Determination of zeros
Part of the Operator Theory: Advances and Applications book series (OT, volume 27)
In the previous chapter we have shown how to find the poles of a meromorphic function from its Laurent series coefficients. In this chapter we shall give similar results for its zeros. As we have seen, the theory used in the previous chapter was essentially the theory which worked for a power series in z or 1/z. In the classical theory for power series, the computation of zeros is easily derived from the theory for its poles with the symmetry properties of chapter 10 for a power series F(z) and its inverse 1/F(z). In this way poles are turned into zeros and the result follows immediately. If F(z) is given by its Laurent series, we could use in the previous chapter the additive splitting of F(z) as
. The poles of Z(z) and
$$Z(z) - \hat Z(z)$$
made up the poles of F(z), but this is not true anymore for its zeros. So, the inversion of Z (z ) and
doesn’t help in this case. What we need now is the multiplicative splitting of F(z) which contained the factors F +(z) and F+(z). Then the zeros of F(z) are distributed over these factors and inverting them turns zeros into poles. How these zeros can then be computed is shown in theorem 15.1. The Rutishauser polynomials related to the poles of 1 /F+(z) will be the dual Rutishauser polynomials related to the zeros of F+(z). Next we have to relate the parameters associated with the power series of F+(z) (and F−(z)) with the parameters associated with the Laurent coefficients of F(z). This result is given in Lemma 15.2. Because the coefficients of the dual Rutishauser polynomials are made up of combinations of these parameters, also the dual Rutishauser polynomials of F(z) and F+ (z) (and F+(z)) will be related (corollary 15.3). From this we shall be able to derive the convergence of the dual Rutishauser polynomials for F(z) (theorem 15.4) and the convergence results concerning the zeros of F(z).
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© Birkhäuser Verlag Basel 1987