  Functional data analysis is becoming increasingly popular to study data from
real-valued random functions. Nevertheless, there is a lack of multiple testing
procedures for such data. These are particularly important in factorial designs
to compare different groups or to infer factor effects. We propose a new class
of testing procedures for arbitrary linear hypotheses in general factorial
designs with functional data. Our methods allow global as well as multiple
inference of both, univariate and multivariate mean functions without assuming
particular error distributions nor homoscedasticity. That is, we allow for
different structures of the covariance functions between groups. To this end,
we use point-wise quadratic-form-type test functions that take potential
heteroscedasticity into account. Taking the supremum over each test function,
we define a class of local test statistics. We analyse their (joint) asymptotic
behaviour and propose a resampling approach to approximate the limit
distributions. The resulting global and multiple testing procedures are
asymptotic valid under weak conditions and applicable in general functional
MANOVA settings. We evaluate their small-sample performance in extensive
simulations and finally illustrate their applicability by analysing a
multivariate functional air pollution data set.
