On Recurrence Relations for Sobolev Orthogonal Polynomials

Matrix interpretation of multiple orthogonality

“…Next we give a result of existence and uniqueness of a type II sequence of vector orthogonal polynomials with respect to a regular vector of linear functionals U, and using a matrix three-term recurrence relations we establish a Favard type theorem. We remark that other characterization for sequences of orthogonal polynomials in terms of matrix three-term recurrence relations can be found in [15,16]. In section 4 we express the resolvent function in terms of the matrix generating function associated to the vector of linear functionals.…”

“…Now we prove the converse of this result which is called the Favard type theorem. Note that in the given literature (see for instance [15,16]), the coefficients of matrix three term recurrence relation, are regular Hermitian matrices, and in (5) this is not the case. …”

Inner products involving differences: the meixner—sobolev polynomials

“…(2) over all p EJP n , dJ1,;(x), i = 0,1, being positive Borel measures on the real line ~ having bounded or unbounded support [8,19]. Expanding pin terms of the Sobolev orthogonal polynomials we obtain the usual Fourier approximation p(x) of f(x) and f'(x).…”

Bases of the space of solutions of some fourth-order linear difference equations: applications in rational approximation

“…Of course one solution is {p n } ∞ n=0 where p n = P n −1 µ P n (x). Another linearly independent solution, is the (not necessarily monic) polynomial p [1] n−1 (deg(p [1] n−1 ) = n − 1) that can be obtained from (4) with the initial conditions Y −1 = 1 and Y 0 = 0. In the theory of orthogonal polynomials the sequence {p [1] n } ∞ n=0 is often called sequence of first associated, numerator or second kind polynomials with respect to the sequence of monic orthogonal polynomials {P n } ∞ n=0 .…”