Covariances of generalized processes with orthogonal values
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- by Lewis Pakula
- Proc. Amer. Math. Soc. 52 (1975), 199-203
- DOI: https://doi.org/10.1090/S0002-9939-1975-0405570-0
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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 $f,g\epsilon \mathcal {D}({\mathbf {R}})$.References
- Xavier Fernique, Processus linéaires, processus généralisés, Ann. Inst. Fourier (Grenoble) 17 (1967), no. fasc. 1, 1–92 (French). MR 221576
- I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
- J. William Helton, Jordan operators in infinite dimensions and Sturm Liouville conjugate point theory, Bull. Amer. Math. Soc. 78 (1971), 57–61. MR 285926, DOI 10.1090/S0002-9904-1972-12850-7
- Isaac Namioka, Partially ordered linear topological spaces, Mem. Amer. Math. Soc. 24 (1957), 50. MR 94681
Bibliographic Information
- © Copyright 1975 American Mathematical Society
- 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