In many Tauberian theorems, the asymptotic properties of functions were investigated with respect to a predefined function (usually in the scale of regularly varying functions). In this paper, we address an alternative problem: Given a generalized function, does it have asymptotics with respect to some regularly varying function? We find necessary and sufficient conditions for the existence of quasiasymptotics of those generalized functions whose Laplace transforms have a bounded argument in a tube domain over the positive orthant. Moreover, we point out a regularly varying function with respect to which quasiasymptotics exists. It turns out that the modulus of a holomorphic function in a tube domain over the positive orthant in the purely imaginary subspace on rays emanating from the origin behaves as a regularly varying function. We use the obtained results to find the quasiasymptotics of the generalized Cauchy problem for convolution equations whose kernels are passive operators.

中文翻译：

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关于多维陶伯论的一个问题

在许多陶伯定理中，相对于预定义函数（通常以规则变化的函数为尺度）研究函数的渐近性质。在本文中，我们解决了一个替代问题：给定一个广义函数，它对于某些规律变化的函数是否具有渐近性？我们找到了那些广义函数的拟症状存在的必要和充分条件，这些广义函数的Laplace变换在正正矢上的管域中具有有限的论点。此外，我们指出存在准症状的规则变化的功能。事实证明，从原点发出的射线上，在纯虚子空间中的正正子上的管域中的全纯函数的模量表现为规则变化的函数。