Citation:
Bulletin of Mathematical Sciences, 4 (1), pp. 83-97

ISSN:
1664-3607

DOI:
10.1007/s13373-013-0047-x

Sponsor:
The second authorwas partially supported by the Research Fellowship Program from Ministerio de Ciencia e
Innovación (MTM 2009-12740-C03-01), Spain. The first, second and third authors were partially supported
by a grant from Ministerio de Economía y Competitividad. Dirección General de Investigación Científica
y Técnica (MTM 2012-36732-C03-01), Spain

We investigate algebraic and analytic properties of sequences of polynomials
orthogonal with respect to the Sobolev type inner product
(f.g)= ∫▒〖f(x)g(x)dμ(x)+∑_(k=1)^K▒∑_(i=0)^(N_k)▒M_█(k.i@) 〗 f^((i) ) (b_k ) g^i (b_k )
where μ is a finite positive BoWe investigate algebraic and analytic properties of sequences of polynomials
orthogonal with respect to the Sobolev type inner product
(f.g)= ∫▒〖f(x)g(x)dμ(x)+∑_(k=1)^K▒∑_(i=0)^(N_k)▒M_█(k.i@) 〗 f^((i) ) (b_k ) g^i (b_k )
where μ is a finite positive Borel measure belonging to the Nevai class, the mass
points bk are located outside the support of μ, and Mk,i are complex numbers such
that Mk,Nk
= 0. First, we study the existence as well as recurrence relations for such polynomials. When the values Mk,i are nonnegative real numbers, we can deduce the
coefficients of the recurrence relation in terms of the connection coefficients for the
sequences of polynomials orthogonal with respect to the Sobolev type inner product
and those orthogonal with respect to the measure μ. The matrix of a symmetric multiplication
operator in terms of the above sequence of Sobolev type orthogonal polynomials
is obtained from the Jacobi matrix associated with the measure μ. Finally, we
focus our attention on some outer relative asymptotics of such polynomials, which are
deduced by using the above connection formulas[+][-]