Almost-interpolatory Chebyshev quadrature

Author:
K. Salkauskas

Journal:
Math. Comp. **27** (1973), 645-654

MSC:
Primary 41A55; Secondary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1973-0340908-0

MathSciNet review:
0340908

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Abstract: The requirement that a Chebyshev quadrature formula have distinct real nodes is not always compatible with the requirement that the degree of precision of an *n*-point formula be at least equal to *n*. This condition may be expressed as ${\left \| d \right \|_\nu } = 0,1 \leqq p$, where $d = ({d_1}, \cdots ,{d_n})$ with \[ {d_j} = \frac {{{\mu _0}(\omega )}}{n}\sum \limits _{i = 1}^n {x_i^j - {\mu _j}(\omega ),\quad j = 1,2, \cdots ,n,} \] ${\mu _j}(\omega ),j = 0,1, \cdots$, are the moments of the weight function $\omega$ used in the quadrature, and ${x_1}, \cdots ,{x_n}$ are the nodes. In those cases when ${\left \| d \right \|_2}$ does not vanish for a real choice of nodes, it has been proposed that a real minimizer of ${\left \| d \right \|_2}$ be used to supply the nodes. It is shown in this paper that, in such cases, minimizers of ${\left \| d \right \|_p},1 \leqq p < \infty$, always lead to formulae that are degenerate in the sense that the nodes are not all distinct. The results are valid for a large class of weight functions.

- R. E. Barnhill, J. E. Dennis Jr., and G. M. Nielson,
*A new type of Chebyshev quadrature*, Math. Comp.**23**(1969), 437–441. MR**242367**, DOI https://doi.org/10.1090/S0025-5718-1969-0242367-4
S. N. Bernstein, "On quadrature formulas of Cotes and Chebyshev," - Witold Hurewicz and Henry Wallman,
*Dimension Theory*, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR**0006493** - Vladimir Ivanovich Krylov,
*Approximate calculation of integrals*, The Macmillan Co., New York-London, 1962, 1962. Translated by Arthur H. Stroud. MR**0144464**

*Dokl. Akad. Nauk SSSR*, v. 14, 1937, pp. 323-326. (Russian)

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Keywords:
Chebyshev quadrature,
minimum norm rules

Article copyright:
© Copyright 1973
American Mathematical Society