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Mathematics of Computation

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

 

 

The error norm of certain Gaussian quadrature formulae


Author: G. Akrivis
Journal: Math. Comp. 45 (1985), 513-519
MSC: Primary 65D32
DOI: https://doi.org/10.1090/S0025-5718-1985-0804940-0
MathSciNet review: 804940
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Abstract: We consider Gauss quadrature formulae $ {Q_n}$, $ n \in {\mathbf{N}}$, approximating the integral $ I(f): = \smallint _{ - 1}^1w(x)f(x)\;dx$, $ w = W/{p_i}$, $ i = 1,2$, with $ W(x) = {(1 - x)^\alpha }{(1 + x)^\beta }$, $ \alpha ,\beta = \pm 1/2$ and $ {p_1}(x) = 1 + {a^2} + 2ax$, $ {p_2}(x) = (2b + 1){x^2} + {b^2}$, $ b > 0$. In certain spaces of analytic functions the error functional $ {R_n}: = I - {Q_n}$ is continuous. In [1] and [2] estimates for $ \left\Vert {{R_n}} \right\Vert$ are given for a wide class of weight functions. Here, for a restricted class of weight functions, we calculate the norm of $ {R_n}$ explicitly.


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DOI: https://doi.org/10.1090/S0025-5718-1985-0804940-0
Article copyright: © Copyright 1985 American Mathematical Society