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On polynomials orthogonal with respect to certain Sobolev inner products. (English) Zbl 0734.42016

Orthogonal polynomials p n λ (x) for the Sobolev inner product

<f,g>=f(x)g(x)dϕ(x)+λf ' (x)g ' (x)dψ(x)

are investigated. An interesting notion of coherent pairs of Borel measures ϕ and ψ is introduced. If p n (x) are the orthogonal polynomials with respect to the measure ϕ, then (ϕ,ψ) is a coherent pair when there exist non-zero constants C k such that

p n ' (x)p m ' (x)dψ(x)=d min(m,n) C m C n ·

It is shown that for a coherent pair (ϕ,ψ) the Sobolev orthogonal polynomials p n λ (x) can be expanded in terms of the orthogonal polynomials p n (x) in such a way that the expansion coefficients (except the last) are independent of n and themselves orthogonal polynomials in the variable λ. Several examples are worked out and an application involving the evaluation of Sobolev-Fourier coefficients is given.


MSC:
42C05General theory of orthogonal functions and polynomials
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems