The problem of testing linear hypotheses for the means of random functions is considered. This includes checking if the mean is zero, checking if two sample means are the same, and checking if the two means have a constant difference or ratio. The random function is defined on a multidimensional compact domain and several independent realizations are observed at random design points, possibly with heteroscedastic error. The number of design points of each realization of the random function can be bounded or arbitrarily large. For two-sample tests, the samples are allowed to be unbalanced and dependent. The testing approach is based on a non-asymptotic Gaussian approximation bound for the estimated Fourier coefficients. A pivotal chi-square type statistics is proposed.