We introduce an algorithm for the least squares solution of a rectangular linear system Ax = b, in which A may be arbitrarily ill-conditioned. We assume that a complementary matrix Z is known such that A − AZ * A is numerically low rank. Loosely speaking, Z * acts like a generalized inverse of A up to a numerically low rank error. We give several examples of (A, Z) combinations in function approximation, where we can achieve high-order approximations in a number of nonstandard settings: the approximation of functions on domains with irregular shapes, weighted least squares problems with highly skewed weights, and the spectral approximation of functions with localized singularities. The algorithm is most efficient when A and Z * have fast matrix-vector multiplication and when the numerical rank of A − AZ * A is small.

Changes in food quality can play a substantial role in the vulnerability of hosts to infectious diseases. In this study, we focused on the genetic differentiation of the water flea Daphnia magna towards food of different quality (by manipulating C:N:P ratios) and its impact on the interaction with a virulent infectious disease, "White Fat Cell Disease (WFCD)". Via a resurrection ecology approach, we isolated two Daphnia subpopulations from different depths in a sediment core, which were exposed to parasites and a nutrient ratio gradient in a common garden experiment. Our results showed a genetic basis for sensitivity towards food deprivation. Both fecundity and host survival was differently affected when fed with low-quality food. This strongly impacted the way both subpopulations interacted with this parasite. A historical reconstruction of nutrient changes in a sediment core reflected an increase in organic material and phosphorus concentration (more eutrophic conditions) over time in the studied pond. These results enable us to relate patterns of genetic differentiation in sensitivity towards food deprivation to an increasing level of eutrophication of the subpopulations, which ultimately impacts parasite virulence effects. This finding was confirmed via a dynamic energy budgets (DEB), in which energy was partitioned for the host and the parasite. The model was tailored to our study by integrating (1) increased growth and a fecundity shift in the host upon parasitism and (2) differences of food assimilation in the subpopulations showing that a reduced nutrient assimilation resulted in increased parasite virulence. The combination of our experiment with the DEB model shows that it is important to consider genetic diversity when studying the impact of nutritional stress on species interactions, especially in the context of changing environments and emerging infectious diseases.

Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighbourhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series which is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the L 2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the L 2 -norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson-and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.

Functions on a bounded domain in scientific computing are often approximated using piecewise polynomial approximations on meshes that adapt to the shape of the geometry. We study the problem of function approximation using splines on a simple, regular grid that is defined on a bounding box. This approach allows the use of high order and highly structured splines as a basis for piecewise polynomials. The methodology is analogous to that of Fourier extensions, using Fourier series on a bounding box, which leads to spectral accuracy for smooth functions. However, Fourier extension approximations involve solving a highly illconditioned linear system, and this is an expensive step. The computational complexity of recent algorithms is O N log 2 (N ) in 1 dimension and O N 2 log 2 (N ) in two dimensions. We show that the compact support of B-splines enables improved complexity for multivariate approximations, namely O(N ) in 1-D, O N 3 2 in 2-D and more generally O N 3(d−1) d in d-D with d > 1. This comes at the cost of achieving only algebraic rates of convergence. Our statements are corroborated with numerical experiments and Julia code is available.

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