  We consider an unknown response function $f$ defined on $\Delta=[0,1]^d$,
$1\le d\le\infty$, taken at $n$ random uniform design points and observed with
Gaussian noise of known variance. Given a positive sequence $r_n\to 0$ as
$n\to\infty$ and a known function $f_0 \in L_2(\Delta)$, we propose, under
general conditions, a unified framework for the goodness-of-fit testing problem
for testing the null hypothesis $H_0: f=f_0$ against the alternative $H_1:
f\in\CF, \|f-f_0\|\ge r_n$, where $\CF$ is an ellipsoid in the Hilbert space $
L_2(\Delta)$ with respect to the tensor product Fourier basis and $\|\cdot\|$
is the norm in $ L_2(\Delta)$. We obtain both rate and sharp asymptotics for
the error probabilities in the minimax setup. The derived tests are inherently
non-adaptive.
  Several illustrative examples are presented. In particular, we consider
functions belonging to ellipsoids arising from the well-known multidimensional
Sobolev and tensor product Sobolev norms as well as from the less-known
Sloan-Wo$\rm\acute{z}$niakowski norm and a norm constructed from multivariable
analytic functions on the complex strip.
  Some extensions of the suggested minimax goodness-of-fit testing methodology,
covering the cases of general design schemes with a known product probability
density function, unknown variance, other basis functions and adaptivity of the
suggested tests, are also briefly discussed.
