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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Almost-interpolatory Chebyshev quadrature

Author: K. Salkauskas
Journal: Math. Comp. 27 (1973), 645-654
MSC: Primary 41A55; Secondary 65D30
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\Vert d \right\Vert _\nu} = 0,1 \leqq p$, where $ d = ({d_1}, \cdots ,{d_n})$ with

$\displaystyle {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\Vert d \right\Vert _2}$ does not vanish for a real choice of nodes, it has been proposed that a real minimizer of $ {\left\Vert d \right\Vert _2}$ be used to supply the nodes. It is shown in this paper that, in such cases, minimizers of $ {\left\Vert d \right\Vert _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.

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Keywords: Chebyshev quadrature, minimum norm rules
Article copyright: © Copyright 1973 American Mathematical Society