与长球面波函数相关的特征值的某些上界
Certain upper bounds on the eigenvalues associated with prolate spheroidal wave functions
AI总结：
本文通过分析长球面波函数相关的积分算子，推导了其特征值幅度的非渐近上界，填补了该领域长期存在的理论空白。
摘要：
长球面波函数（PSWFs）在从物理学（如波动现象、流体动力学）到工程学（如信号处理、滤波器设计）的各个领域中发挥着重要作用。PSWFs重要性的主要原因之一是它们是计算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。这是因为PSWFs是积分算子的特征函数，该算子表示时间限制后接低通滤波。 不言而喻，该算子的行为由其特征值的衰减率决定。因此，研究这一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，特别是相关积分算子特征值的衰减率，至少在半个世纪前就已被认识到。然而，令人惊讶的是，尽管有大量的数值经验和几种渐近展开式，几十年来一直缺乏关于特征值幅度的非平凡显式上界。 本文的主要目标是填补PSWFs理论中的这一空白。我们分析了与PSWFs相关的积分算子，推导了其特征值幅度的相当紧的非渐近上界。通过几个数值实验说明了我们的结果。
英文摘要：
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.
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