Covariances of generalized processes with orthogonal values

Author:
Lewis Pakula

Journal:
Proc. Amer. Math. Soc. **52** (1975), 199-203

MSC:
Primary 60G20

DOI:
https://doi.org/10.1090/S0002-9939-1975-0405570-0

MathSciNet review:
0405570

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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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© Copyright 1975
American Mathematical Society