Harmonic Analysis of Generalized Stochastic Processes on Locally Compact Abelian Groups

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].