论文标题
复杂订单的规模不变运算符和自相似过程
Complex-Order Scale-Invariant Operators and Self-Similar Processes
论文作者
论文摘要
导数和集成运算符是线性运算符的经过良好研究的示例，可以按比例扩展到固定的乘法因子。即，它们是规模不变的。分数订单衍生物（集成运营商）也属于该家族。在本文中，我们通过在傅立叶域中构造分数运算符将分数运算符扩展到复杂订单的运算符。我们详细分析了这些操作员，并特别强调输出的衰减特性。我们进一步使用这些操作员介绍一个复杂值的稳定过程，这些过程与复杂值的赫斯特指数是自相似的。这些过程通过函数的Schwartz空间上的特征函数表示。除了自相似性和平稳性外，我们还研究了拟议过程的规律性（根据Sobolev空间）。
Derivatives and integration operators are well-studied examples of linear operators that commute with scaling up to a fixed multiplicative factor; i.e., they are scale-invariant. Fractional order derivatives (integration operators) also belong to this family. In this paper, we extend the fractional operators to complex-order operators by constructing them in the Fourier domain. We analyze these operators in details with a special emphasis on the decay properties of the outputs. We further use these operators to introduce a family of complex-valued stable processes that are self-similar with complex-valued Hurst indices. These processes are expressed via the characteristic functionals over the Schwartz space of functions. Besides the self-similarity and stationarity, we study the regularity (in terms of Sobolev spaces) of the proposed processes.
