Hölder-logarithmic stability in Fourier synthesis

Logarithmic Fourier decay for self conformal measures

We prove that the Fourier transform of a self conformal measure on $\mathbb{R}$ decays to $0$ at infinity at a logarithmic rate, unless the following holds: The underlying IFS is smoothly conjugated to an IFS that both acts linearly on its attractor and contracts by scales that are not Diophantine. Our key technical result is an effective version of a local limit Theorem for cocycles with moderate deviations due to Benoist-Quint (2016), that is of independent interest.

矩形傅里叶级数的Zhizhiashvili定理的临界扩展

A Critical-Scale Extension of Zhizhiashvili's Theorem for Rectangular Fourier Series

针对Zhizhiashvili对数模定理的端点问题，通过可和端点Dini条件替代严格幂次，证明了矩形部分和几乎处处Pringsheim收敛的端点Dini准则，并得到双对数端点类。

矩形傅里叶和的一个Oskolkov--Zhizhiashvili准则

An Oskolkov--Zhizhiashvili Criterion for Rectangular Fourier Sums

针对d维环面上三角傅里叶级数的对称矩形部分和，在Zhizhiashvili临界尺度下证明了可和端点准则，改进了经典充分条件。

支撑在良逼近数上的测度的尖锐傅里叶衰减估计

Sharp Fourier decay estimates for measures supported on the well-approximable numbers

本文在良逼近数上构造了傅里叶变换近乎最优衰减的测度，从而对Kaufman的先前构造实现了对数级别的改进。

Lacunary Fourier and Walsh-Fourier series near L^1

We prove that, for functions in the Orlicz class LloglogLloglogloglogL, lacunary subsequences of the Fourier and the Walsh-Fourier series converge almost everywhere. Our integrability condition is less stringent than the homologous assumption in the almost everywhere convergence theorems of Lie (Fourier case) and Do-Lacey (Walsh-Fourier case), where the quadruple logarithmic term is replaced by a triple logarithm. Our proof of the Walsh-Fourier case is self-contained and, in antithesis to Do and Lacey's argument, avoids the use of Antonov's lemma, arguing directly via novel weak-L^p bounds for…

关于带变号伪微分算子的发展方程的存在性与唯一性结果及其在对数 Laplace 算子和二阶微分算子（无椭圆性）中的应用

An existence and uniqueness result to evolution equations with sign-changing pseudo-differential operators and its applications to logarithmic Laplacian operators and second-order differential operators without ellipticity

本文通过扩展 Fourier 变换定义域构建新弱形式，解决了带变号伪微分算子的发展方程的存在与唯一性问题，并应用于对数 Laplace 算子和无椭圆性的二阶微分算子。

Stability estimates for reconstruction from the Fourier transform on the ball

We prove Hölder-logarithmic stability estimates for the problem of finding an integrable function $v$ on $\mathbb{R}^d$ with a super-exponential decay at infinity from its Fourier transform $\mathcal{F} v$ given on the ball $B_r$. These estimates arise from a Hölder-stable extrapolation of $\mathcal{F} v$ from $B_r$ to a larger ball. We also present instability examples showing an optimality of our results.

Pitt's inequalities and uncertainty principle for generalized Fourier transform

We study the two-parameter family of unitary operators \[ \mathcal{F}_{k,a}=\exp\Bigl(\frac{iπ}{2a}\,(2\langle k\rangle+{d}+a-2 )\Bigr) \exp\Bigl(\frac{iπ}{2a}\,Δ_{k,a}\Bigr), \] which are called $(k,a)$-generalized Fourier transforms and defined by the $a$-deformed Dunkl harmonic oscillator $Δ_{k,a}=|x|^{2-a}Δ_{k}-|x|^{a}$, $a>0$, where $Δ_{k}$ is the Dunkl Laplacian. Particular cases of such operators are the Fourier and Dunkl transforms. The restriction of $\mathcal{F}_{k,a}$ to radial functions is given by the $a$-deformed Hankel transform $H_{λ,a}$. We obtain necessary and sufficient cond…

离散傅里叶集中算子的迹亏界

Trace-defect bounds for discrete Fourier concentration operators

本文研究离散傅里叶集中算子的迹亏，通过平移半范数给出其上下界，并应用于特征值计数与离散 Landau 渐近的恢复。