Mariano Gasca and Thomas Sauer, Polynomial interpolation in several variables, in Multivariate polynomial interpolation, Adv. Comput. Math. 12 (2000), no. 4, 377-410.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....y; z) z z n ) z z 0 )T (x; y; z) But Tn is a polynomial of degree n, so that Tn 0. The analog of this proposition holds for higher dimensional sphere, as shown in [6] That Bezout s theorem can be used to establish the uniqueness of the interpolation is folklore (see, for example, [5]) We present the proposition here since its proof gives a prelude of the factorization that leads to our main result. The points in the above proposition have little symmetry on S , since no two latitudes have the same number of points. It is worthwhile to emphasis that the position of the ....

M. Gasca and T. Sauer, Polynomial interpolation in several variables, Adv. in Comput. Math., 12 (2000), 377-410.

....so that they have a common component. But q is irreducible, it follows that P (x) q(x)P 1 (x) which is (3.3) Using this process repeatedly proves the theorem. Bezout s theorem has been used in polynomial interpolation of several variables by many authors; see, for example, the recent survey [5] and the references there in. Theorem 3.7, however, does not seem to have been stated before. One natural question is whether Theorem 3.4 holds true for arbitrary points on the circles. Despite Theorem 3.7, we believe that this is not the case. The factorization theorem given in Theorem 1.3 is ....

M. Gasca and T. Sauer, Polynomial interpolation in several variables, Adv. in Comput. Math., 12 (2000), p. 377-410.

....the so called minimal interpolation. In this generality, the notion of Lagrange interpolation is as follows: for a given set of X and data ff(x i ) x i 2 Xg, nd a subspace P 2 , often depending on the point set X , such that there is a unique polynomial P 2 P that agrees with f on X (see [3, 5] and the references therein) In our case, we will de ne a polynomial subspace explicitly for the points that we specify. 2. Interpolation on the sphere and on the ball Polynomials on S that are symmetric with respect to x d 1 are related to polynomials . This fact has been used for ....

....2.1. In this example, the points on S are symmetric with respect to the equator. Evidently, it is possible to extend such a result to d dimensional. The use of the Bezout theorem in proving the poisedeness of interpolation is folklore. For interpolation by polynomials of d variables, see [5] for example. Let us mention that one set of interpolation points on S is given in [6] For S the points are located on n 1 latitudes, say S (z i ) i = 0; 1; n, and the k th latitude S (z k ) contains 2k 1 points. However, since the number of points on the circles are all ....

M. Gasca and T. Sauer, Polynomial interpolation in several variables, Adv. in Comput. Math., 12 (2000), p. 377-410.

....full rank. This shows that is a nontrivial polynomial in . Invoking the analytic function Lemma 2, is nonzero almost everywhere, except for a measure zero subset of . Remark 3: An alternative proof of Theorem 3 can be constructed by using the theory of Lagrange interpolation in several variables [5], 14] 16] The advantage of such an approach is that it affords geometric insight that facilitates the construction of full rank examples and counter examples. The disadvantage is that the proof requires a long and delicate argument. Proof of Corollary 1: It is again sufficient to consider the ....

M. Gasca and T. Sauer, "Polynomial interpolation in several variables," Adv. Comput. Math., vol. 12, no. 4, pp. 377--410, 2000.

....Matem atica Aplicada Fundamental Universidad de Valladolid Introduction. Polynomial interpolation in one variable has a well known and good theory. For multivariate polynomials, and even in case of bivariate polynomials, the situation becomes more difficult and different phenomena appear (see [7]) Polynomial interpolation in the subspace K[X;Y ] n of polynomials of degree n corresponds to a linear map L : K[X;Y ] n Gamma K N (1.2) Where the coordinates of L are evaluation of polynomials, or some partial derivatives, at prescribed points of K 2 . Lagrange interpolation corresponds ....

M. Gasca and T. Sauer. Polynomial interpolation in several variables. To appear in Adv.Comp.Math.

.... the so called blossoming approach, we refer to the tutorial of de Rose, Goldman, and Lounsbery [139] and the survey of Seidel [154] Triangulation methods were described in the survey of Schumaker [152] For interpolation by bivariate polynomials, we refer to the survey of Gasca and Sauer [76]. Finally, we note that a characterization (di erent from Theorem 2.1) of the smoothness of polynomial pieces on adjacent triangles, without using B ezier Bernstein techniques, was proved by Davydov, N urnberger, and Zeilfelder [63] 3 Dimension of spline spaces In this section, we summarize ....

M. Gasca and Th. Sauer, Polynomial interpolation in several variables, Advances in Comp. Math, to appear.

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Mariano Gasca and Thomas Sauer, Polynomial interpolation in several variables, in Multivariate polynomial interpolation, Adv. Comput. Math. 12 (2000), no. 4, 377-410.

No context found.

Mariano Gasca and Thomas Sauer, Polynomial interpolation in several variables, in Multivariate polynomial interpolation, Adv. Comput. Math. 12 (2000), no. 4, 377-410.

No context found.

M. Gasca and T. Sauer, Polynomial interpolation in several variables, Adv. Comput. Math. 12 (2000), 377-410.

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