Harmonic Analysis and Applications

June 4-8, 2018


"Chain sequences, continued fractions and $L^1$-algebras associated with orthogonal polynomials"

Kahler, Stefan Alexander

A fruitful way to bring harmonic analysis into the theory of orthogonal polynomials---or, from the opposite point of view, to construct certain associated Banach algebras from orthogonal polynomials and special functions---is the concept of a polynomial hypergroup. It is based on a linearization property and accompanied by corresponding $L^1$-algebras, and many specific sequences of orthogonal polynomials such as classical examples like Chebyshev polynomials of the first and second kind or Legendre polynomials induce such polynomial hypergroups. Several concepts of harmonic analysis such as the Fourier and the Plancherel transformation are available and take a unified and rather concrete form; nevertheless, the behavior is strongly dependent on the underlying polynomials, which yields a great variety and an abundance of examples.\\ In this talk, we consider two very different classes. The associated symmetric Pollaczek polynomials are a two-parameter generalization of the ultraspherical polynomials and given by the recurrence $p_0(x)=1$, $p_1(x)=x$, \begin{equation*} x p_n(x)=p_{n+1}(x)+\frac{(n+\nu)(n+\nu+2\alpha)}{(2n+2\nu+2\alpha+2\lambda+1)(2n+2\nu+2\alpha+2\lambda-1)}p_{n-1}(x) \end{equation*} for suitable parameters $\alpha,\lambda,\nu$; in particular, they contain the classical examples mentioned above. While these polynomials are orthogonal with respect to an absolutely continuous measure, the second polynomials under consideration---the little $q$-Legendre polynomials---come with a purely discrete orthogonalization measure. We are interested in amenability properties of the corresponding $L^1$-algebras, which can be very different from locally compact groups: while in the group case $L^1$-algebras are known to be always weakly amenable (and to be amenable if and only if the underlying group is amenable), even the nonexistence of nonzero bounded point derivations is often not satisfied for polynomial hypergroups. We present full characterizations of (right character) amenability, weak amenability and the global nonexistence of nonzero bounded point derivations for the class of associated symmetric Pollaczek polynomials, which extends earlier results on the classes of symmetric Pollaczek polynomials (i. e., the special case $\nu=0$) and associated ultraspherical polynomials (i. e., the special case $\lambda=0$) presented at Strobl16. Moreover, we give a certain uniform boundedness result for the little $q$-Legendre polynomials and obtain that their $L^1$-algebras are spanned by their idempotents (which is a stronger property than weak amenability). Despite the aforementioned deep structural differences between the two classes under consideration, in both cases our strategy relies on the theory of continued fractions and chain sequences, and the talk will particularly focus on these parts of the argumentation.

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