Abstract

The creation of the theory of canonic expansions (CEs) is related with the names Loeve, Kolmogorov, Karhunen, and Pugachev and dates back to the 1940–1950s. The development of the theory of CEs and wavelet CEs is considered in application to the problems of the analysis, modeling, and synthesis of stochastic systems (SSs) and technologies. The direct and inverse Pugachev theorems about CEs are extended to the case of stochastic linear functionals within the framework of the correlational theory of stochastic functions (SFs). The CEs of linear and quasi-linear SFs are derived. Particular attention is paid to the problems of the equivalent regression linearization of strongly nonlinear transformations by CEs. The nonlinear regression algorithms on the basis of CEs are proposed. The theory of wavelet CEs within the specified domain of the change of the argument on the basis of Haar wavelets is developed. For stochastic elements (SEs), the direct and inverse Pugachev theorems are formulated and the correlational theory of joint CEs for two SEs is developed together with the theory of linear transformations. The solution of linear operator equations by the CEs of SEs in linear spaces with a basis is given. Special attention is focused on the CEs of SEs in Banach spaces with a basis. Some elements of the general theory of distributions for the CEs of SFs and SEs are developed. Particular attention is paid to the method based on CEs with independent components. Some new methods for the calculation of Radon–Nikodym derivatives are proposed. The considered applications of CEs and wavelet CEs to analysis, modeling, and synthesis problems are as follows: SSs and technologies, modeling, identification and recognition filtering, metrological and biometric technologies and systems, and synergic organizational technoeconomic systems (OTESs). The conclusion contains inferences and propositions for further studies. The list of references contains 43 items.

中文翻译：

---

发展随机正则展开理论及其应用

摘要

正则展开理论 (CE) 的创立与 Loeve、Kolmogorov、Karhunen 和 Pugachev 等人有关，可以追溯到 1940-1950 年代。 CE 和小波 CE 理论的发展被考虑应用于随机系统（SS）和技术的分析、建模和综合问题。关于 CE 的正普加乔夫定理和反普加乔夫定理被扩展到随机函数 (SF) 相关理论框架内的随机线性泛函的情况。推导了线性和准线性 SF 的 CE。特别关注CE 强非线性变换的等效回归线性化问题。提出了基于CE的非线性回归算法。在Haar小波的基础上发展了参数变化指定域内的小波CE理论。对于随机元素（SE），制定了正向和逆普加乔夫定理，并与线性变换理论一起发展了两个 SE 的联合 CE 的相关理论。给出了有基线性空间中线性算子方程的SE 的CE 解。特别关注有基础的 Banach 空间中 SE 的 CE。发展了 SF 和 SE 的 CE 分布一般理论的一些要素。特别关注基于具有独立组件的 CE 的方法。提出了一些计算 Radon-Nikodym 导数的新方法。 CE 和小波 CE 在分析、建模和综合问题中的应用如下：SS 和技术、建模、识别和识别过滤、计量和生物识别技术和系统以及协同组织技术经济系统 (OTES)。结论包含推论和进一步研究的建议。参考文献列表包含 43 项。