Existence and local behavior of nondiagonal bivariate quadratic function approximation (Q1423951)

Let \(f(x,y)\) be a given bivariate function having a power series expansion at the origin \((0,0)\) and \(E(x,y)\) a bivariate series beginning by a certain number of zero terms necessary to calculate the coefficients of bivariate polynomials \(a_2,a_1,a_0\) of respective degrees \(m,n,k\) from a linear system \(a_2f^2+a_1f+a_0=E\) obtained by equating the powers \(x^iy^j\). The quadratic bivariate Hermite-Padé approximant \(u(x,y)\) to \(f\) is then the solution of \(a_2u^2+a_1u+a_0=0\). The authors prove the existence and the uniqueness of these off-diagonal approximants. Numerical values of errors \(f-u\) are compared to the errors produced by diagonal bivariate Chisholm approximants and by the Taylor bivariate truncated series for the functions \(e^{-x-y}\) and \(\ln(1+x+y)\).