Almost-convergent sequences with respect to polynomial hypergroups. (English) Zbl 1299.40002
G. G. Lorentz [Acta Math., Uppsala 80, 167–190 (1948; Zbl 0031.29501)] gave a characterisation of sequences for which all Banach limits coincide (almost invariant sequences) as those sequences \((\alpha_n)\) for which the arithmetic means \(\frac 1{k+1}(\alpha_n+\ldots +\alpha_{n+k})\) converge uniformly in \(n\) to some value \(d\) as \(k\to\infty\). If \(\mathbb N_0\) is equipped with a certain polynomial hypergroup structure analogous results can be stated with respect to invariance of operators \(T_m\) defined via convolution instead of the shift operator. A sequence \((\alpha(n))_{n\in\mathbb N_0}\) is then called almost invariant if all \(T_m\)-invariant means of \((\alpha_n)\) coincide. This property is shown to be equivalent to the uniform convergence of averages built by these generalised shift operators to the \(T_m\)-invariant mean of \((\alpha_n)\).
Reviewer: Martin Bluemlinger (Wien)
MSC:
| 40A05 | Convergence and divergence of series and sequences |
| 43A62 | Harmonic analysis on hypergroups |
Citations:
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