Bivariate Hermite Interpolation and Numerical Curves

In this paper, Hermite interpolation by bivariate algebraic polynomials of total degree ?nis considered. The interpolation parameters are the values of a function and its partial derivatives up to some ordern_ν−1 at the nodesz_ν=(x_ν, y_ν),ν=1, …, s, wheren_νis the multiplicity ofz_ν. The sequence ={n₁, …, n_s; n} of multiplicities associated with the degree of interpolating polynomials is investigated. Some results of the paper were announced in [GHS93].     

Extrema of Multivariate Polynomials, Their Gradients and Directional Derivatives

O. D. Kellogg in [K28] established a connection between the supremum norms of a homogeneous polynomial and its gradient. We completed this result with a characterization of extrema in the bivariate case in [H94a] and announced the extension for multivariate homogeneous polynomials. The extension is presented in this paper. The generalization of the completed result to the case of arbitrary multivariate polynomials is also given here. The bivariate case of this contains, as special cases, the Bernstein and Markoff inequalities. Next, a well-known equality, involving suprema over directions of derivatives, is discussed. This relation turned out to be a dual to the above result in the homogeneous case (see [H94b]). On this basis, the sets of directions of suprema of the equality are characterized. July 21, 1999. Date revised: May 22, 1999. Date accepted: June 14, 2000.     

The Gasca–Maeztu conjecture for n=5

An \(n\) -poised set in two dimensions is a set of nodes admitting unique bivariate interpolation with polynomials of total degree at most \(n\) . We are interested in poised sets with the property that all fundamental polynomials are products of linear factors. Gasca and Maeztu (Numer Math 39:1–14, 1982) conjectured that every such set necessarily contains \(n+1\) collinear nodes. Up to now, this had been confirmed only for \(n\le 4\) , the case \(n=4\) having been proved for the first time by Busch (Rev Un Mat Argent 36:33–38, 1990). In the present paper, we prove the case \(n=5\) with new methods that might also be useful in deciding the still open cases for \(n\ge 6\) .     

On the poisedness of Bojanov–Xu interpolation

In this paper, we consider the bivariate Hermite interpolation introduced by Bojanov and Xu [SIAM J. Numer. Anal. 39(5) (2002) 1780–1793]. The nodes of the interpolation with Π_2k-δ, where δ=0 or 1, are the intersection points of 2k+1 distinct rays from the origin with a multiset of k+1-δ concentric circles. Parameters are the values and successive radial derivatives, whenever the corresponding circle is multiple. The poisedness of this interpolation was proved only for the set of equidistant rays [Bojanov and Xu, 2002] and its counterparts with other conic sections [Hakopian and Ismail, East J. Approx. 9 (2003) 251–267]. We show that the poisedness of this (k+1-δ)(2k+1) dimensional Hermite interpolation problem is equivalent to the poisedness of certain 2k+1 dimensional Lagrange interpolation problems. Then the poisedness of Bojanov–Xu interpolation for a wide family of sets of rays satisfying some simple conditions is established. Our results hold also with above circles replaced by ellipses, hyperbolas, and pairs of parallel lines.Next a conjecture [Hakopian and Ismail, J. Approx. Theory 116 (2002) 76–99] concerning a poisedness relation between the Bojanov–Xu interpolation, with set of rays symmetric about x-axis, and certain univariate lacunary interpolations is established. At the end the poisedness for a wide class of lacunary interpolations is obtained.     

On a new property of <Emphasis Type="Italic">n</Emphasis>-poised and <Emphasis Type="Italic">G</Emphasis><Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets

In this paper we consider n-poised planar node sets, as well as more special ones, called G C _n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ? is called k-node line for a node set \(\mathcal X\) if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C _n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of \(\mathcal X\) is used by each node in \(\mathcal X\setminus M, \)meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C _n set \(\mathcal {X}\) is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C _n sets. At the end we present a conjecture concerning any k-node line.     

On a class of Hermite interpolation problems

In this paper we characterize sets of solvability of Hermite multivariate interpolation problems with the sum of multiplicities less than or equal to 2n + 1, where n is the degree of the polynomial space. This can be viewed as a natural generalization of a well-known result of Severi (1921). This revised version was published online in June 2006 with corrections to the Cover Date.     

A new proof of the Gasca–Maeztu conjecture for

In the Chung–Yao construction of poised nodes for bivariate polynomial interpolation [K.C. Chung, T.H. Yao, On lattices admitting unique Lagrange interpolations, SIAM J. Numer. Anal. 14 (1977) 735–743], the interpolation nodes are intersection points of some lines. The Berzolari–Radon construction [L. Berzolari, Sulla determinazione di una curva o di una superficie algebrica e su alcune questioni di postulazione, Lomb. Ist. Rend. 47 (2) (1914) 556–564; J. Radon, Zur mechanischen Kubatur, Monatsh. Math. 52 (1948) 286–300] seems to be more general, since in this case the nodes of interpolation lie (almost) arbitrarily on some lines. In 1982 Gasca and Maeztu conjectured that every poised set allowing the Chung–Yao construction is of Berzolari–Radon type. So far, this conjecture has been confirmed only for polynomial spaces of small total degree n≤4, the result being evident for n≤2 and not hard to see for n=3. For the case n=4 two proofs are known: one of J.R. Busch [J.R. Busch, A note on Lagrange interpolation in , Rev. Un. Mat. Argentina 36 (1990) 33–38], and another of J.M. Carnicer and M. Gasca [J.M. Carnicer, M. Gasca, A conjecture on multivariate polynomial interpolation, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) Ser. A Mat. 95 (2001) 145–153]. Here we present a third proof which seems to be more geometric in nature and perhaps easier. We also present some results for the case of n=5 and for general n which might be useful for later consideration of the problem.     

Two‐level preconditioners for serendipity finite element matrices

In this paper an approach to construct algebraic multilevel preconditioners for serendipity finite element matrices is presented. Two‐level preconditioners constructed in the paper allow to obtain multilevel preconditioners in serendipity case using multilevel preconditioners for linear finite element matrices. Copyright © 2006 John Wiley & Sons, Ltd.     

The Moore–Penrose inverse of block magic rectangles

As is known, a semi-magic square is an n?×?n matrix having the sum of entries in each row and each column equal to a constant. This note generalizes this notion and introduce a special class of block matrices called block magic rectangles. It is proved that the Moore–Penrose inverse of a block magic rectangle is also a block magic rectangle.