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Infinite-dimensional, holomorphic functions have been studied in detail overthe last several decades, due to their relevance to parametric differentialequations and computational uncertainty quantification. The approximation ofsuch functions from finitely many samples is of particular interest, due to thepractical importance of constructing surrogate models to complex mathematicalmodels of physical processes. In a previous work, [5] we studied theapproximation of so-called Banach-valued,$(\boldsymbol{b},\varepsilon)$-holomorphic functions on theinfinite-dimensional hypercube $[-1,1]^{\mathbb{N}}$ from $m$ (potentiallyadaptive) samples. In particular, we derived lower bounds for the adaptive$m$-widths for classes of such functions, which showed that certain algebraicrates of the form $m^{1/2-1/p}$ are the best possible regardless of thesampling-recovery pair. In this work, we continue this investigation byfocusing on the practical case where the samples are pointwise evaluationsdrawn identically and independently from a probability measure. Specifically,for Hilbert-valued $(\boldsymbol{b},\varepsilon)$-holomorphic functions, weshow that the same rates can be achieved (up to a small polylogarithmic oralgebraic factor) for essentially arbitrary tensor-product Jacobi(ultraspherical) measures. Our reconstruction maps are based on least squaresand compressed sensing procedures using the corresponding orthonormal Jacobipolynomials. In doing so, we strengthen and generalize past work that hasderived weaker nonuniform guarantees for the uniform and Chebyshev measures(and corresponding polynomials) only. We also extend various best $s$-termpolynomial approximation error bounds to arbitrary Jacobi polynomialexpansions. Overall, we demonstrate that i.i.d.\ pointwise samples arenear-optimal for the recovery of infinite-dimensional, holomorphic functions.

Optimal approximation of infinite-dimensional holomorphic functions II: recovery from i.i.d. pointwise samples