Vector orthogonal relations. Vector QD-algorithm

摘要

Vector-Padé approximants to a function F = (ƒ1; … ƒd) from C to Cd have been defined, uniquely, without any auxiliary choice than the degrees of the numerator and the denominator (the same for all the components ƒi), as in the scalar case [1,5]. The denominators are associated to polynomials Psr, which are given by vector orthogonal properties (R) and which satisfy for each s, recurrence relations of order d + 1 (i.e. with d + 2 terms), called relations (D).We study here consequences of (R) and (D): first we prove an algorithm similar to the generalized MNA-algorithm; then we define a vector QD-algorithm which links two diagonals (Psr)r and (Ps + 1r)r.Conversely if a family (Pr)r ⩾ 0 verifying (D) is given, it is possible to build (Psr)r ⩾ 0,s ⩾ 0, and d linear functionals Cα, α = 1,…, d, such that P0r = Pr and (Psr) verify the orthogonal relations (R), with respect to the Cα.