A tool for locating zeros of orthogonal polynomials in Sobolev inner product spaces

AbstractIn the theory of polynomials orthogonal with respect to an inner product of the form 〈f,〉 = ∫∞0f(x)g(x)dψ(x)+Σmk=1Akf(ik)(0)g(ik)(0), one is confronted with the following situation: for certain values of the parameters, the orthogonal polynomial of degree n does not have all its zeros inside the support of the distribution function dψ. This paper gives a method to investigate the zero distribution by looking at a type of limiting polynomial. For the case m = 2 it is shown that there are exactly two zeros outside the true interval of orthogonality for A1,A2 large; moreover, it is proved that these zeros are nonreal (complex conjugates) in the case i1 + 1 = i2. Also several examples are given