  This article studies global testing of the slope function in functional
linear regression model in the framework of reproducing kernel Hilbert space.
We propose a new testing statistic based on smoothness regularization
estimators. The asymptotic distribution of the testing statistic is established
under null hypothesis. It is shown that the null asymptotic distribution is
determined jointly by the reproducing kernel and the covariance function. Our
theoretical analysis shows that the proposed testing is consistent over a class
of smooth local alternatives. Despite the generality of the method of
regularization, we show the procedure is easily implementable. Numerical
examples are provided to demonstrate the empirical advantages over the
competing methods.
