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Mathematics of Computation

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

 

 

Chebyshev-type quadrature and partial sums of the exponential series


Author: Arno Kuijlaars
Journal: Math. Comp. 64 (1995), 251-263
MSC: Primary 65D32; Secondary 41A55
DOI: https://doi.org/10.1090/S0025-5718-1995-1250771-6
MathSciNet review: 1250771
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Abstract: Chebyshev-type quadrature for the weight functions

$\displaystyle {w_a}(t) = \frac{{1 - at}}{{\pi \sqrt {1 - {t^2}} }},\quad - 1 < t < 1,\quad - 1 < a < 1,$

is related to a problem concerning partial sums of the exponential series, namely the problem to extend the nth partial sum to a polynomial of degree 2N having all zeros on the circle $ \vert z\vert = \vert a\vert N$. Using this connection, we show that the minimal number N of nodes needed for Chebyshev-type quadrature of degree n for $ {w_a}(t)$ satisfies an inequality $ {C_1}n \leq N \leq {C_2}n$ with positive constants $ {C_1},{C_2}$. As an application we prove that the minimal number N of nodes for Chebyshev-type quadrature of degree n on a torus embedded in $ {{\mathbf{R}}^3}$ satisfies an inequality $ {C_1}{n^2} \leq N \leq {C_2}{n^2}$.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1995-1250771-6
Keywords: Chebyshev-type quadrature, partial sums, distribution of zeros, multidimensional quadrature
Article copyright: © Copyright 1995 American Mathematical Society