Non-Gaussian analysis and hypergroups

Abstract

UDC 517.515 In recent years two approaches to the generalization of the Gaussian infinite-dimensional analysis (white noise analysis) to non-Gaussian measures have appeared: one of them is based on spectral theory for families of commuting self-adjoint operators [1, 2] and the other proceeds from biorthogonal expansions [3, 4]. In this note we show that in the one-dimensional model case the second approach can be extensively generalized if the characters of an Ll-hypergroup are used in its construction instead of exponential functions. For the properties of hypergroups applied below see [5-7] (here we use the term "Ll-hypergroup" instead of "hypercomplex system with locally compact base" according to [7]). 1. Consider a commutative Ll-hypergroup H I with locally compact basis Q, nonnegative structure measure, and multiplicative measure din(x) (x e Q); H I is assumed to be normal (Q 9 x ~-~ x* • Q is