Dyadic Hermite interpolation on a rectangular mesh (Q1293226)

The univariate bisection algorithm of the second author [Numer. Algorithms 2, 187 (1992; Zbl 0754.65011)] is extended to two variables. Assume that the values of a function \(f\) and its first derivatives \(p=f_x\) and \(q= f_y\) are given at the vertices of a rectangle \(R=I\times J\). The authors study a formula (depending on five parameters) which calculates from these data, values of \(f,p,q\) at the midpoint of \(R\) and at the midpoints of all edges of \(R\). This formula gives by iteration values for \(f,p,q\) at all dyadic points in \(R\). The question arises, of when this densely defined function can be extended to a \(C^1\) function on \(R\). A matrix representation of the interpolation formula and estimates for spectral radii are used to derive conditions on the parameters for which such a \(C^1\) extension is possible for all Hermite data. Several examples illustrate how the interpolation works for different sets of parameters.