Chebyshev-type quadrature and partial sums of the exponential series
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- by Arno Kuijlaars PDF
- Math. Comp. 64 (1995), 251-263 Request permission
Abstract:
Chebyshev-type quadrature for the weight functions \[ {w_a}(t) = \frac {{1 - at}}{{\pi \sqrt {1 - {t^2}} }},\quad - 1 < t < 1,\quad - 1 < a < 1,\] 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 $|z| = |a|N$. 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 ${C_1}{n^2} \leq N \leq {C_2}{n^2}$.References
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Additional Information
- © Copyright 1995 American Mathematical Society
- 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