  We consider the problem of estimating the slope function in a functional
regression with a scalar response and a functional covariate. This central
problem of functional data analysis is well known to be ill-posed, thus
requiring a regularised estimation procedure. The two most commonly used
approaches are based on spectral truncation or Tikhonov regularisation of the
empirical covariance operator. In principle, Tikhonov regularisation is the
more canonical choice. Compared to spectral truncation, it is robust to
eigenvalue ties, while it attains the optimal minimax rate of convergence in
the mean squared sense, and not just in a concentration probability sense. In
this paper, we show that, surprisingly, one can strictly improve upon the
performance of the Tikhonov estimator in finite samples by means of a linear
estimator, while retaining its stability and asymptotic properties by combining
it with a form of spectral truncation. Specifically, we construct an estimator
that additively decomposes the functional covariate by projecting it onto two
orthogonal subspaces defined via functional PCA; it then applies Tikhonov
regularisation to the one component, while leaving the other component
unregularised. We prove that when the covariate is Gaussian, this hybrid
estimator uniformly improves upon the MSE of the Tikhonov estimator in a
non-asymptotic sense, effectively rendering it inadmissible. This domination is
shown to also persist under discrete observation of the covariate function. The
hybrid estimator is linear, straightforward to construct in practice, and with
no computational overhead relative to the standard regularisation methods. By
means of simulation, it is shown to furnish sizeable gains even for modest
sample sizes.
