Interpolation

Komornik, Vilmos

doi:10.1007/978-1-4471-7316-8_8

Vilmos Komornik⁹

Part of the book series: Springer Undergraduate Mathematics Series ((SUMS))

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Abstract

In order to shorten tedious astronomical computations, for centuries multiplication was transformed into addition by using the trigonometrical identity

$$\displaystyle{ 2\cos \alpha \cos \beta =\cos (\alpha +\beta ) +\cos (\alpha -\beta ). }$$

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Notes

1.
We say that a is a root of multiplicity m of a function f if
$$\displaystyle{ f^{(\,j)}(a) = 0\quad \text{for}\quad j = 0,\ldots,m - 1,\quad \text{and}\quad f^{(m)}(a)\neq 0. }$$
2.
We recall that in this book the symbol ≡ means that equality holds for all points where both sides are defined.
3.
In higher dimensions this equivalence holds only for domains having a sufficiently regular boundary. See, e.g., Grisvard [211].
4.
The general case follows by an affine change of variable.
5.
Observe that the main terms of p and ω eliminate each other.
6.
More precisely, we take the coefficients of x ^k−2 in p, q and of x ^k−1 in r.
7.
See the definition of multiplicity in the footnote of page p. 193.
8.
The integrals of the odd powers of x vanish.
9.
We recall that in this chapter the letter I denotes always a compact interval.
10.
We use the expressions “almost all” and “almost every” in Lebesgue’s sense. See, e.g., Burkill [69], Halmos [220], Rudin [430], or [285].
11.
Fejér used the expression step parabolas to emphasize that their graphs have horizontal tangents at the nodes.
12.
We use the relations $\arccos x_{k} = \frac{2k-1} {2n} \pi$.
13.
See, e.g., Laurent [316] for a general exposition.
14.
If (a_{ k}) is of order d, then for j > d the binomial coefficients vanish.
15.
The result of this exercise is the starting point of the finite element method for the numerical solution of partial differential equations. See, e.g., Ciarlet [107], Raviart–Thomas [405].

Bibliography

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P.R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., Princeton, N.J., 1950.
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V. Komornik, Lectures on Functional Analysis and the Lebesgue integral, Springer, Universitext, 2016.
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P.A. Raviart, J.M. Thomas, Introduction à l’analyse numérique des équations aux dérivées partielles, Masson, Paris, 1983.
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Department of Mathematics, University of Strasbourg, Strasbourg, France
Vilmos Komornik

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Komornik, V. (2017). Interpolation. In: Topology, Calculus and Approximation. Springer Undergraduate Mathematics Series. Springer, London. https://doi.org/10.1007/978-1-4471-7316-8_8

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DOI: https://doi.org/10.1007/978-1-4471-7316-8_8
Published: 05 April 2017
Publisher Name: Springer, London
Print ISBN: 978-1-4471-7315-1
Online ISBN: 978-1-4471-7316-8
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