On the Lagrange interpolation polynomials of entire functions (Q791060)

We investigate the growth of an entire function f and estimate the error term when approximating f in the complex plane by Lagrange interpolation polynomials over zeros of orthogonal polynomials. In particular, Lagrange interpolation at the zeros of Hermite polynomials is considered. Our main result is: Theorem. Let W be the class of all weight functions of the form \(w_ Q(x)=\exp \{-2Q(x)\},x\in {\mathbb{R}}\), where (i) Q(x) is an even differentiable function, increasing for \(x>0\); (ii) there exists \(\rho<1\) such that \(x^{\rho}Q'(x)\) is increasing; and (iii) the unique positive sequence \(\{q_ n\}\) determined by \(q_ nQ'(q_ n)=n\) satisfies \(q_{2n}/q_ n=C>1\) for \(n=1,2,3,..\). for some constant C independent of n. Let f be an entire function, and \(M(R)=\max_{| z| =R}| f(z)|\). Let \(w_ Q\in W\). Then, there exists a constant \(A\in(0,1)\), depending on Q only, such that whenever \(\lim \sup_{R\to \infty}(\log \quad M(R)/2Q(R))\leq A\) we have for any \(z\in {\mathbb{C}}\), \(\lim \sup_{n\to \infty}| f(z)-{\mathcal L}_ n(w_ Q;f;z)|^{1/n}<1,\) where \(L_ n(w_ Q;f;z)=\sum^{n}_{k=1}f(z\chi_{kn})\ell_{kn}(z)\), \(\chi_{kn}\) are the zeros of the nth orthogonal polynomial associated with \(w_ Q\), and \(\ell_{kn}(z)\) are the fundamental polynomials of Lagrange interpolation.