ZEROS OF LINEAR-COMBINATIONS OF ORTHOGONAL POLYNOMIALS

Let psi be a distribution function on [-1, 1] from the Szego-class, wh ich contains in particular all Jacobi weights, and let (p(n)) be the m onic polynomials orthogonal with respect to d psi. Let m(n)is an eleme nt of N, n is an element of N, be non-decreasing with lim(n-->infinity ) (n-m(n)) = infinity, l(n)is an element of N with 0 less than or equa l to m(n), and mu(j,n) is an element of R for j = 0,...,m(n),n is an e lement of N. It is shown that for each sufficiently large n, Sigma(j=0 )(m(n))mu(j,n) P-n-j(x) has n-l(n) simple zeros in (-1,1) and l(n) zer os in C\[-1,1] if for n greater than or equal to n(0), Sigma(j=0)(m(n) )mu(j,n)Z(m(n)-j) has m(n)-l(n) zeros in the in the disc \z\ less than or equal to r < 1, l(n) zeros outside of the disc \z\ greater than or equal to R > 1 and Sigma(j=0)(m(n))\mu(j,n)\q(j) less than or equal t o const., where q > 2 max {r, 1/R}. If m(n) is constant for n greater than or equal to n(0) then the statement holds even for such polynomia ls (P-n) orthogonal with respect to a distribution d psi satisfying th e weak assumption psi' > 0 a.e, on [-1,1]. For linear combinations of polynomials orthogonal on the unit circle corresponding results are de rived.