ON POLYNOMIALS ORTHOGONAL WITH RESPECT TO CERTAIN SOBOLEV INNER PRODUCTS

We are concerned with polynomials {pn(λ)} that are orthogonal with respect to the Sobolev inner product 〈 f, g 〉λ = ∝ fg dθ{symbol} + λ ∝ f′g′ dψ, where λ is a non-negative constant. We show that if the Borel measures dθ{symbol} and dψ obey a specific condition then the Pn(λ)'s can be expanded in the polynomials orthogonal with respect to dθ{symbol} in such a manner that, subject to correct normalization, the expansion coefficients, except for the last, are independent of n and are themselves orthogonal polynomials in λ. We explore several examples and demonstrate how our theory can be used for efficient evaluation of Sobolev-Fourier Coefficients. © 1991.