Biorthogonal polynomials, quadrature and reproducing kernels
Part of the Operator Theory: Advances and Applications book series (OT, volume 27)
The recurrence relations we have introduced so far can be interpreted as recurrence relations for polynomials that are orthogonal in a formal sense with respect to an indefinite bilinear form. Some of these orthogonalities and some other properties of these polynomials will be given in section 8.1. It is a well known fact that the zeros of orthogonal polynomials are very useful as abscissas for the construction of Gauss quadrature rules. Some ideas on this application will be given in section 8.2. In section 8.3 it is shown how some classical results on reproducing kernels in separable Hilbert spaces can be formally generalized to our situation. All this is developed in our viewpoint of the basic algorithms of chapter 3 that generated the polynomials of the first kind
for m fixed and n = 0,1,2,… which corresponds to a horizontal path in an (m,n) grid. In other works related to Pade approximation, a similar theory is given, but for polynomials on diagonals in an (m,n) grid. The most complete compilation in this area is probably given by Brezinski [BRE] and Draux [DRA]. Another way to show the difference between our approach and the more classical ones is that we started from algorithms of chapter 3 that are fundamental for the solution of Toeplitz systems, while in classical Padé approximation recurrence relations are related to the Euclidean algorithm and methods to solve Hankel systems. So ours is a “Toeplitz approach” while the more classical one is a “Hankel approach”. In section 8.4 we shall briefly mention some results on the latter.
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© Birkhäuser Verlag Basel 1987