex closed and semialgebraic set and let $f$ be a polynomial positive on $X$. We prove that there exists an exponent $N\geq 1$, such that for any $ξ\in\mathbb{R}^n$ the function $φ_N(x)=e^{N|x-ξ|^2}f(x)$ is strongly convex on $X$. When $X$ is unbounded we have to assume also that the leading form of $f$ is positive in $\mathbb{R}^n\setminus\{0\}$. We obtain strong convexity of $\varPhi_N(x)=e^{e^{N|x|^2}}f(x)$ on possibly unbounded $X$, provided $N$ is sufficiently large, assuming only that $f$ is positive on $X$. We apply these results for searching critical…

Distances sets that are a shift of the integers and Fourier basis for planar convex sets

The aim of this paper is to prove that if a planar set $A$ has a difference set $Δ(A)$ satisfying $Δ(A)\subset \Z^++s$ for suitable $s$ than $A$ has at most 3 elements. This result is motivated by the conjecture that the disk has not more than 3 orthogonal exponentials. Further, we prove that if $A$ is a set of exponentials mutually orthogonal with respect to any symmetric convex set $K$ in the plane with a smooth boundary and everywhere non-vanishing curvature, then $ # (A \cap {[-q,q]}^2) \leq C(K) q$ where $C(K)$ is a constant depending only on $K$. This extends and clarifies in the plane t…

Convex bodies with a point of curvature do not have Fourier bases

We prove that no smooth symmetric convex body $Ω$ with at least one point of non-vanishing Gaussian curvature can admit an orthogonal basis of exponentials. (The non-symmetric case was proven by Kolountzakis). This is further evidence of Fuglede's conjecture, which states that such a basis is possible if and only if $Ω$ can tile $R^d$ by translations.

频率位于指定格点的平行六面体的指数基

Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice

该研究针对频率限制在指定格点的情形，推导了平行六面体存在指数Riesz基的两个充分条件，分别推广了正交基情形的充要条件、通过生成矩阵谱范数给出约束。

Fuglede conjecture holds for convex planar domains

Let $Ω$ be a compact convex domain in the plane. We prove that $L^2(Ω)$ has an orthogonal basis of exponentials if and only if $Ω$ tiles the plane by translation.

Bases of complex exponentials with restricted supports

The complex exponentials with integer frequencies form a basis for the space of square integrable functions on the unit interval. We analyze whether the basis property is maintained if the support of the complex exponentials is restricted to possibly overlapping subsets of the unit interval. We show, for example, that if $S_1, \ldots, S_K \subset [0,1]$ are finite unions of intervals with rational endpoints that cover the unit interval, then there exists a partition of $\mathbb{Z}$ into sets $Λ_1, \ldots, Λ_K$ such that $\bigcup_{k=1}^K \{ e^{2πi λ(\cdot)} χ_{S_k} : λ\in Λ_k \}$ is a Riesz bas…

Exponential Riesz bases, discrepancy of irrational rotations and BMO

We study the basis property of systems of exponentials with frequencies belonging to 'simple quasicrystals'. We show that a diophantine condition is necessary and sufficient for such a system to be a Riesz basis in L^2 on a finite union of intervals. For the proof we extend to BMO a theorem of Kesten about the discrepancy of irrational rotations of the circle.

A sharpening of a problem on Bernstein polynomials and convex functions

We present an elementary proof of a conjecture proposed by I. Rasa in 2017 which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive by A. Komisarski and T. Rajba very recently by the use of stochastic convex orderings.

A problem of I. Raşa on Bernstein polynomials and convex functions

We present an elementary proof of a conjecture by I. Raşa which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive very recently by the use of stochastic convex orderings. Moreover, we derive the corresponding results for Mirakyan-Favard-Szász operators and Baskakov operators.

$L^2$上两个区间中的指数里斯基基

Exponential Riesz bases in $L^2$ on two interval

本文研究了两个区间上指数系统成为里斯基基的充分条件，并揭示了此类系统可能存在的一个额外点效应。

Non-symmetric Convex Polytope and Gabor Orthonormal Bases

In this paper, we show that the non-symmetric convex polytope cannot be served as a window function to produce a Gabor orthonormal basis by any time-frequency sets.

Finding duality for Riesz bases of exponentials on multi-tiles

It is known that if $Ω\subset \mathbb{R}^{d}$ belongs to a class of multi-tiling domains when translated by a lattice $Λ$, there exists a Riesz basis of exponentials for $L^{2}(Ω)$ constructed using $k$ translates of the dual lattice $Λ^*$. In this paper, we give an explicit construction of the corresponding biorthogonal dual Riesz basis. We also extend the iterative reconstruction algorithm introduced in prior work to this setting.

Riesz bases of exponentials and multi-tiling in finite abelian groups

Motivated by the open problem of exhibiting a subset of Euclidean space which has no exponential Riesz basis, we focus on exponential Riesz bases in finite abelian groups. We point out that that every subset of a finite abelian group has such a basis, removing interest in the existence question in this context. We then define tightness quantities for subsets to measure the conditioning of Riesz bases; for normalized tightness quantities, a value of one corresponds to an orthogonal basis, and a value of infinity corresponds to nonexistence of a basis. As an application, we obtain new weak evide…

$L^p$ 空间中指数函数与一致有界函数的无条件 Schauder 框架

Unconditional Schauder frames of exponentials and of uniformly bounded functions in $L^p$ spaces

本文研究 $L^p$ 空间中指数函数与一致有界函数的无条件 Schauder 框架，证明 Gaposhkin 定理在框架情形下失效，但非空内部集合中仍无指数无条件框架。