  The problem of data-driven identification of coherent observables of
measure-preserving, ergodic dynamical systems is studied using kernel integral
operator techniques. An approach is proposed whereby complex-valued observables
with approximately cyclical behavior are constructed from a pair eigenfunctions
of integral operators built from delay-coordinate mapped data. It is shown that
these observables are $\epsilon$-approximate eigenfunctions of the Koopman
evolution operator of the system, with a bound $\epsilon$ controlled by the
length of the delay-embedding window, the evolution time, and appropriate
spectral gap parameters. In particular, $ \epsilon$ can be made arbitrarily
small as the embedding window increases so long as the corresponding
eigenvalues remain sufficiently isolated in the spectrum of the integral
operator. It is also shown that the time-autocorrelation functions of such
observables are $\epsilon$-approximate Koopman eigenvalue, exhibiting a
well-defined characteristic oscillatory frequency (estimated using the Koopman
generator) and a slowly-decaying modulating envelope. The results hold for
measure-preserving, ergodic dynamical systems of arbitrary spectral character,
including mixing systems with continuous spectrum and no non-constant Koopman
eigenfunctions in $L^2$. Numerical examples reveal a coherent observable of the
Lorenz 63 system whose autocorrelation function remains above 0.5 in modulus
over approximately 10 Lyapunov timescales.
