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- W2088892400 abstract "Let K 1 (s,t) and K 2 (s,t), − T s, T, be real, symmetric, continuous and strictly positive-definite kernels, and denote by K 1 and K 2 the corresponding integral operators. Let x(t) be a sample function of either of two zero-mean processes with covariances K 1 (s,t) and K 2 (s,t). We prove a generalized version of the following: If the integral equation $(K_{2}psi_{i})(t)=lambda_{i}(K_{1}psi_{i}(t),qquad -T leqq t leqq T$,$ has formal solutions λ i and ψ i (t) which may contain δ-functions, and if {K 1 ψ i } forms a complete set in L 2 [-T,T], then (i) the two kernels have the following simultaneous diagonalization: $eqalignno{& K_{1}(s,t) Sigma_{i}(K_{1} psi_{i})(s)(K_{1}psi_{i})(t)cr & K_{2}(s,t) Sigma_{i}lambda_{i}(K_{1} psi_{i})(s)(K_{1}psi_{i})(t)}$ uniformly on [-T,T] × [-T, T], and (ii) the sample function has an expansion $x(t) Sigma_{i}(x,psi_{i})(K_{i} psi_{i})(t)$ in the stochastic mean, uniformly in t, and the coefficients are simultaneously orthogonal, i.e., $E_{1}{(x, psi_{i})(x,psi_{i})} = delta_{ij}, qquad E_{2}{(x,psi_{i})(x, psi_{i}}=lambda_{i} delta_{ij}$, where (x,ψ i ) is obtained by formally integrating ψ i (t) against x(t)." @default.
- W2088892400 created "2016-06-24" @default.
- W2088892400 creator A5034063717 @default.
- W2088892400 date "1967-05-06" @default.
- W2088892400 modified "2023-09-26" @default.
- W2088892400 title "Integral Equation for Simultaneous Diagonalization of Two Covariance Kernels" @default.
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- W2088892400 doi "https://doi.org/10.1002/j.1538-7305.1967.tb01719.x" @default.
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