Formulas for bivariate hyperosculatory interpolation

Abstract:

For a given function $f(x,y)$, bivariate hyperosculatory interpolation formulas are obtained by employing a suitably constructed binary nic ${p_n}(x,y)$ that is fitted to the values of $f(x,y)$ and its first and second partial derivatives at the m points $({x_i},{y_i})$ of a rectangular $h \times k$ Cartesian grid, where ($({x_i},{y_i}) = ({x_0} + {p_i}h,{y_0} + {q_i}k),{p_i}$ and ${q_i}$ are small integers $\geqq 0,i = 0(1)m - 1,m \geqq 2$. In terms of the variables (p, q), where $x = {x_0} + ph,y = {y_0} + qk$ (and $f(x,y) = F(p,q)$), we have ${p_n}(x,y) = {P_n}(p,q)$. Often, for ${P_n}(p,q)$ having a specified desirable form, this problem turns out to be insoluble for every configuration of the points $({x_i},{y_i})$. When this is not the case, it generally requires considerable investigation to find a practical configuration of points $({x_i},{y_i})$ for which there is a solution of the form ${P_n}(p,q)$. Formulas are found for choices of ${P_n}(p,q)$, and soluble configurations of points $({x_i},{y_i})$, that have dominant remainder terms in \[ {h^r}{k^s}{f_{x \ldots x(r\;{\text {times}})y \ldots y(s\;{\text {times}})}}({x_0},{y_0})\] whose orders $r + s$ are as high as possible. Three two-point formulas, two three-point formulas and one four-point formula, including all remainder terms through the order \[ r + s = \left ( {\begin {array}{*{20}{c}} {n,} \hfill & {{\text {for}}\;m = 2} \hfill \\ {n + 1,} \hfill & {{\text {for}}\;m = 3,4} \hfill \\ \end {array} } \right ),\] are given here in convenient matrix form.