A bivariate generalization of Hermite’s interpolation formula

Abstract

One of the most commonly used methods for deriving formulas for bivariate interpolation is that of extending to two variables the formulas of Lagrange, Aitken, Newton, Gauss, Stirling, Everett, Bessel, etc., in which forward, backward and central-differences are used. These formulas have the property that the resulting interpolation polynomial agrees with the interpolated function, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x comma y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(x,\,y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , at each of the node points of a Cartesian grid. In this study, we shall investigate the existence of a wider class of interpolation formulas, together with their associated error terms, than those obtainable by the method just described. To this end, we develop a bivariate osculatory interpolation polynomial which not only agrees with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x comma y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(x,\,y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in function values at each of the node points of a Cartesian grid but which also enjoys the property that agreement in values of partial and mixed partial derivatives up to specified, arbitrary orders is obtainable at these points. The result is essentially a bivariate generalization of Hermite’s interpolation formula.