Rational hyperholomorphic functions in (\mathbb R^4)

An extension of Herglotz's theorem to the quaternions

A classical theorem of Herglotz states that a function $n\mapsto r(n)$ from $\mathbb Z$ into $\mathbb C^{s\times s}$ is positive definite if and only there exists a $\mathbb C^{s\times s}$-valued positive measure $dμ$ on $[0,2π]$ such that $r(n)=\int_0^{2π}e^{int}dμ(t)$for $n\in \mathbb Z$. We prove a quaternionic analogue of this result when the function is allowed to have a number of negative squares. A key tool in the argument is the theory of slice hyperholomorphic functions, and the representation of such functions which have a positive real part in the unit ball of the quaternions. We st…

Rational tetra-inner functions and the special variety of the tetrablock

The set \[ \overline{\mathbb{E}}= \{ x \in {\mathbb{C}}^3: \quad 1-x_1 z - x_2 w + x_3 zw \neq 0 \mbox{ whenever } |z| < 1, |w| < 1 \} \] is called the tetrablock and has intriguing complex-geometric properties. It is polynomially convex, nonconvex and starlike about $0$. It has a group of automorphisms parametrised by ${\mathrm{Aut}~} {\mathbb{D}} \times {\mathrm{Aut}~} {\mathbb{D}} \times {\mathbb{Z}}_2$ and its distinguished boundary $b\overline{\mathbb{E}}$ is homeomorphic to the solid torus $\overline{\mathbb{D}} \times {\mathbb{T}}$. It has a special subvariety \[\mathcal{R}_{\mathbb{\ov…

Asymptotic sharpness of a Bernstein-type inequality for rational functions in H^{2}

A Bernstein-type inequality in the standard Hardy space H^{2} of the unit disc \mathbb{D}=\{z\in\mathbb{C}:\,|z|<1\}, for rational functions in \mathbb{D} having at most n poles all outside of \frac{1}{r}\mathbb{D}, 0

Gabor frames for rational functions

We study the frame properties of the Gabor systems $$\mathfrak{G}(g;α,β):=\{e^{2πi βm x}g(x-αn)\}_{m,n\in\mathbb{Z}}.$$ In particular, we prove that for Herglotz windows $g$ such systems always form a frame for $L^2(\mathbb{R})$ if $α,β>0$, $αβ\leq1$. For general rational windows $g\in L^2(\mathbb{R})$ we prove that $\mathfrak{G}(g;α,β)$ is a frame for $L^2(\mathbb{R})$ if $0<α,β$, $αβ<1$, $αβ\not\in\mathbb{Q}$ and $\hat{g}(ξ)\neq0$, $ξ>0$, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspa…

逆Fueter定理的一个变体与二阶广义多解析Cauchy-Kovalevskaya延拓

A variation of the inverse Fueter theorem and the generalized polyanalytic Cauchy-Kovalevskaya extension of order 2

本文研究二阶轴向多解析函数的广义Cauchy-Kovalevskaya延拓，建立其与Fueter定理的联系，推导Fueter映射分解的可逆性及相关积分表示公式，是超复分析的重要成果。

有理函数的通用框架集

Universal frame set for rational functions

本文证明了对于任意ε>0和自然数M，存在一个上Beurling密度小于1+ε的通用集Λ，使得任何良好行为的有理函数g的框架系统在L²(R)中成立。

Time-frequency analysis on the adeles over the rationals

We show that the construction of Gabor frames in $L^{2}(\mathbb{R})$ with generators in $\mathbf{S}_{0}(\mathbb{R})$ and with respect to time-frequency shifts from a rectangular lattice $α\mathbb{Z}\timesβ\mathbb{Z}$ is equivalent to the construction of certain Gabor frames for $L^{2}$ over the adeles over the rationals and the group $\mathbb{R}\times\mathbb{Q}_{p}$. Furthermore, we detail the connection between the construction of Gabor frames on the adeles and on $\mathbb{R}\times\mathbb{Q}_{p}$ with the construction of certain Heisenberg modules.

非阿基米德有理映射的鲁厄ζ函数

Ruelle's zeta function for non-Archimedean rational maps

本文研究了非阿基米德有理映射的鲁厄ζ函数及其相关性质，证明了其解析性并探讨了朱利亚集的形状。