C. de Boor

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On Ascertaining Inductively the Dimension of the Joint Kernel of Certain Commuting Linear Operators

Advances in Applied Mathematics, 1996

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Approximation by Polynomials

Series on Number Theory and Its Applications, 2016

2. The Weierstrass approximation theorem 3. Estimates for the Bernstein polynomials 4. Weierstras... more 2. The Weierstrass approximation theorem 3. Estimates for the Bernstein polynomials 4. Weierstrass' original proof 5. The Stone-Weierstrass approximation theorem 6. Chebyshev's theorems 7. Approximation by polynomials and trigonometric polynomials 8. The nonexistence of a continuous linear projection 9. Approximation of functions of higher regularity 10. Inverse theorems References Introductory remarks These notes comprise the main part of a course on approximation theory presented at Uppsala University in the Fall of 1998, viz. the part on polynomial approximation. The material is mainly classical. As sources I used Cheney [1966], Dzjadyk [1977], Korovkin [1959], and Lorentz [1953], as well as papers listed in the bibliography. The emphasis is on explaining the main ideas behind the most important techniques. The last part of the course was on rational approximation and is not included here. I followed mainly Cheney [1966, Chapter 5, pp. 150-167]. I also discussed Padé approximation briefly, following Cheney [1966, Chapter 5, pp. 173-177] and the introduction in Rudälv [1998]. I am grateful to Tsehaye Kahsu Araaya for remarks to the manuscript. 2 a k 2 − d(0, A) 2 , since 1 2 a j + 1 2 a k belongs to A in view of the convexity. We see that the right-hand side tends to zero as j, k → ∞. This implies that (a j) is a Cauchy sequence, and it must therefore have a limit in H. The limit cannot depend on the sequence, for if we take two sequences (a j) and (b j) and mix them, the new sequence (a 0 , b 0 , a 1 , b 1 , a 2 , ...) must converge by the same argument. We call the limit π(0); by translation we define π(x) ∈ A. Exercise 1.3. Prove that the mapping π: H → A is continuous; more precisely that π(x) − π(y) x − y , x, y ∈ H. Prove also that the set A is contained in a half-space as soon as x / ∈ A: in the real case every a ∈ A must satisfy (a − x|π(x) − x) π(x) − x 2. What about the complex case? Exercise 1.4. Prove that if A is a closed linear subspace, then x−π(x) is orthogonal to π(x). Prove that we get two idempotent mappings π and I − π, and determine all possible relations between the subspaces ker π, ker(I − π), im π, im(I − π). So Hilbert space is an easy case where the best approximant is unique. However, there are other interesting cases when we can prove uniqueness of the best approximant. Now we may perhaps dare to say that approximation theory is the study of approximating sequences, best approximants and their uniqueness or nonuniqueness in cases where X is a space of interesting functions, and A is some subspace of nice functions, like polynomials, trigonomentric polynomials,... 2. The Weierstrass approximation theorem The best starting point for these lectures is the classical Weierstrass 1 approximation theorem. It says that for any continuous real-valued function f on the interval [0, 1] and any integer k 1 there is a polynomial p k such that |f (x) − p k (x)| 1/k for all

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Divided Differences

Starting with a novel definition of divided differences, this essay derives and discusses the bas... more

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Topics in Multivariate Approximation Theory

: Lectures delivered at the S.R.C. Numerical Analysis Summer School and Workshop at the Universit... more : Lectures delivered at the S.R.C. Numerical Analysis Summer School and Workshop at the University of Lancaster, England, July 19 - August 20, 1981. Topics include tensor products, multivariate polynomial interpolation, esp. Kergin Interpolation, and the recent developments of multivariate B-splines. (Author)

FFT as Nested Multiplication, with a Twist

SIAM Journal on Scientific and Statistical Computing, 1980

A simple , yet complete and detailed description of the Fast Fourier Tra nsform for general N is ... more A simple , yet complete and detailed description of the Fast Fourier Tra nsform for general N is given , with the a im of maki ng the un derlying idea quite apparent. To help with this didactic goal, a simple twist , i.e., a shifting of information from rows to columns during the calculations, is introduced which allows to give a simple meaning to intermediate results and assures that the final results need no further reordering .

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Spline Functions and Surfaces

,., 10e 4ot av a, For App o e ac-4* 10 74 331) A %r q :' C 2G5C3 1AGENCY USE CNLY (Leav'e Mlan712... more

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Appendix to ‘Splines and Histograms’ by I.J. Schoenberg

International series of numerical mathematics, 1973

It is the purpose of this appendix to point out the tensor product structure of the construction ... more It is the purpose of this appendix to point out the tensor product structure of the construction in part II, thus relating it to known results concerning interpolation by tensor product splines. The additional algebraic machinery required seems worth knowing since one may use it to advantage for the efficient calculation of bivariate histosplines.

