A Sequential Approach to Mild Distributions

The Banach Gelfand Triple ( S 0 , L 2 , S 0 ′ ) ( R d ) consists of S 0 ( R d ) , ∥ · ∥ S 0 , a very specific Segal algebra as algebra of test functions, the Hilbert space L 2 ( R d ) , ∥ · ∥ 2 and the dual space S 0 ′ ( R d ) , whose elements are also called “mild distributions”. Together they provide a universal tool for Fourier Analysis in its many manifestations. It is indispensable for a proper formulation of Gabor Analysis, but also useful for a distributional description of the classical (generalized) Fourier transform (with Plancherel’s Theorem and the Fourier Inversion Theorem as core statements) or the foundations of Abstract Harmonic Analysis, as it is not difficult to formulate this theory in the context of locally compact Abelian (LCA) groups. A new approach presented recently allows to introduce S 0 ( R d ) , ∥ · ∥ S 0 and hence ( S 0 ′ ( R d ) , ∥ · ∥ S 0 ′ ) , the space of “mild distributions”, without the use of the Lebesgue integral or the theory of tempered distributions. The present notes will describe an alternative, even more elementary approach to the same objects, based on the idea of completion (in an appropriate sense). By drawing the analogy to the real number system, viewed as infinite decimals, we hope that this approach is also more interesting for engineers. Of course it is very much inspired by the Lighthill approach to the theory of tempered distributions. The main topic of this article is thus an outline of the sequential approach in this concrete setting and the clarification of the fact that it is just another way of describing the Banach Gelfand Triple. The objects of the extended domain for the Short-Time Fourier Transform are (equivalence classes) of so-called mild Cauchy sequences (in short ECmiCS). Representatives are sequences of bounded, continuous functions, which correspond in a natural way to mild distributions as introduced in earlier papers via duality theory. Our key result shows how standard functional analytic arguments combined with concrete properties of the Segal algebra S 0 ( R d ) , ∥ · ∥ S 0 can be used to establish this natural identification.

BV functions in a Gelfand triple for differentiable measure and its applications

In this paper, we introduce a definition of BV functions for (non-Gaussian) differentiable measure in a Gelfand triple which is an extension of the definition of BV functions in [Ann. Probab. 40 (2012), 1759–1794], using Dirichlet form theory. By this definition, we can analyze the reflected stochastic quantization problem associated with a self-adjoint operator

HARNACK INEQUALITIES AND APPLICATIONS FOR MULTIVALUED STOCHASTIC EVOLUTION EQUATIONS

By the method of coupling and Girsanov transformation, Harnack inequalities (F.-Y. Wang, 1997) and strong Feller property are proved for the transition semigroup associated with the multivalued stochastic evolution equation on a Gelfand triple. The concentration property of the invariant measure for the semigroup is investigated. As applications of Harnack inequalities, explicit upper bounds of the Lp-norm of the density, contractivity, compactness and entropy-cost inequality for the semigroup are also presented.

Time-frequency partitions for the Gelfand triple $(S_0,L^2,S_0')$

We give a new characterization of the Gelfand triple of function spaces in $(S_0, L^2, S_0')$ by means of a family of time-frequency localization operators. The localization operators are defined by the short-time Fourier transform and determine the local time-frequency behavior, whereas the global time-frequency distribution is characterized by a sequence space norm. We also show that the alternative time-frequency localization method with the Weyl transform fails to yield a similar characterization of time-frequency distribution.

Derivations on white noise functionals

The Gaussian space (E*, μ) is a natural infinite dimensional analogue of Euclidean space with Lebesgue measure and a special choice of a Gelfand triple gives a fundamental framework of white noise calculus [2] as distribution theory on Gaussian space. It is proved in Kubo-Takenaka [7] that (E) is a topological algebra under pointwise multiplication. The main purpose of this paper is to answer the fundamental question: what are the derivations on the algebra (E)?