On the bivariate Shepard-Lidstone operators

On the existence of Hermite-Birkhoff quadrature formulas of Gaussian type☆

In this paper, we provide an integral error formula for a certain scale of mean value interpolations which includes the multivariate polynomial interpolation schemes of Kergin and Hakopian. This formula involves only derivatives of order one higher than the degree of the interpolating polynomial space, and from it we can obtain sharp $L_\infty$-estimates. These $L_\infty$-estimates are precisely those that numerical analysts want, to guarantee that a scheme based on such an interpolation has the maximum possible order.

/pdf/integral-error-formulae-for-the-scale-of-mean-value-jfc9f0pe2v.pdf

Integral error formulæ for the scale of mean value interpolations which includes Kergin and Hakopian interpolation

We shall show that quadrature formulas of double precision, also using evaluation functionals of derivatives of the function instead of evaluation functionals of the function only, exist if the order of the derivatives is prescribed in the form of a certain pyramidal display. This type of Gaussian formulas is derived through an extremal property of certain polynomials extending the extremal property of the classical Gaussian formulas. The results generalize to the multiple node case.

A New Class of Gaussian Quadrature Formulas Based on Birkhoff Type Data

https://www.ams.org/mcom/1990-54-190/S0025-5718-1990-1010595-7/S0025-5718-1990-1010595-7.pdf

Comparison of Birkhoff type quadrature formulae

The Birkhoff interpolation polynomial $P(X,t)$, $0 \leqq t \leqq 1$ of a given function depends upon the knots of interpolation, $X:0 \leqq x_1 < x_2 < \cdots < x_m \leqq 1$. There is a natural extension of $P(X,t)$ to the case when some of the knots coincide. This extension is proved to be continuous. The method of proof is by “decoalescence”.

Continuity of the Birkhoff Interpolation

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Continuity of the Birkhoff Interpolation