Harmonic Analysis of Generalized Stochastic Processes on Locally Compact Abelian Groups
Harmonic Analysis of Generalized Stochastic Processes on Locally Compact Abelian Groups
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局部紧阿贝尔群广义随机过程的调和分析
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发表时间：
2003
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通讯作者：
W. Hörmann
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作者：
H. Feichtinger;W. Hörmann
简介 局部紧群上的普通函数将群元素映射为复数，而随机过程可以理解为映射到希尔伯特空间。广义函数的思想是将函数的知识减少到特定平均值的知识。它引出了广义函数的概念，即测试函数空间上的连续线性泛函。这两种思想的结合是局部紧阿贝尔群上的广义随机过程的基础：G 上的测试函数空间上的希尔伯特空间值有界线性算子。施瓦茨空间 S(Rm) 的性质，特别是它在傅立叶变换下的不变性，使其成为描述广义函数的非常合适的工具，但其广义化，即所谓的施瓦茨-布鲁哈特空间非常复杂（参见[19]）和 lca 的结构理论。需要组来描述空间。相比之下，第一作者发现的函数空间S0(G)（参见[3]、[19]）可以在不使用结构理论的情况下以简单的方式定义一般lca。组。而且它是一个 Banach 空间（极大地简化了对偶空间上自然拓扑的描述），傅立叶变换将 S0(G) 映射到 S0(Ĝ)，即对偶群 Ĝ 上的相应空间。使用庞特里亚金对偶定理，可以扩展傅里叶变换，以获得从 S ′ 0(G) 到 S ′ 0(Ĝ) 的广义傅里叶变换。由于 Schwartz-Bruhat 的空间 S(G) 在 S0(G) 中是稠密的，因此显然缓和分布 σ ∈ S ′(G) 的概念更为普遍。然而，如果人们对导数不感兴趣，那么 S ' 0(G) 的概念足够通用，并且具有许多实际优点，主要是因为它的简单性。我们只提一下，对于 G = T 的情况，空间 S0(G) 与绝对收敛傅里叶级数的代数 A(G) 重合，它可以被视为任意 lca 的自然推广。组。欧几里得空间上的广义随机过程的概念在[12]、[7]和[8]中得到了发展，其中使用了具有紧支持的无限常可微函数的空间D。在[14]中，具有紧支持的连续函数的空间K(G)用作测试函数的空间。我们认为，这些函数空间的主要缺点（除了可以通过开发适当的积分或分布理论可以克服的技术问题之外）是它们在傅立叶变换下不是不变的，并且它们只是拓扑向量空间。我们所知道的关于广义随机过程的唯一工作是在 G = R 的情况下使用在傅立叶变换下不变的测试函数空间的[13]。其中函数空间是在 C 上定义的，或者在 C 上有技术困难，但似乎不可能将此定义扩展到局部紧凑阿贝尔群。此外，测试函数是解析函数，这使得无法描述对应对偶空间的支持的概念。观察到 S0(G) 在概念上和技术上是一个更方便的空间，并且观察到大多数相关概念出现在 lca 的广义随机过程理论中。群可以在这个概念的基础上得到证明，从而引出本文。感兴趣的读者可能会在第二作者的论文中找到有关广义随机过程的更多详细信息，特别是 R 上过程的维格纳分布的定义 [11]。
Introduction Whereas ordinary functions on a locally compact group map the group elements into the complex numbers, a stochastic process can be understood as a mapping into a Hilbert space. The idea of a generalized function is to reduce the knowledge about the function to that of certain averages. It leads to the concept of generalized functions as continuous linear functionals on spaces of test functions. The combination of both ideas is the basis for generalized stochastic processes on locally compact Abelian groups: Hilbert space valued bounded linear operators on spaces of test functions on G. The properties of the Schwartz space S(Rm), in particular its invariance under the Fourier transform, make it a very suitable tool for the description of generalized functions, but its generalization, the so-called Schwartz-Bruhat space is very complicated (cf. [19]) and structure theory of lca. groups is required to describe the space. In contrast, the function space S0(G) discovered by the first author (cf. [3], [19]) can be defined without the use of structure theory in a simple way for general lca. groups. Moreover it is a Banach space (which greatly simplifies the description of the natural topology on the dual space), and the Fourier transform maps S0(G) onto S0(Ĝ), the corresponding space on the dual group Ĝ. Using Pontryagin’s duality theorem it is then possible to extend the Fourier transform in order to obtain a generalized Fourier transform from S ′ 0(G) onto S ′ 0(Ĝ). Since the space S(G) of Schwartz-Bruhat is dense in S0(G) it is clear that the concept of tempered distributions σ ∈ S ′(G) is more general. However, if one is not interested in derivatives the concept of S ′ 0(G) is general enough and has many practical advantages, mainly due to its simplicity. Let us only mention that for the case G = T the space S0(G) coincides with A(G), the algebra of absolutely convergent Fourier series, of which it can be seen as a natural generalization for arbitrary lca. groups. The concept of generalized stochastic processes over Euclidean spaces has been developed in [12], [7] and [8], where the space D of infinitely often differentiable functions with compact support was used. In [14] the space K(G) of continuous functions with compact support serves as the space of test functions. The main disadvantage of these function spaces (besides technical questions that may be overcome by developing the appropriate integration or distribution theory) is in our opinion the fact that they are not invariant under Fourier transform and that they are only topological vector spaces. The only work on generalized stochastic processes we know that uses a test function space that is invariant under Fourier transform is [13] for the case G = R. There the function space is defined over C or with technical difficulties over C, but it seems impossible to extend this definition to locally compact Abelian groups. In addition the test functions are analytic functions which makes it impossible to describe the concept of a support for the corresponding dual space. The observation that S0(G) is a conceptually and technically much more convenient space and the observation that most of the relevant concepts arising in the theory of generalized stochastic processes over lca. groups can be proved on the basis of this concept lead to this paper. The interested reader may find more details about generalized stochastic processes, and in particular the definition of the Wigner distribution for a process over R in the thesis of the second author [11].