## Error bounds for Kronrod extension of generalizations of Micchelli-Rivlin quadrature formula for analytic functions

Rada M. Mutavdžić, Aleksandar V. Pejčev, and Miodrag M. Spalević

### Abstract

We consider the Kronrod extension of generalizations of the Micchelli-Rivlin quadrature formula for the Fourier-Chebyshev coefficients with the highest algebraic degree of precision. For analytic functions, the remainder term of these quadrature formulas can be represented as a contour integral with a complex kernel. We study the kernel on elliptic contours with foci at the points $\mp 1$ and the sum of semi-axes $\rho>1$ for the mentioned quadrature formulas. We derive $L^\infty$-error bounds and $L^1$-error bounds for these quadrature formulas. Finally, we obtain explicit bounds by expanding the remainder term. Numerical examples that compare these error bounds are included.

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### Key words

Kronrod extension of generalizations of the Micchelli-Rivlin quadrature formula, Chebyshev weight function of the first kind, error bound, remainder term for analytic functions, contour integral representation

### AMS subject classifications

65D32, 65D30, 41A55

### Links to the cited ETNA articles

 [12] Vol. 45 (2016), pp. 371-404 Sotirios E. Notaris: Gauss-Kronrod quadrature formulae - A survey of fifty years of research

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