### A Characterization of Multivariate Quasi-interpolation Formulas and its Applications.

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We compare the yields of two methods to obtain Bernstein type pointwise estimates for the derivative of a multivariate polynomial in a domain where the polynomial is assumed to have sup norm at most 1. One method, due to Sarantopoulos, relies on inscribing ellipses in a convex domain K. The other, pluripotential-theoretic approach, mainly due to Baran, works for even more general sets, and uses the pluricomplex Green function (the Zaharjuta-Siciak extremal function). When the inscribed ellipse method...

La generalización de las fórmulas de interpolación de Lagrange y Newton a varias variables es uno de los temas habituales de estudio en interpolación polinómica. Dos clases de configuraciones geométricas particularmente interesantes en el plano fueron obtenidas por Chung y Yao en 1978 para la fórmula de Lagrange y por Gasca y Maeztu en 1982 para la de Newton. Estos últimos autores conjeturaron que toda configuración de la primera clase es de la segunda, y probaron que el recíproco no es cierto....

We give an elementary proof of the product formula for the multivariate transfinite diameter using multivariate Leja sequences and an identity on vandermondians.

A given set W = W X of n-variable class C 1 functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum $\sum _{\chi}\left({\int}_{{\mathbb{R}}^{n}}\nabla f\xb7\nabla {W}_{\chi}^{*}\right){W}_{\chi}$ converges to f with respect to the norm ${\u2225\nabla (\xb7)\u2225}_{{L}^{2}\left({\mathbb{R}}^{n}\right)}$ . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = W x of compactly supported class C 2−ɛ functions on ℝn such that [...]...

There are two grounds the spline theory stems from – the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We use the general variational approach called $\mathrm{\mathit{s}\mathit{m}\mathit{o}\mathit{o}\mathit{t}\u210e\mathit{i}\mathit{n}\mathit{t}\mathit{e}\mathit{r}\mathit{p}\mathit{o}\mathit{l}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}}$ introduced by Talmi and Gilat and show that it covers not only the cubic spline and its 2D and 3D analogues but also the well known tension spline (called also spline...

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal residual method with a measure of the residual corresponding to the error in a specified solution norm. The residual norm can be designed such that the resulting low-rank approximations are optimal with respect to particular norms of interest, thus allowing to take...