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Representation of weakly harmonizable processes.

M M Rao1

  • 1Department of Mathematics, University of California, Riverside, California 92521.

Proceedings of the National Academy of Sciences of the United States of America
|September 1, 1981
PubMed
Summary
This summary is machine-generated.

Weakly harmonizable processes are generalized using contractive linear operators in Hilbert spaces. This study characterizes these processes and their associated spectra, resolving a long-standing problem in the field.

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Area of Science:

  • Stochastic processes
  • Functional analysis
  • Harmonic analysis

Background:

  • Weakly stationary processes are characterized by unitary families.
  • Existing literature lacks a spectral characterization for weakly harmonizable processes.

Purpose of the Study:

  • To generalize weakly stationary processes to weakly harmonizable processes.
  • To characterize weakly harmonizable processes using linear operators and Fourier integrals.
  • To provide a spectral characterization for weakly harmonizable processes.

Main Methods:

  • Representation of weakly harmonizable processes using positive definite contractive linear operators in Hilbert spaces.
  • Characterization of the vector Fourier integral of a measure on a reflexive space.
  • Development of spectral analysis for weakly harmonizable processes.

Main Results:

  • Established a generalization of weakly stationary processes.
  • Provided a novel characterization of weakly harmonizable processes via linear operators and Fourier integrals.
  • Demonstrated the existence of associated spectra for these processes.

Conclusions:

  • Weakly harmonizable processes can be effectively represented by contractive linear operators.
  • The study successfully characterizes these processes and confirms the existence of their spectra.
  • Resolved a problem posed by Rozanov concerning the spectral properties of these processes.