Bivariate polynomial interpolation on the square at new nodal sets

As known, the problem of choosing ‘‘good’’ nodes is a central one in polynomial interpolation. While the problem is essentially solved in one dimension (all good nodal sequences are asymptotically equidistributed with respect to the arc-cosine metric), in several variables it still represents a substantially open question. In this work we consider new nodal sets for bivariate polynomial interpolation on the square. First, we consider fast Leja points for tensor-product interpolation. On the other hand, for interpolation in P 2n on the square we experiment four families of points which are (asymptotically) equidistributed with respect to the Dubiner metric, which extends to higher dimension the arc-cosine metric. One of them, nicknamed Padua points, gives numerically a Lebesgue constant growing like log square of the degree. 2004 Elsevier Inc. All rights reserved. 0096-3003/$ see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.07.001 q Work supported by the research project CPDA028291 ‘‘Efficient approximation methods for nonlocal discrete transforms’’ of the University of Padova, the ex-60% funds of the University of Verona, and by the GNCS-INdAM. * Corresponding author. E-mail address: [email protected] (S. De Marchi). 262 M. Caliari et al. / Appl. Math. Comput. 165 (2005) 261–274 1. Optimal and near-optimal interpolation points Let X R be compact. We call optimal polynomial interpolation points a set X N X of cardinality N, such that the Lebesgue constant KnðXNÞ 1⁄4 max x2X knðx;XN Þ; knðx;XNÞ :1⁄4 XN i1⁄41 j‘iðx;XN Þj; ð1Þ defined for all sets XN = {x1, . . . ,xN} X which are unisolvent for polynomial interpolation of degree n, attains its minimum on XN 1⁄4 X N . Here, kn(x;XN) is the Lebesgue function of XN, the ‘i are the fundamental Lagrange polynomials of degree n, and N is the dimension of the corresponding polynomial space, i.e. N 1⁄4 nþd d , or N = (n + 1) for the tensor-product case (cf. e.g. [2,4,10]). To be more precise, the fundamental Lagrange polynomials are defined as the ratio ‘iðx;XN Þ 1⁄4 VDM X ðiÞ N VDMðXN Þ ; ð2Þ where VDM denotes the Vandermonde determinants with respect to any given basis of the corresponding polynomial space, and where X ðiÞ N represents the set XN in which x replaces xi. It comes easy to see that tensor-product Lagrange polynomials are simply the product of univariate Lagrange polynomials. As well-known optimal points are not known explicitly, therefore in applications people consider near-optimal points, i.e. roughly speaking, points whose Lebesgue constant increases asymptotically like the optimal one. Moreover, letting En(XN) = kf Pnk1,X, where Pn is the interpolating polynomial of degree 6 n on X of a given continuous function f, and E n 1⁄4 kf P nk1;X the best uniform approximation error, then EnðXN Þ6 ð1þ KnðXN ÞÞE n; which represents an estimate for the interpolation error. Thus, near-optimal nodes minimize also (asymptotically) the interpolation error. In the one-dimensional case, as well-known, Chebyshev, Fekete, Leja as well as the zeros of Jacobi orthogonal polynomials are near-optimal points for polynomial interpolation, and their Lebesgue constants increase logarithmically in the dimension N of the corresponding polynomial space (cf. [5,13]). All these points have asymptotically the arc-cosine distribution, that is they are asymptotically equidistributed w.r.t. the arc-cosine metric. We now recall the definition of two important univariate nodal sets: it Fekete and Leja points. Definition 1. Given XN = {x1, . . . ,xN} [a,b] let VDMðXN Þ1⁄4detðx j i Þ16i;j6N be the classical Vandermonde determinant. The Fekete points are the set F = {f1, ... , fN} such that M. Caliari et al. / Appl. Math. Comput. 165 (2005) 261–274 263 jVDMðF NÞj 1⁄4 max XN 1⁄2a;b jVDMðXN Þj: Definition 2. Let k1 arbitrarily chosen in [a,b]. The points ks2 [a,b], s = 2, . . . ,N, such that Ys 1 k1⁄41 jks kkj 1⁄4 max x21⁄2a;b Ys 1