标题： 与椭球波函数相关的特征值的某些上界 显示英文标题

标题： Certain upper bounds on the eigenvalues associated with prolate spheroidal wave functions

摘要： 拟长球面波函数（PSWFs）在多个领域中起着重要作用，从物理学（例如波现象、流体动力学）到工程学（例如信号处理、滤波器设计）。PSWFs重要性的主要原因是它们是处理带限函数的自然且高效的工具，而带限函数经常出现在上述领域中。这是由于PSWFs是表示时间限制后进行低通滤波的积分算子的特征函数。毋庸置疑，该算子的行为由其特征值的衰减率所决定。因此，研究这种衰减率在相关理论和应用中起着关键作用——例如，在构造求积公式、插值、滤波器设计等方面。PSWFs的重要性，特别是与积分算子相关的特征值衰减率的重要性，至少在半个世纪前就已经被认识到。然而，尽管有大量数值经验以及几种渐近展开式的存在，但几十年来一直缺乏对特征值大小的非平凡显式上界。本文的主要目标是填补PSWF理论中的这一空白。我们分析与PSWFs相关的积分算子，以推导出其特征值大小的相当紧的非渐近上界。我们的结果通过几个数值实验进行了说明。 显示英文摘要

摘要： Prolate spheroidal wave functions (PSWFs) play an important role in various areas, from physics (e.g. wave phenomena, fluid dynamics) to engineering (e.g. signal processing, filter design). One of the principal reasons for the importance of PSWFs is that they are a natural and efficient tool for computing with bandlimited functions, that frequently occur in the abovementioned areas. This is due to the fact that PSWFs are the eigenfunctions of the integral operator, that represents timelimiting followed by lowpassing. Needless to say, the behavior of this operator is governed by the decay rate of its eigenvalues. Therefore, investigation of this decay rate plays a crucial role in the related theory and applications - for example, in construction of quadratures, interpolation, filter design, etc. The significance of PSWFs and, in particular, of the decay rate of the eigenvalues of the associated integral operator, was realized at least half a century ago. Nevertheless, perhaps surprisingly, despite vast numerical experience and existence of several asymptotic expansions, a non-trivial explicit upper bound on the magnitude of the eigenvalues has been missing for decades. The principal goal of this paper is to close this gap in the theory of PSWFs. We analyze the integral operator associated with PSWFs, to derive fairly tight non-asymptotic upper bounds on the magnitude of its eigenvalues. Our results are illustrated via several numerical experiments.