  Under the frequency domain framework for weakly dependent functional time
series, a key element is the spectral density kernel which encapsulates the
second-order dynamics of the process. We propose a class of spectral density
kernel estimators based on the notion of a flat-top kernel. The new class of
estimators employs the inverse Fourier transform of a flat-top function as the
weight function employed to smooth the periodogram. It is shown that using a
flat-top kernel yields a bias reduction and results in a higher-order accuracy
in terms of optimizing the integrated mean square error (IMSE). Notably, the
higher-order accuracy of flat-top estimation comes at the sacrifice of the
positive semi-definite property. Nevertheless, we show how a flat-top estimator
can be modified to become positive semi-definite (even strictly positive
definite) in finite samples while retaining its favorable asymptotic
properties. In addition, we introduce a data-driven bandwidth selection
procedure realized by an automatic inspection of the estimated correlation
structure. Our asymptotic results are complemented by a finite-sample
simulation where the higher-order accuracy of flat-top estimators is manifested
in practice.
