  In this work, we present a generalized methodology for analyzing the
convergence of quasi-optimal Taylor and Legendre approximations, applicable to
a wide class of parameterized elliptic PDEs with finite-dimensional
deterministic and stochastic inputs. Such methods construct an optimal index
set that corresponds to the sharp estimates of the polynomial coefficients. Our
analysis furthermore represents a new approach for estimating best $M$-term
approximation errors by means of coefficient bounds, without using Stechkin
inequality. The framework we propose for analyzing asymptotic truncation errors
is based on an extension of the underlying multi-index set into a continuous
domain, and then an approximation of the cardinality (number of integer
multi-indices) by its Lebesgue measure. Several types of isotropic and
anisotropic (weighted) multi-index sets are explored, and rigorous proofs
reveal sharp asymptotic error estimates in which we achieve sub-exponential
convergence rates of the form $M \exp(-(\kappa M)^{1/N})$, with $\kappa$ a
constant depending on the shape and size of multi-index sets) with respect to
the total number of degrees of freedom. Through several theoretical examples,
we explicitly derive the constant $\kappa$ and use the resulting sharp bounds
to illustrate the effectiveness of Legendre over Taylor approximations, as well
as compare our rates of convergence with current published results.
Computational evidence complements the theory and shows the advantage of our
generalized framework compared to previously developed estimates.
