  In this paper, we introduce a method known as polynomial frame approximation
for approximating smooth, multivariate functions defined on irregular domains
in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and
relies only on orthogonal polynomials on a bounding tensor-product domain. In
particular, the domain of the function need not be known in advance. When
restricted to a subdomain, an orthonormal basis is no longer a basis, but a
frame. Numerical computations with frames present potential difficulties, due
to the near-linear dependence of the truncated approximation system.
Nevertheless, well-conditioned approximations can be obtained via
regularization, for instance, truncated singular value decompositions. We
comprehensively analyze such approximations in this paper, providing error
estimates for functions with both classical and mixed Sobolev regularity, with
the latter being particularly suitable for higher-dimensional problems. We also
analyze the sample complexity of the approximation for sample points chosen
randomly according to a probability measure, providing estimates in terms of
the corresponding \textit{Nikolskii inequality} for the domain. In particular,
we show that the sample complexity for points drawn from the uniform measure is
quadratic (up to a log factor) in the dimension of the polynomial space,
independently of $d$, for a large class of nontrivial domains. This extends a
well-known result for polynomial approximation in hypercubes.
