(M,N)-coherent pairs of order (m,k) and Sobolev orthogonal polynomials

A pair of regular linear functionals (U,V) is said to be a (M,N)-coherent pair of order (m,k) if their corresponding sequences of monic orthogonal polynomials {Pn(x)}n≥0 and {Qn(x)}n≥0 satisfy a structure relation such as ∑i=0Mai,nPn+m−i(m)(x)=∑i=0Nbi,nQn+k−i(k)(x),n≥0, where ai,n and bi,n are complex numbers such that aM,n≠0 if n≥M, bN,n≠0 if n≥N, and ai,n=bi,n=0 when i>n. In the first part of this work we focus our attention on the algebraic properties of an (M,N)-coherent pair of order (m,k). To be more precise, let us assume that m≥k. If m=k then U and V are related by a rational factor (in the distributional sense); if m>k then U and V are semiclassical and they are again related by a rational factor. In the second part of this work we deal with a Sobolev type inner product defined in the linear space of polynomials with real coefficients, P, as 〈p(x),q(x)〉λ=∫Rp(x)q(x)dμ0(x)+λ∫Rp(m)(x)q(m)(x)dμ1(x),p,q∈P, where λ is a positive real number, m is a positive integer number and (μ0,μ1) is a (M,N)-coherent pair of order m of positive Borel measures supported on an infinite subset of the real line, meaning that the sequences of monic orthogonal polynomials {Pn(x)}n≥0 and {Qn(x)}n≥0 with respect to μ0 and μ1, respectively, satisfy a structure relation as above with k=0, ai,n and bi,n being real numbers fulfilling the above mentioned conditions. We generalize several recent results known in the literature in the framework of Sobolev orthogonal polynomials and their connections with coherent pairs (introduced in [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 their extensions. In particular, we show how to compute the coefficients of the Fourier expansion of functions on an appropriate Sobolev space (defined by the above inner product) in terms of the sequence of Sobolev orthogonal polynomials {Sn(x;λ)}n≥0.