  In this paper, we are concerned with multivariate Gegenbauer approximation of
functions defined in the $d$-dimensional hypercube. Two new and sharper bounds
for the coefficients of multivariate Gegenbauer expansion of analytic functions
are presented based on two different extensions of the Bernstein ellipse. We
then establish an explicit error bound for the multivariate Gegenbauer
approximation associated with an $\ell^q$ ball index set in the uniform norm.
We also consider the multivariate approximation of functions with finite
regularity and derive the associated error bound on the full grid in the
uniform norm. As an application, we extend our arguments to obtain some new
tight bounds for the coefficients of tensorized Legendre expansions in the
context of polynomial approximation of parameterized PDEs.
