On stratified extensions of Gauss–Laguerre and Gauss–Hermite quadrature formulas

Degree optimal average quadrature rules for the generalized Hermite weight function

“…Note that when γ = 0, the modified anti-Gauss and average rules given by (9) and (11) agree with the anti-Gauss and average rules introduced by Laurie in [13]. Ehrich [6] investigated the properties of (9) and (11) for the Laguerre and classical Hermite weight functions, and he obtained the degree optimal average rules for these weights. Throughout this paper the error of anti-Gauss rule (9) will be denoted bỹ…”

“…As a different approach, Laurie [13] introduced the so-called antiGauss rules and corresponding average rules to estimate the error of the Gauss quadrature rules. The anti-Gauss and average rules and their modified versions introduced by Ehrich [6], which will be defined below, exist for any nonnegative weight function.…”

“…This means that the degree of this rule may be greater than 2n + 1 for some value(s) of γ. Ehrich [6] proved that for the classical Hermite weight function there exists a unique value of γ, namely γ = 1/n, such that the corresponding modified average rule has the precise degree of exactness 2n + 3. He also obtained a similar result for the generalized Laguerre measure [6,Theorem 3].…”

“…This means that the degree of this rule may be greater than 2n + 1 for some value(s) of γ. Ehrich [6] proved that for the classical Hermite weight function there exists a unique value of γ, namely γ = 1/n, such that the corresponding modified average rule has the precise degree of exactness 2n + 3. He also obtained a similar result for the generalized Laguerre measure [6,Theorem 3]. The main purpose of this paper is to determine the modified average rules of the highest possible degree associated with the generalized Hermite weight function (8), and to investigate the properties of these and related modified anti-Gauss and average rules.…”

On generalized averaged Gaussian formulas

“…The more general definition in (1.3) has been used by Ehrich [4] to construct modified formulas. The averaged formula…”