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

Orthogonal polynomials \(p^{\lambda}_ n(x)\) 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 \(p_ n(x)\) are the orthogonal polynomials with respect to the measure \(\phi\), then \((\phi,\psi)\) is a coherent pair when there exist non-zero constants \(C_ k\) such that \[ \int p'_ n(x)p'_ m(x)d\psi (x)=\frac{d_{\min (m,n)}}{C_ mC_ n}. \] It is shown that for a coherent pair (\(\phi\),\(\psi\)) the Sobolev orthogonal polynomials \(p^{\lambda}_ n(x)\) 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 \(\lambda\). Several examples are worked out and an application involving the evaluation of Sobolev-Fourier coefficients is given.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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