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- W4221142653 abstract "Given a stationary continuous-time process $f(t)$, the Hilbert-Schmidt operator $A_{tau}$ can be defined for every finite $tau$cite{Vautard1989SingularSA}. Let $lambda_{tau,i}$ be the eigenvalues of $A_{tau}$ with descending order. In this article, a Hilbert space $mathcal{H}_f$ and the (time-shift) continuous one-parameter semigroup of isometries $mathcal{K}^s$ are defined. Let ${v_i, iinmathbb{N}}$ be the eigenvectors of $mathcal{K}^s$ for all $sgeq 0$. Let $f = displaystylesum_{i=1}^{infty}a_iv_i + f^{perp}$ be the orthogonal decomposition with descending $|a_i|$. We prove that $displaystylelim_{tautoinfty}lambda_{tau,i} = |a_i|^2$. The continuous one-parameter semigroup ${mathcal{K}^s: sgeq 0}$ is equivalent, almost surely, to the classical Koopman one-parameter semigroup defined on $L^2(X,nu)$, if the dynamical system is ergodic and has invariant measure $nu$ on the phase space $X$." @default.
- W4221142653 created "2022-04-03" @default.
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- W4221142653 date "2022-02-17" @default.
- W4221142653 modified "2023-09-29" @default.
- W4221142653 title "Bridging Koopman Operator and time-series auto-correlation based Hilbert-Schmidt operator" @default.
- W4221142653 doi "https://doi.org/10.48550/arxiv.2202.08755" @default.
- W4221142653 hasPublicationYear "2022" @default.
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