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On recurrence relations for Sobolev orthogonal polynomials. (English) Zbl 0824.33006

The authors study Sobolev inner products on the linear space \({\mathcal P}\) of polynomials of the following form: \[ (p,q)= \sum_{k=0}^ N \int_{\mathcal R} p^{(k)} (x) \overline {q^{(k)} (x)} d\mu_ k (x), \qquad p,q\in {\mathcal P}, \tag{1} \] where \(N\geq 1\) is an integer and the \(\mu_ k\) are positive Borel measures. One of the problems in the theory is the existence of a ‘symmetrizing polynomial’ \(h: {\mathcal R}\to {\mathcal R}\) in the sense \((hp, q)= (p, hq)\), \(p,q\in {\mathcal P}\). The authors solve this problem completely and give properties of the polynomial \(h\); the existence is equivalent to the measures \(\mu_ k\) with \(1\leq k\leq N\) all being atomic with a finite number of mass points. They also show that then there exists a unique polynomial \(H\) of minimal degree \(m\) with \(H(0) =0\) such that the Sobolev orthogonal polynomials satisfy a linear recurrence relation of length \(2m+1\) of the form \[ H(x) \varphi_ n (x)= \sum_{k= n-m}^{n+m} b_{n,k} \varphi_ k (x) \qquad (n\geq m). \] The connection between the support of these atomic measures and the zeros of the derivatives of \(h\) is given and some examples are treated, including a method to calculate \(h\), given the atomic measures.
Finally the authors look into the existence of a polynomial \(g\) in the atomic case, that satisfies \[ g(x) \varphi_ n (x)= A_{N,n} (x) q_ n (x)+ B_{N,n} (x) q_{n-1} (x), \] where \(\varphi_ n\) is the Sobolev orthogonal polynomial with respect to (1), \(q_ n\) the monic orthogonal polynomial associated with \(\mu_ 0\) and \(\deg A_{N<n}= \deg g\), \(\deg B_{N,n}= \deg g-1\).
Moreover, they give a second-order differential equation for the Sobolev orthogonal polynomial \(\varphi_ n\) in the case that \(\mu_ 0\) leads to a semi-classical orthogonal polynomial sequence.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
28A25 Integration with respect to measures and other set functions
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