Regular Variation and Differential Equations
Regular Variation and Differential Equations
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DOI：
10.1007/bfb0103952
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发表时间：
2000-04
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影响因子：
--
通讯作者：
V. Marić
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文献类型：
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作者：
V. Marić
规则变化的概念是由Jovan Karamata在他1930年的著名论文“Sur une mode des crosissance reguliére des fonctions”中发现的。卡拉马塔的目标是陶伯理论，这是那个时代的一个亮点，标志着杰出的分析家的工作，主要是GH哈代，JL利特尔伍德和E。朗道，在N. 1932年维纳的一般陶伯定理。然而，除了证明Tauberian定理首先为拉普拉斯，Stieltjes和后来的更一般的积分变换的卷积型，经常变化很快就适用于阿贝尔定理，使事实上渐近行为的积分和系列，傅立叶的特别是。在分析中的进一步应用包括Mercerian定理，解析数论，复分析，特别是整函数。与W.费勒在1968年的著名论文[14]中指出，正则变分被认为是概率论及其应用的主要工具。L.德哈安的工作在1970年，[23]，在那里他介绍了一个实质性的推广正规变化，目的再次主要是在概率论。这可以在JL Geluk和L. de哈安of 1987，[18].第一个文件连接经常变化和微分方程是一个VG Avakumović的1947年，”Sur l 'équation différentielle de Al-Fermi”。他的论文并没有引起太多的注意--正则变函数在当时离微分方程理论还很遥远，直到M。Tomić和作者于1976年开始，[37]。最早研究线性方程组的是E. Omey于1981年，[55]。Karamata理论的最完整的介绍及其推广以及大多数应用都包含在1987年由NH Bingham，CM Goldie和JL Teugels [9]撰写的书中。微分方程的初步结果见附录2。第一本关于这一主题的专著是E. 1976年的Seneta，[60]。这篇论文的核心是基于Miodrag Tomić和作者的共同成果。对本书主题的重要贡献是所包含的JL格鲁克和E。Omey和HC霍华德和作者的联合。虽然米奥德拉格·托米奇在形式上不是本书的作者，但他的影响和思想贯穿了整本书。这既适用于导致一些定理的证明，也适用于证明所需的许多特殊技术和设备，
The notion of regular variation was discovered by Jovan Karamata in his famous paper of 1930" Sur une mode des croissance reguliére des fonctions". Karamata's aim was Tauberian theory, one of the highlights of the epoch marked by the work of eminent analysts, predominantly that of GH Hardy, JL Littlewood and also of E. Landau, culminating in N. Wiener's general Tauberian theorem in 1932. However, in addition to proving Tauberian theorems first for Laplace-Stieltjes and later for the more general integral transforms of convolution type, regular variation was soon applied in Abelian theorems, giving in fact asymptotic behavior of integrals and series, the Fourier ones in particular. Further applications in analysis include Mercerian theorems, analytic number theory, complex analysis-entire functions in particular. With W. Feller's well known treatise of 1968,[14], regular variation was recognized as a major tool in the probability theory and its applications. A new impetus to the subject was provided by the L. de Haan work in 1970,[23], where he introduced a substantial generalization of regular variation, aiming again primarily at the probability theory. This can be found in the monograph of JL Geluk and L. de Haan of 1987,[18]. The first paper connecting regular variation and the differential equation is the one of VG Avakumović of 1947," Sur l'équation différentielle de Thomas-Fermi". His paper did not attract much attention-regularly varying functions were totally distant from the theory of differential equation at that time, until the investigations of M. Tomić and the author started in 1976,[37]. The first study of the linear equations is that of E. Omey in 1981,[55]. The most complete presentation of Karamata theory and its generalizations as well as the majority of the applications are contained in the book of 1987 by NH Bingham, CM Goldie and JL Teugels [9]. Rudimentary results on differential equations form its Appendix 2. The first monograph on the subject is the one of E. Seneta of 1976,[60]. The core of this treatise is based on joint results of Miodrag Tomić and the author. Significant contribution to the main theme of the book are the included results of JL Geluk and E. Omey and the joint ones of HC Howard and the author. Although Miodrag Tomić is formally not an author of this book, the whole text is permeated with his influence and ideas. This holds both for conjectures leading to a number of theorems and for many special techniques and devices needed for the proofs where general methods