Total Positive of the Spine Collocation Matrix

Indiana University Mathematics Journal

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Ideal interpolation: Mourrain's condition vs. D-invariance

Approximation and Probability, 2006

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Gauss Elimination by Segments and Multivariate Polynomial Interpolation

Approximation and Computation: A Festschrift in Honor of Walter Gautschi, 1994

The construction of a polynomial interpolant to data given at nite pointsets in IR d (or, most ge... more The construction of a polynomial interpolant to data given at nite pointsets in IR d (or, most generally, to data speci ed by nitely many linear functionals) is considered, with special emphasis on the linear system to be solved. Gauss elimination by segments (i.e., by groups of columns rather than by columns) is proposed as a reasonable means for obtaining a description of all solutions and for seeking out solutions with`good' properties. A particular scheme, due to Amos Ron and the author, for choosing a particular polynomial interpolating space in dependence on the given data points, is seen to be singled out by requirements of degree-reduction, dilation-invariance, and a certain orthogonality requirement. The close connection, between this particular construction of a polynomial interpolant and the construction of an H-basis for the ideal of all polynomials which vanish at the given data points, is also discussed.

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Computational Aspects of Optimal Recovery

Optimal Estimation in Approximation Theory, 1977

This paper offers a Fortran program for the calculation of the optimal recovery scheme of Micchel... more This paper offers a Fortran program for the calculation of the optimal recovery scheme of Micchelli, Rivlin and Winograd [15]. A short derivation of that recovery scheme is given first, in order to make the paper selfcontained and also to provide an alternative to the original derivation in [15]. For the latter reason, a derivation of the related envelope construction of Gaffney and Powell [8] is also given. From a computational point of view, these schemes are special cases of the following computational problem: to construct an extension with prescribed norm of a linear functional on some finite dimensional linear subspace to all of IL1[a,b].

Box splines defined

Applied Mathematical Sciences, 1993

In this chapter, we give various equivalent definitions of the box spline, and derive its basic p... more

Tensor Products and Commutative Matrices

Journal of the Society for Industrial and Applied Mathematics, 1964

Convergence of cubic spline interpolation with the not-a-knot condition

It is shown that cubic spline interpolation with the not-a-knot side condition converges to any C... more

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What are the limits of Lagrange projectors?

The behavior of interpolants as interpolation sites coalesce is explored in the suitably restrict... more The behavior of interpolants as interpolation sites coalesce is explored in the suitably restricted context of multivariate polynomial interpolation. A Lagrange projector P τ is, by definition, a linear map on some linear space X of (scalar-valued) functions on some domain T that associates each function f ∈ X with the unique element P τ f in ran P τ 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 .

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How does Agee’s smoothing method work

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B-Splines without Divided Differences

ABSTRACT

Collocation at Gaussian Points

SIAM Journal on Numerical Analysis, 1973

Approximations to an isolated solution of an mth order nonlinear ordinary differential equation w... more Approximations to an isolated solution of an mth order nonlinear ordinary differential equation with m linear side conditions are determined. These approximations are piecewise polynomial functions of order $m + k$ (degree less than $m + k$) possessing $m - 1$ continuous derivatives. They are determined by collocation, i.e., by the requirement that they satisfy the differential equation at k points in each subinterval, together with the m side conditions. If the solution of the sufficiently smooth differential equation problem has $m + 2k$ continuous derivatives and if the collocation points are the zeroes of the kth Legendre polynomial relative to each subinterval, then the global error in these approximations is $O(| \Delta |^{m + k} )$ with $| \Delta |$ the maximum subinterval length. Moreover, at the ends of each subinterval, the approximation and its first $m - 1$ derivatives are $O(| \Delta |^{2k} )$ accurate. The solution of the nonlinear collocation equations may itself be approximated by solving ...

The least solution for the polynomial interpolation problem

Mathematische Zeitschrift, 1992

We consider the following problem: given a subspace Λ of the dual Π of the space Π of svariate po... more We consider the following problem: given a subspace Λ of the dual Π of the space Π of svariate polynomials, find a space P ⊂ Π which is correct for Λ in the sense that each continuous linear functional on Λ can be interpolated by a unique p ∈ P . We provide a map, Λ → Λ ↓ ⊂ Π, which we call the least map, that solves this interpolation problem and give a comprehensive discussion of its properties. This least solution, Λ ↓ , is a homogeneous space and is shown to have minimal degree among all possible solutions. It is the unique minimal degree solution which is dual (in a natural sense) to all minimal degree solutions. It also interacts nicely with various maps applied to Λ, such as convolution, translation, change of variables, and, particularly, differentiation. Our approach is illustrated by detailed examples, concerning finite-dimensional Λ's spanned by point-evaluations or line integrals. Methods which facilitate the identification of the least solution are established. The paper is complemented by [BR3], in which an algorithmic approach for obtaining Λ ↓ is presented whose computational aspects are detailed.

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Approximation power of smooth bivariate PP functions

Mathematische Zeitschrift, 1988

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