On the number of zeros of iterated operators on analytic Legendre expansions (Q1607766)

Let \(E_R\) be an ellipse with foci at \(\pm 1\) and with sum of the semiaxes \(R>1\). Let \(f(z)=\sum^\infty_{n=0} c_nP_n(z)\) be an analytic in \(E_R\) function where \(P_n\), is the \(n\)th Legendre polynomial. The author has proved the following Theorem 2.1 If \(1<T<R\) then the number of zeros of \((L^kf)(z)\) in \(E_T\) is \(O(k\ln k)\) where \(L=(1-z^2) D^2-2zD\), \(D=d/dz\), and \(L^k=L (L^{k-1})\).