Optimal stability of the Lagrange formula and conditioning of the Newton formula

A pointwise condition number associated to a representation of an interpolation operator is introduced. It is proved that the Lagrange formula is optimal with respect to this conditioning. For other representations of the interpolation operator, an upper bound for the conditioning is derived. A quantitative measure in terms of the Skeel condition number is used to compare the conditioning with the Lagrange representation. The conditioning of the Newton representation is considered for increasing nodes and for nodes in Leja order. For the polynomial Newton formula with n+1 equidistant nodes in increasing order, it is proved that 3n is the best uniform bound of its conditioning and it is attained at the last node. Numerical experiments are included

On the stability of the representation of finite rank operators

The stability of the representation of finite rank operators in terms of a basis is analyzed. A conditioning is introduced as a measure of the stability properties. This conditioning improves some other conditionings because it is closer to the Lebesgue function. Improved bounds for the conditioning of the Fourier sums with respect to an orthogonal basis are obtained, in particular, for Legendre, Chebyshev, and disk polynomials. The Lagrange and Newton formulae for the interpolating polynomial are also considered

Cubic pencils of lines and bivariate interpolation

AbstractCubic pencils of lines are classified up to projectivities. Explicit formulae for the addition of lines on the set of nonsingular lines of the pencils are given. These formulae can be used for constructing planar generalized principal lattices, which are sets of points giving rise to simple Lagrange formulae in bivariate interpolation. Special attention is paid to the irreducible nonsingular case, where elliptic functions are used in order to express the addition in a natural form

Optimal interval length for the collocation of the Newton interpolation basis

It is known that the Lagrange interpolation problem at equidistant nodes is ill-conditioned. We explore the influence of the interval length in the computation of divided differences of the Newton interpolation formula. Condition numbers are computed for lower triangular matrices associated to the Newton interpolation formula at equidistant nodes. We consider the collocation matrices L and PL of the monic Newton basis and a normalized Newton basis, so that PL is the lower triangular Pascal matrix. In contrast to L, PL does not depend on the interval length, and we show that the Skeel condition number of the (n + 1) × (n + 1) lower triangular Pascal matrix is 3n. The 8-norm condition number of the collocation matrix L of the monic Newton basis is computed in terms of the interval length. The minimum asymptotic growth rate is achieved for intervals of length 3

Central orderings for the Newton interpolation formula

The stability properties of the Newton interpolation formula depend on the order of the nodes and can be measured through a condition number. Increasing and Leja orderings have been previously considered (Carnicer et al. in J Approx Theory, 2017. https://doi.org/10.1016/j.jat.2017.07.005; Reichel in BIT 30:332–346, 1990). We analyze central orderings for equidistant nodes on a bounded real interval. A bound for conditioning is given. We demonstrate in particular that this ordering provides a more stable Newton formula than the natural increasing order. We also analyze of a central ordering with respect to the evaluation point, which provides low bounds for the conditioning. Numerical examples are included

A Marsden Type Identity for Periodic Trigonometric Splines

AbstractAn extension of Marsden′s identity for periodic trigonometric splines is obtained by a bivariate approach to that space. A basis of these spaces, whose elements have minimal or quasi-minimal support, is studied

Global maps of soil temperature.

Research in global change ecology relies heavily on global climatic grids derived from estimates of air temperature in open areas at around 2 m above the ground. These climatic grids do not reflect conditions below vegetation canopies and near the ground surface, where critical ecosystem functions occur and most terrestrial species reside. Here, we provide global maps of soil temperature and bioclimatic variables at a 1-km <sup>2</sup> resolution for 0-5 and 5-15 cm soil depth. These maps were created by calculating the difference (i.e. offset) between in situ soil temperature measurements, based on time series from over 1200 1-km <sup>2</sup> pixels (summarized from 8519 unique temperature sensors) across all the world's major terrestrial biomes, and coarse-grained air temperature estimates from ERA5-Land (an atmospheric reanalysis by the European Centre for Medium-Range Weather Forecasts). We show that mean annual soil temperature differs markedly from the corresponding gridded air temperature, by up to 10°C (mean = 3.0 ± 2.1°C), with substantial variation across biomes and seasons. Over the year, soils in cold and/or dry biomes are substantially warmer (+3.6 ± 2.3°C) than gridded air temperature, whereas soils in warm and humid environments are on average slightly cooler (-0.7 ± 2.3°C). The observed substantial and biome-specific offsets emphasize that the projected impacts of climate and climate change on near-surface biodiversity and ecosystem functioning are inaccurately assessed when air rather than soil temperature is used, especially in cold environments. The global soil-related bioclimatic variables provided here are an important step forward for any application in ecology and related disciplines. Nevertheless, we highlight the need to fill remaining geographic gaps by collecting more in situ measurements of microclimate conditions to further enhance the spatiotemporal resolution of global soil temperature products for ecological applications

Least Supported Bases and Local Linear Independence

this paper to generate some preference for the least supported bases by showing that a basis is least supported if and only if it is locally linearly independent. The concept of local linear independence introduced here coincides with the concept defined in spline spaces. So, for instance, the B-spline basis of the space of polynomial splines is least supported since the functions in this basis are locally linearly independent [cf. Theorem 4.18 of Schumaker (1981)] and also in the more general case of Tchebycheffian spline functions, the Tchebycheffian B-spline basis is least supported because these functions are locally linearly independent by Lemma 6.3 of Schumaker (1976). In multivariate spline spaces it is also usual that the basic functions are not minimally supported. For this reason, in Chui and He (1988) [see also p. 138 of Chui (1988)] it is introduced the concept of a quasiminimally supported function. On the other hand, certain box-splines form a least supported basis as follows from the proof of its local linear independence in Dahmen and Micchelli (1985). Also the bases of bivariate splines obtained in Carnicer and Pe~na (1993b) are least supported because, by Theorem 3.1 and Theorem 4.4 of the same paper, they are locally linearly independent. The layout of this paper is as follows: In Sect. 2, we begin by defining an order relation among the supports of bases and, if it exists, a least supported basis will be a basis whose support is the least element for the order relation. We shall also introduce the concept of quasiminimally supported function of degre