Sobolev orthogonal polynomials and [formula omitted]-coherent pairs of measures

Abstract

We introduce the notion of (M,N)−coherent pair of measures as a generalization of the concept of coherent pair of measures introduced by Iserles et al. [A. Iserles, P.E. Koch, S.P. Nørsett, J.M. Sanz-Serna, On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory 65(2) (1991) 151–175], and subsequently generalized by several authors. A pair of measures (dμ0,dμ1) is called (M,N)-coherent if the corresponding orthogonal polynomial sequences (Pn)n and (Qn)n (resp.) satisfy a (non-zero) structure relation such as ∑i=0Nri,nPn−i(x)=∑i=0Msi,nQn−i+1′(x) for all n=0,1,2,…, where M and N are fixed non-negative integer numbers, and ri,n and si,n are given real parameters satisfying some natural conditions. We prove that the regular moment linear functionals associated to an (M,N)-coherent pair are semiclassical and they are related by a rational modification (in the usual sense of distribution theory). We also discuss the converse statement. Under the assumption that (dμ0,dμ1) form an (M,N)-coherent pair, we study the sequence (Snλ)n of the monic orthogonal polynomials with respect to the Sobolev inner product 〈f,g〉λ:=∫−∞+∞fgdμ0+λ∫−∞+∞f′g′dμ1, defined in the space of all polynomials with real coefficients, where λ≥0. An efficient algorithm is stated to compute the coefficients in the Fourier–Sobolev type series f(x)∼∑n=0∞cnλSnλ(x) with respect to 〈⋅,⋅〉λ for suitable smooth functions f such that f∈Lμ02(R) and f′∈Lμ12(R). Finally, some illustrative computational examples are presented.