little $q$-Legendre 多项式的调和分析
Harmonic analysis of little $q$-Legendre polynomials
AI总结：
本文研究 little $q$-Legendre 多项式的调和分析，利用 Fourier 变换与连分数等工具证明其 $L^1$-代数具有弱顺从性与右特征顺从性但不顺从，提供了首个此类多项式超群实例。
摘要：
许多正交多项式类满足特定的线性化性质，从而产生多项式超群结构，这为 Fourier 分析、调和分析和泛函分析提供了优雅且富有成果的联系。从相反的观点来看，这允许将某些 Banach 代数视为与底层正交多项式相关联的 $L^1$-代数。个体行为强烈依赖于这些底层多项式。我们研究 little $q$-Legendre 多项式，它们关于离散测度正交。我们将证明它们的 $L^1$-代数具有这样的性质：每个元素都可以用幂等元的线性组合来逼近。这特别意味着这些 $L^1$-代数是弱顺从的（即，到对偶模的每个有界导子都是内导子），已知任何局部紧群的 $L^1$-代数都共享这一性质；在多项式超群背景下，弱顺从性很少被满足且特别令人感兴趣，因为它对应于底层多项式导数及其关于多项式基的（Fourier）展开的某种性质。据我们所知，little $q$-Legendre 多项式提供了第一个这样的多项式超群例子：其 $L^1$-代数是弱顺从且右特征顺从的，但不是顺从的。作为关键工具，我们建立了特征的某些一致有界性质。我们的策略依赖于超群上的 Fourier 变换、Plancherel 同构、连分数、特征估计和渐近行为。
英文摘要：
Many classes of orthogonal polynomials satisfy a specific linearization property giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to Fourier analysis, harmonic analysis and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as $L^1$-algebras, associated with underlying orthogonal polynomials. The individual behavior strongly depends on these underlying polynomials. We study the little $q$-Legendre polynomials, which are orthogonal with respect to a discrete measure. We will show that their $L^1$-algebras have the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these $L^1$-algebras are weakly amenable (i. e., every bounded derivation into the dual module is an inner derivation), which is known to be shared by any $L^1$-algebra of a locally compact group; in the polynomial hypergroup context, weak amenability is rarely satisfied and of particular interest because it corresponds to a certain property of the derivatives of the underlying polynomials and their (Fourier) expansions w.r.t. the polynomial basis. To our knowledge, the little $q$-Legendre polynomials yield the first example of a polynomial hypergroup whose $L^1$-algebra is weakly amenable and right character amenable but fails to be amenable. As a crucial tool, we establish certain uniform boundedness properties of the characters. Our strategy relies on the Fourier transformation on hypergroups, the Plancherel isomorphism, continued fractions, character estimations and asymptotic behavior.
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