Jacobi-Sobolev Orthogonal Polynomials: Asymptotics for N-Coherence of Measures

Let us introduce the Sobolev-type inner product (f, g) = (f, g)(1) + lambda(f ', g')(2), where lambda > 0 and (f, g)(1) = integral(1)(-1)f(x)g(x)(1 - x)(alpha) (1 + x)(beta) (1 + x)(beta)dx, (f, g)(2) = integral(1)(-1) f(x)g(x)((1 - x)(a+1) (1 - x)(alpha+1) (1 - x)(beta+1))/(Pi(M)(k=1)vertical bar x -xi k vertical bar(Nk+1))dx + Sigma(M)(k=1) Sigma(Nk)(i=0) M(k,i)f((i)) (xi(k))(g)((i))(xi(k)), with alpha, beta > -1, vertical bar xi(k)vertical bar > 1, and M-k,M-i > 0, for all k, i. A Mehler-Heine-type formula and the inner strong asymptotics on (-1, 1) as well as some estimates for the polynomials orthogonal with respect to the above Sobolev inner product are obtained. Necessary conditions for the norm convergence of Fourier expansions in terms of such Sobolev orthogonal polynomials are given.