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

Orthogonal polynomials ${p}_{n}^{\lambda }\left(x\right)$ for the Sobolev inner product

$=\int f\left(x\right)g\left(x\right)d\phi \left(x\right)+\lambda \int {f}^{\text{'}}\left(x\right){g}^{\text{'}}\left(x\right)d\psi \left(x\right)$

are investigated. An interesting notion of coherent pairs of Borel measures $\phi$ and $\psi$ is introduced. If ${p}_{n}\left(x\right)$ are the orthogonal polynomials with respect to the measure $\phi$, then $\left(\phi ,\psi \right)$ is a coherent pair when there exist non-zero constants ${C}_{k}$ such that

$\int {p}_{n}^{\text{'}}\left(x\right){p}_{m}^{\text{'}}\left(x\right)d\psi \left(x\right)=\frac{{d}_{min\left(m,n\right)}}{{C}_{m}{C}_{n}}·$

It is shown that for a coherent pair ($\phi$,$\psi$) the Sobolev orthogonal polynomials ${p}_{n}^{\lambda }\left(x\right)$ can be expanded in terms of the orthogonal polynomials ${p}_{n}\left(x\right)$ in such a way that the expansion coefficients (except the last) are independent of n and themselves orthogonal polynomials in the variable $\lambda$. Several examples are worked out and an application involving the evaluation of Sobolev-Fourier coefficients is given.

##### MSC:
 42C05 General theory of orthogonal functions and polynomials 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems