论文标题
小$ q $ - legendre多项式的谐波分析
Harmonic analysis of little $q$-Legendre polynomials
论文作者
论文摘要
许多类别的正交多项式满足了特定的线性化特性，从而产生了多项式高组结构，该结构提供了与傅立叶分析，谐波分析和功能分析的优雅而富有成果的联系。从相反的角度来看，这允许某些Banach代数为$ l^1 $ - 代数，与基础正交多项式相关。个体行为在很大程度上取决于这些基本的多项式。我们研究了小$ q $ - legendre多项式，相对于离散度量是正交的。我们将证明他们的$ l^1 $ - 代数具有通过iDempotents的线性组合近似每个元素的属性。这特别意味着这些$ l^1 $ - 代数是弱的（即，向双模块中的每个有界推导都是内部派生的），这是由本地紧凑型组的任何$ l^1 $ algebra共享的；在多项式高组上下文中，很少满足弱的舒适性，并且特别感兴趣，因为它与基础多项式的衍生物及其（傅立叶）扩展W.R.T.相对应。多项式基础。据我们所知，Little $ Q $ - legendre多项式产生了一个多项式高组的第一个例子，其$ l^1 $ -Algebra是薄弱的，弱的且正确的角色，但无法修复。作为至关重要的工具，我们建立了字符的某些统一界限。我们的策略依赖于超级组的傅立叶变换，plancherel同构，持续分数，角色估计和渐近行为。
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.
