Dirichlet级数的Hardy空间中的均值、阶与收敛性

中文摘要 AI 辅助

设$\mathscr{H}^2$表示系数平方可和的Dirichlet级数构成的Hilbert空间。由Bohr定理可知，若$\mathscr{H}^2$中的函数$f$有到右半平面的有界解析延拓，则其范数可由均值\\[\\|f\\|_{\mathscr{H}^2}^2 = \lim_{\sigma\to 0^+} \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T \lvert f(\sigma+it) \rvert^2\\,dt\\]计算。我们研究解析延拓有界这一假设可在多大程度上放宽，并在两个不同方向上获得最优结果。第一个方向涉及解析延拓的阶，并作为推论得到$\mathscr{H}^2$中经典Landau--Schnee收敛定理的一个版本。第二个方向涉及收敛横坐标，解决了McCarthy提出的一个问题。

英文摘要

Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. It follows from a theorem of Bohr that if $f$ in $\mathscr{H}^2$ has a bounded analytic continuation to the right half-plane, then its norm can be computed from the mean values \[\|f\|_{\mathscr{H}^2}^2 = \lim_{σ\to 0^+} \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T \lvert f(σ+it) \rvert^2\,dt.\] We investigate to what extent the assumption that the analytic continuation be bounded can be relaxed, and obtain optimal results in two different directions. The first direction concerns the order of the analytic continuation, and yields as a corollary a version of the classical Landau--Schnee convergence theorem for $\mathscr{H}^2$. The second direction concerns the abscissa of convergence, and resolves a problem posed by McCarthy.

发表机构

• Norwegian University of Science and Technology（挪威科技大学）
• KU Leuven（荷语鲁汶大学）

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