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A multivariate divided differences
 Approximation Theory VIII
, 1995
"... “Est enim fere ex pulcherrimis quæ solvere desiderem.” (It is among the most beautiful I could desire to solve.) [Newton 1676] Abstract. Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate) ..."
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“Est enim fere ex pulcherrimis quæ solvere desiderem.” (It is among the most beautiful I could desire to solve.) [Newton 1676] Abstract. Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate)
What Are the Limits of Lagrange Projectors?
"... Abstract. The behavior of interpolants as interpolation sites coalesce is explored in the suitably restricted context of multivariate polynomial interpolation. A Lagrange projector Pτ is, by definition, a linear map on some linear space X of (scalarvalued) functions on some domain T that associates ..."
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Abstract. The behavior of interpolants as interpolation sites coalesce is explored in the suitably restricted context of multivariate polynomial interpolation. A Lagrange projector Pτ is, by definition, a linear map on some linear space X of (scalarvalued) functions on some domain T that associates each function f ∈ X with the unique element Pτf in ranPτ that agrees with f at a given (finite) set τ. This note concerns the nature of limits of such Lagrange projectors, the limits taken in the (bounded) pointwise sense, with respect to some norm on X, and with the cardinality n: = #τ of the set τ of interpolation sites kept fixed. Thus, this note does not deal with the convergence of an interpolation process as the interpolation sites become become dense. Rather, the interest focuses on what might or might not happen as τ approaches some set σ with #σ < #τ. 1. Pointwise Limits of Linear Projectors of Finite Rank In this section, some basic facts concerning linear projectors and bounded pointwise convergence are recalled for the reader’s convenience. Any linear projector P of finite rank on the linear space X over the commutative field F with algebraic
On multivariate Newton interpolation at discrete Leja points
 Dolomites Research Notes on Approximation (DRNA), this issue. 39 De
"... The basic LU factorization with row pivoting, applied to a rectangular Vandermondelike matrix of an admissible mesh on a multidimensional compact set, extracts from the mesh the socalled Discrete Leja Points, and provides at the same time a Newtonlike interpolation formula. Working on the mesh, w ..."
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The basic LU factorization with row pivoting, applied to a rectangular Vandermondelike matrix of an admissible mesh on a multidimensional compact set, extracts from the mesh the socalled Discrete Leja Points, and provides at the same time a Newtonlike interpolation formula. Working on the mesh, we obtain also a good approximate estimate of the interpolation error. 1 Introduction. In the last two years, starting from the seminal paper of Calvi and Levenberg [8], it has been recognized that the socalled “admissible meshes ” play a central role in the construction of multivariate polynomial approximation processes on compact sets. This concept is essentially a matter of polynomial inequalities.
Ideal interpolation
, 2004
"... A linear interpolation scheme is termed ‘ideal’ when its errors form a polynomial ideal. The paper surveys basic facts about ideal interpolation and raises some questions. ..."
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A linear interpolation scheme is termed ‘ideal’ when its errors form a polynomial ideal. The paper surveys basic facts about ideal interpolation and raises some questions.
� � ��p�
"... �ÆÆ � a0,...,am� n + 1�����Æ yi = f(xi), i = 0,...,n. p(xi) = f(xi) i = 0,...,n. p��m��p(x) = a0 +a1x+···+amx m, am ̸ = 0. a0 +a1x0 +a2x 2 0 +···+amx n 0 = y0 (1) p(xi) = f(xi) i = 0,...,n. (2) Å � �n = m,Å����(2) pn(x) a0 +a1x1 +a2x 2 1 +···+amx n 1 = y1 a0 +a1xn +a2x 2 n +···+amx n n = yn. 1� = ..."
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�ÆÆ � a0,...,am� n + 1�����Æ yi = f(xi), i = 0,...,n. p(xi) = f(xi) i = 0,...,n. p��m��p(x) = a0 +a1x+···+amx m, am ̸ = 0. a0 +a1x0 +a2x 2 0 +···+amx n 0 = y0 (1) p(xi) = f(xi) i = 0,...,n. (2) Å � �n = m,Å����(2) pn(x) a0 +a1x1 +a2x 2 1 +···+amx n 1 = y1 a0 +a1xn +a2x 2 n +···+amx n n = yn. 1� = f(x) �� � ����n��ℓi� ℓi(xi) = 1�ℓi(xj) = 0, i ̸ = j. n∑ yiℓi(x) i=0 1 ℓi��Åℓi(x)�� � nÅ ��� � 2�ÆÆ���(1). x0,...,xi−1,xi+1,...,xn ℓi(x) = c(x−x0)···(x−xi−1)(x−xi+1)···(x−xn), �c����ℓi(xj) = ℓi(x) 1, 1 c = (xi −x0)···(xi −xi−1)(xi −xi+1)···(xi −xn). = (x−x0)···(x−xi−1)(x−xi+1)···(x−xn) (xi −x0)···(xi −xi−1)(xi −xi+1)···(xi −xn). �ω(x): = (x−x0)···(x−xn).Å ℓi� ω(x) ℓi(x) = (x−xi)ω ′(xi). ��LangrangeÆÆ� n ∑ ω(x) pn(x) = yi (x−xi)ω i=0 ′(xi). � � �1.�{xi: i = 0,...,n} [a,b]���n+1� � C[a,b].Å