Title: Ultradifferentiable classes of entire functions Show Chinese title

Title: 超可微整个函数类

Abstract: We study classes of ultradifferentiable functions defined in terms of small weight sequences violating standard growth and regularity requirements. First, we show that such classes can be viewed as weighted spaces of entire functions for which the crucial weight is given by the associated weight function of the so-called conjugate weight sequence. Moreover, we generalize results from M. Markin from the so-called small Gevrey-setting to arbitrary convenient families of (small) sequences and show how the corresponding ultradifferentiable function classes can be used to detect boundedness of normal linear operators on Hilbert spaces (associated to an evolution equation problem). Finally, we study the connection between small sequences and the recent notion of dual sequences introduced in the PhD-thesis of J. Jim\'{e}nez-Garrido. Show Chinese abstract

Abstract: 我们研究了一类以小权序列定义的超可微函数类，这些序列违反了标准的增长和正则性要求。 首先，我们证明这类函数类可以被视为整个函数的加权空间，其中关键权重由所谓的共轭权序列的相伴权函数给出。 此外，我们将M. Markin在所谓的小Gevrey设定下的结果推广到任意方便的小序列族，并展示了对应的超可微函数类如何用于检测希尔伯特空间上正规线性算子的有界性（与一个演化方程问题相关）。 最后，我们研究了小序列与J. Jiménez-Garrido在其博士论文中引入的对偶序列概念之间的联系。