Norm approximation by Taylor polynomials in Hardy and Bergman spaces

For positive [Formula: see text] and real [Formula: see text] let [Formula: see text] denote the weighted Bergman spaces of the unit ball [Formula: see text] introduced in [R. Zhao and K. Zhu, Theory of Bergman Spaces on the Unit Ball in [Formula: see text], Mémoires de la Société Mathématique de France, Vol. 115 (2008)]. It is well known that, at least in the case [Formula: see text], all functions in [Formula: see text] can be approximated in norm by their Taylor polynomials if and only if [Formula: see text]. In this paper we show that, for [Formula: see text] with [Formula: see text], we always have [Formula: see text] as [Formula: see text], where [Formula: see text] and [Formula: see text] is the [Formula: see text]th Taylor polynomial of [Formula: see text]. We also show that for every [Formula: see text] in the Hardy space [Formula: see text], [Formula: see text], we always have [Formula: see text] as [Formula: see text], where [Formula: see text]. This generalizes and improves a result in [J. McNeal and J. Xiong, Norm convergence of partial sums of [Formula: see text] functions, Internat. J. Math. 29 (2018) 1850065, 10 pp.].

Convergence rates of high dimensional Smolyak quadrature

We analyse convergence rates of Smolyak integration for parametric maps u: U → X taking values in a Banach space X, defined on the parameter domain U = [−1,1]N. For parametric maps which are sparse, as quantified by summability of their Taylor polynomial chaos coefficients, dimension-independent convergence rates superior to N-term approximation rates under the same sparsity are achievable. We propose a concrete Smolyak algorithm to a priori identify integrand-adapted sets of active multiindices (and thereby unisolvent sparse grids of quadrature points) via upper bounds for the integrands’ Taylor gpc coefficients. For so-called “(b,ε)-holomorphic” integrands u with b∈lp(∕) for some p ∈ (0, 1), we prove the dimension-independent convergence rate 2/p − 1 in terms of the number of quadrature points. The proposed Smolyak algorithm is proved to yield (essentially) the same rate in terms of the total computational cost for both nested and non-nested univariate quadrature points. Numerical experiments and a mathematical sparsity analysis accounting for cancellations in quadratures and in the combination formula demonstrate that the asymptotic rate 2/p − 1 is realized computationally for a moderate number of quadrature points under certain circumstances. By a refined analysis of model integrand classes we show that a generally large preasymptotic range otherwise precludes reaching the asymptotic rate 2/p − 1 for practically relevant numbers of quadrature points.

Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation

A new way to compute the Taylor polynomial of a matrix exponential is presented which reduces the number of matrix multiplications in comparison with the de-facto standard Paterson-Stockmeyer method for polynomial evaluation. Combined with the scaling and squaring procedure, this reduction is sufficient to make the Taylor method superior in performance to Padé approximants over a range of values of the matrix norms. An efficient adjustment to make the method robust against overscaling is also introduced. Numerical experiments show the superior performance of our method to have a similar accuracy in comparison with state-of-the-art implementations, and thus, it is especially recommended to be used in conjunction with Lie-group and exponential integrators where preservation of geometric properties is at issue.