A TOOL FOR LOCATING ZEROS OF ORTHOGONAL POLYNOMIALS IN SOBOLEV INNER-PRODUCT SPACES

In the theory of polynomials orthogonal with respect to an inner product of the form [f, g] = integral-infinity/0 f(x)g(x)dpsi(x) + SIGMA(k=1)(m)A(k)f(i(k))(0)g(i(k))(0), one if 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 dpsi. 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.