Linear extrapolation by rational functions, exponentials and logarithmic functions (Q1092586)

Let K(.,.) denote a complex-valued function of two variables, \(L_ 0,...,L_ N\), and \(\Lambda_ 0,...,\Lambda_ N\) two families of linear forms acting on the first and the second variable of K, respectively. Then \(\det (L_ j\Lambda_ kK)=\det (\Lambda_ kL_ jK)\) by transposition. Choosing for K the Cauchy kernel \((z,w)\to (z-w)^{-1}\), for \(L_ j\) the point evaluations at \((z_ j)_{0\leq j\leq N}\), and for \(\Lambda^ k_ j\) the evaluations of the derivatives of order k \((0\leq k\leq m_{j-1})\) of K at the points \((w_ j)_{1\leq j\leq M}\) of C, the associated Vandermonde determinant is calculated explicitly in terms of the difference products \(\prod_{0\leq k<j\leq N}(z_ j-z_ k)\quad and\quad \prod_{1\leq k<j\leq M}(w_ k-w_ j)^{m_ km_ j}.\) As an application, it is shown that some classical polynomial interpolation procedures can be extended to rational functions with given poles by recurrence relations of nearly the same complexity. Moreover, a study of the exponential kernel K: (z,w)\(\to e^{zw}\) is included. Finally, the authors establish a regularity criterion for certain linear sequence transformations.