投影内函数动力学与迈耶 - 布拉施克型晶体测度
Projected Inner-Function Dynamics and Crystalline Measures of Meyer-Blaschke type
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中文总结 AI 辅助
研究从单位圆递归出发，通过投影傅里叶系数阵列构造晶体测度，确定模型空间作用，表明全纯时指数衰减可获相关测度，有限布拉施克乘积能给出相关多项式及恒等式，推广迈耶构造。
中文摘要 AI 辅助
我们推广了伊夫·迈耶（Yves Meyer）从布拉施克因子幂次产生的稀疏晶体测度的构造。从单位圆上的递归$f_n = \theta^n f_0$开始，其中$\theta$是内函数，我们通过将傅里叶系数阵列$\widehat{f_n}(k)$的元素放置在频率$k + \alpha n$处将其投影到实轴上。我们确定了模型空间在此构造中的作用：在迈耶的单因子布拉施克递归中，当$kn < 0$时系数阵列$\widehat{f_n}(k)$消失的要求等同于$f_0\in K_{zb_\lambda}$，对于一般内函数条件$f_0\in K_{z\theta}$产生具有局部有限支撑和傅里叶侧多项式增长的纯原子拉东测度。我们还表明，当$f_0$在包含单位圆的环域内全纯时，指数傅里叶衰减足以获得具有多项式增长的纯原子拉东测度，尽管不一定具有局部有限支撑。对于有限布拉施克乘积，系数递归给出一个明确的湮灭指数多项式，其零点集控制逆傅里叶变换的支撑和分离。这产生了迈耶 - 布拉施克型晶体测度和具有采样及有限截断结果的泊松恒等式。
英文摘要
We generalize a construction of Yves Meyer of sparse crystalline measures arising from powers of a Blaschke factor. Starting from a recursion $f_n=θ^n f_0$ on the unit circle where $θ$ is an inner function, we project the Fourier coefficient array $\widehat{f_n}(k)$ to the real line by placing its entries at the frequencies $k+αn$. We identify the role of model spaces in this construction: in Meyer's one-factor Blaschke recursion, the requirement that the coefficient array $\widehat{f_n}(k)$ vanish whenever $kn<0$ is equivalent to $f_0\in K_{zb_λ}$, and for general inner functions the condition $f_0\in K_{zθ}$ yields a purely atomic Radon measure with locally finite support and polynomial growth on the Fourier side. We also show that, when $f_0$ is holomorphic in an annulus containing the unit circle, exponential Fourier decay is sufficient to obtain a purely atomic Radon measure of polynomial growth, though not necessarily locally finite support. For finite Blaschke products, the coefficient recursion gives an explicit annihilating exponential polynomial whose zero set controls the support and separation of the inverse Fourier transform. This yields Meyer-Blaschke-type crystalline measures and Poisson identities with sampling and finite-truncation consequences.
