  Motivated by nonlinear approximation results for classes of parametric
partial differential equations (PDEs), we seek to better understand so-called
library approximations to analytic functions of countably infinite number of
variables. Rather than approximating a function of interest in a single space,
a library approximation uses a collection of spaces and the best space may be
chosen for any point in the domain. In the setting of this paper, we use a
specific library which consists of local Taylor approximations on sufficiently
small rectangular subdomains of the (rescaled) parameter domain,
$Y:=[-1,1]^\mathbb{N}$. When the function of interest is the solution of a
certain type of parametric PDE, recent results (Bonito et al, 2020,
arXiv:2005.02565) prove an upper bound on the number of spaces required to
achieve a desired target accuracy. In this work, we prove a similar result for
a more general class of functions with anisotropic analyticity. In this way we
show both where the theory developed in (Bonito et al 2020) depends on being in
the setting of parametric PDEs with affine diffusion coefficients, and also
expand the previous result to include more general types of parametric PDEs.
