Covariances of generalized processes with orthogonal values

Abstract

A general form for the covariance of a generalized process with orthogonal values is found in the case where the covariance B B depends on test functions and their first derivatives. Specifically, if B ( ϕ , ϕ ) = ∫ ϕ 2 d μ 0 + ∫ ϕ ϕ ′ d μ 1 + ∫ ϕ 2 d μ 2 ⩾ 0 for ϕ ϵ D ( R ) 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 μ 0 , μ 1 , μ 2 {\mu _0},{\mu _1},{\mu _2} , then there exist Radon measures ν 0 , ν 1 , ν 2 {\nu _0},{\nu _1},{\nu _2} such that B ( ϕ , ϕ ) = ∫ ϕ 2 d ν 0 + ∫ ϕ ϕ ′ d ν 1 + ∫ ϕ ′ 2 d ν 2 B(\phi ,\phi ) = \int {{\phi ^2}d{\nu _0} + \int {\phi \phi ’d{\nu _1} + \int {\phi {’^2}d{\nu _2}} } } and, moreover, ∫ f 2 d ν 0 + ∫ f g d ν 1 + ∫ g 2 d ν 2 ⩾ 0 \int {{f^2}d{\nu _0} + \int {fgd{\nu _1} + \int {{g^2}d{\nu _2} \geqslant 0} } } for all f , g ϵ D ( R ) f,g\epsilon \mathcal {D}({\mathbf {R}}) .