Spectral integrals of Bernoulli generalized functionals

Abstract

Let $\mathcal{S}\subset {\mathcal{L}}^2\subset {\mathcal{S}}^{\ast }$ be the Gel'fand triple over the Bernoulli space, where elements of ${\mathcal{S}}^{\ast }$ are called Bernoulli generalized functionals. In this paper, we define integrals of Bernoulli generalized functionals with respect to a spectral measure (projection operator-valued measure) in the framework of $\mathcal{S}\subset {\mathcal{L}}^2\subset {\mathcal{S}}^{\ast }$, and examine their fundamental properties. New notions are introduced, several results are obtained and examples are also shown.

Keywords:

• Bernoulli functional
• Gel'fand triple
• spectral measure
• spectral integral
• white noise theory

Acknowledgements

The authors are extremely grateful to the referees for their valuable comments and suggestions on improvement of the first version of the present paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by National Natural Science Foundation of China (Grant No. 11861057).