Toeplitz matrix techniques and convergence of complex weight Padé approximants

摘要

One considers diagonal Padé approximants about ∞ of functions of the form f(z)=∫−11(z−x)−1w(x)dx,z∉[−1,1], where w is an integrable, possibly complex-valued, function defined on [-1, 1].Convergence of the sequence of diagonal Padé approximants towards f is established under the condition that there exists a weight ω, positive almost everywhere on [-1, 1], such that g(x)=w(x)/ω(x) is continuous and not vanishing on [-1, 1].The rate of decrease of the error is also described.The proof proceeds by establishing the link between the Padé denominators and the orthogonal polynomials related to ω, in terms of the Toeplitz matrix of symbol g(cos Φ).