  A test based on tapering is proposed for use in testing a global linear
hypothesis under a functional linear model. The test statistic is constructed
as a weighted sum of squared linear combinations of Fourier coefficients, a
tapered quadratic form, in which higher Fourier frequencies are down-weighted
so as to emphasize the smooth attributes of the model. A formula is
$Q_n^{OPT}=n\sum_{j=1}^{p_n}j^{-1/2}\|\boldsymbol{Y}_{n,j}\|^2$. Down-weighting
by $j^{-1/2}$ is selected to achieve adaptive optimality among tests based on
tapering with respect to its ``rates of testing,'' an asymptotic framework for
measuring a test's retention of power in high dimensions under smoothness
constraints. Existing tests based on truncation or thresholding are known to
have superior asymptotic power in comparison with any test based on tapering;
however, it is shown here that high-order effects can be substantial, and that
a test based on $Q_n^{OPT}$ exhibits better (non-asymptotic) power against the
sort of alternatives that would typically be of concern in functional data
analysis applications. The proposed test is developed for use in practice, and
demonstrated in an example application.
