# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Covariances of generalized processes with orthogonal valuesHTML articles powered by AMS MathViewer

by Lewis Pakula
Proc. Amer. Math. Soc. 52 (1975), 199-203 Request permission

## Abstract:

A general form for the covariance of a generalized process with orthogonal values is found in the case where the covariance $B$ depends on test functions and their first derivatives. Specifically, if $B(\phi ,\phi ) = \int {{\phi ^2}d{\mu _0} + \int {\phi \phi ’d{\mu _1} + \int {{\phi ^2}d{\mu _2} \geqslant 0} } } \;{\text {for }}\phi \epsilon \mathcal {D}({\mathbf {R}})$ and Radon measures ${\mu _0},{\mu _1},{\mu _2}$, then there exist Radon measures ${\nu _0},{\nu _1},{\nu _2}$ such that $B(\phi ,\phi ) = \int {{\phi ^2}d{\nu _0} + \int {\phi \phi ’d{\nu _1} + \int {\phi {’^2}d{\nu _2}} } }$ and, moreover, $\int {{f^2}d{\nu _0} + \int {fgd{\nu _1} + \int {{g^2}d{\nu _2} \geqslant 0} } }$ for all $f,g\epsilon \mathcal {D}({\mathbf {R}})$.
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