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On polynomials orthogonal with respect to certain Sobolev inner products. (English) Zbl 0734.42016
Orthogonal polynomials $p\sp{\lambda}\sb n(x)$ for the Sobolev inner product $$ <f,g>=\int f(x)g(x)d\phi (x)+\lambda \int f'(x)g'(x)d\psi (x) $$ are investigated. An interesting notion of coherent pairs of Borel measures $\phi$ and $\psi$ is introduced. If $p\sb n(x)$ are the orthogonal polynomials with respect to the measure $\phi$, then $(\phi,\psi)$ is a coherent pair when there exist non-zero constants $C\sb k$ such that $$ \int p'\sb n(x)p'\sb m(x)d\psi (x)=\frac{d\sb{\min (m,n)}}{C\sb mC\sb n}. $$ It is shown that for a coherent pair ($\phi$,$\psi$) the Sobolev orthogonal polynomials $p\sp{\lambda}\sb n(x)$ can be expanded in terms of the orthogonal polynomials $p\sb n(x)$ 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:
42C05General theory of orthogonal functions and polynomials
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:
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