论文标题
二元空间与多项式高组的HAAR度量
Dual spaces vs. Haar measures of polynomial hypergroups
论文作者
论文摘要
许多对称的正交多项式$（p_n（x））_ {n \ in \ mathbb {n} _0} $在$ \ mathbb {n} _0 $上诱导超组结构。 HAAR度量是用$ h（n）加权的计数度量：= 1/\ int_ \ mathbb {r} \！p_n^2（x）\，\ mathrm {d}μ（x）\ geq1 $，其中$ $ $ $ $表示正元化度量。我们观察到，许多自然发生的示例满足了杰出的属性$ h（n）\ geq2 \;（n \ in \ mathbb {n}）$。我们提供了足够的标准，特别表明$ h（n）\ geq2 \;（n \ in \ mathbb {n}）$如果（Hermitian）双空间$ \ wideHat {\ mathbb {n} _0} $ que equals acexpess acexpess abledance abledance abundance abundance abundance。我们还研究了产物非负线化的作用（以及此类扩展所产生的谐波和功能分析的作用）。此外，我们用$ h（1）<2 $构造了两种示例类型。据我们所知，这些就是第一个这样的例子。第一种类型基于karlin-mcgregor多项式，$ \ wideHat {\ mathbb {n} _0} $由两个间隔组成，可以在某种意义上选择“最大值”； $ h $是二次增长。第二种类型依赖于某些紧凑的操作员； $ h $呈指数增长，$ \ wideHat {\ mathbb {n} _0} $是离散的。
Many symmetric orthogonal polynomials $(P_n(x))_{n\in\mathbb{N}_0}$ induce a hypergroup structure on $\mathbb{N}_0$. The Haar measure is the counting measure weighted with $h(n):=1/\int_\mathbb{R}\!P_n^2(x)\,\mathrm{d}μ(x)\geq1$, where $μ$ denotes the orthogonalization measure. We observed that many naturally occurring examples satisfy the remarkable property $h(n)\geq2\;(n\in\mathbb{N})$. We give sufficient criteria and particularly show that $h(n)\geq2\;(n\in\mathbb{N})$ if the (Hermitian) dual space $\widehat{\mathbb{N}_0}$ equals the full interval $[-1,1]$, which is fulfilled by an abundance of examples. We also study the role of nonnegative linearization of products (and of the harmonic and functional analysis resulting from such expansions). Moreover, we construct two example types with $h(1)<2$. To our knowledge, these are the first such examples. The first type is based on Karlin-McGregor polynomials, and $\widehat{\mathbb{N}_0}$ consists of two intervals and can be chosen "maximal" in some sense; $h$ is of quadratic growth. The second type relies on certain compact operators; $h$ grows exponentially, and $\widehat{\mathbb{N}_0}$ is discrete.
