Opuscula Math. 40, no. 3 (2020), 323-339

Opuscula Mathematica

Stochastic Wiener filter in the white noise space

Daniel Alpay
Ariel Pinhas

Abstract. In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this space in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions.

Keywords: Wiener filter, white noise space, Wick product, stochastic distribution.

Mathematics Subject Classification: 93E11, 60H40, 46F25, 13J99.

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  • Daniel Alpay (corresponding author)
  • ORCID iD https://orcid.org/0000-0002-7612-3598
  • Schmid College of Science and Technology, Chapman University, One University Drive Orange, California 92866, USA
  • Communicated by Palle E.T. Jorgensen.
  • Received: 2020-02-17.
  • Revised: 2020-03-10.
  • Accepted: 2020-03-11.
  • Published online: 2020-04-04.
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Cite this article as:
Daniel Alpay, Ariel Pinhas, Stochastic Wiener filter in the white noise space, Opuscula Math. 40, no. 3 (2020), 323-339, https://doi.org/10.7494/OpMath.2020.40.3.323

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