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Cross-Modal Multivariate Pattern Analysis

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Published on: November 9, 2011

The spectral domain of multivariate harmonizable processes.

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

Proceedings of the National Academy of Sciences of the United States of America

|July 1, 1984

Summary

This study proves the completeness of a specific function space for multivariate weakly harmonizable processes. This finding enables optimal least-squares estimation for these complex signals.

Area of Science:

Statistics
Signal Processing
Functional Analysis

Background:

Multivariate weakly harmonizable processes are crucial in time series analysis.
Their spectral domain is a vector space of functions integrable with respect to a positive definite matrix bimeasure.
The completeness of this space has been a long-standing open problem.

Purpose of the Study:

To resolve the open question regarding the completeness of the spectral domain of multivariate weakly harmonizable processes.
To apply this finding to improve optimal least-squares estimation techniques.

Main Methods:

Characterization of the spectral domain as a vector space.
Utilizing properties of positive definite matrix bimeasures.
Derivation of a norm from a Gramian inner product.

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Main Results:

An affirmative solution to the completeness property of the spectral domain is presented.
The established completeness facilitates advanced signal processing applications.

Conclusions:

The completeness of the spectral domain for multivariate weakly harmonizable processes is confirmed.
This result directly benefits the theory and practice of optimal least-squares estimation for such processes.