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This chapter offers a survey of our recent findings concerning error estimates in Gaussian-type quadrature formulas for analytic functions defined on confocal ellipses. Our recent research has focused on a methodology for numerically assessing error terms within Gaussian quadrature formulas, particularly examining a special case involving the Jacobi weight function \(\omega (t) = (1-t)^{\alpha }(1+t)^{\beta }\) , \(\alpha ,\beta >-1\) . We initially addressed the scenario where both \(\alpha \) and \(\beta \) are zero, corresponding to the Legendre weight function. Subsequently, we expanded this investigation to encompass instances where \(\alpha \) and \(\beta \) represent any natural number, thereby relating to the Gegenbauer weight function.

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Error Estimates of Gaussian-type Quadratures—A Survey

  • 章节

  • 发表日期：

    2025年

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  • Davorka R. Jandrlić,
  • Aleksandar V. Pejčev,
  • Miodrag M. Spalević

摘要

This chapter offers a survey of our recent findings concerning error estimates in Gaussian-type quadrature formulas for analytic functions defined on confocal ellipses. Our recent research has focused on a methodology for numerically assessing error terms within Gaussian quadrature formulas, particularly examining a special case involving the Jacobi weight function \(\omega (t) = (1-t)^{\alpha }(1+t)^{\beta }\) , \(\alpha ,\beta >-1\) . We initially addressed the scenario where both \(\alpha \) and \(\beta \) are zero, corresponding to the Legendre weight function. Subsequently, we expanded this investigation to encompass instances where \(\alpha \) and \(\beta \) represent any natural number, thereby relating to the Gegenbauer weight function.