Integrating these differences over all possible frequencies, leads to a global, $L_2$-type, measure of deviation which is used to test the null hypothesis of interest. We derive the limiting distribution of an appropriately centered version of such a test statistic under the null. Second, and because of the slow convergence of the distribution of the test statistic under the null against its derived limiting Gaussian distribution, we develop a novel frequency domain bootstrap procedure to estimate this distribution. The method works under quite general conditions on the underlying functional linear process and its range of applicability is not restricted to the particular test statistic considered. We show that the bootstrap procedure approximates correctly the distribution of the proposed test statistic under the null. Furthermore, consistency of the bootstrap-based test under the alternative is established. Our theoretical deviations are accomplished by a simulation study which shows a very good behavior of the bootstrap procedure in approximating the distribution of interest and that the test based on the bootstrap critical values has good size and power behavior. Furthermore, a real-life data application is presented which together with some test-diagnostics enables an interesting discussion of the test results obtained.

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