Domain of convergence for a series of orthogonal polynomials
Let \(\left\{p_k\left(z\right)\right\}_{k=0}^\infty\) be a family of orthogonal polynomials with respect to an exponential weight \(w\left(x\right)=\exp\left(-Q\left(x\right)\right)\in{\mathcal F}\left(\text{Lip}\frac{1}{2}\right)\). Note that \({\mathcal F}\left(\text{Lip}\frac{1}{2}\right)\) is the Levin-Lubinsky class [\textit{E. Levin} and \textit{D. S. Lubinsky}, Orthogonal polynomials for exponential weights. New York, NY: Springer (2001; Zbl 0997.42011)]. So, \(p_n\left(x\right)=p_n\left(w^2,x\right)\) are polynomials of degree \(n\), \(n=0,1,2,\dots\), such that \[ \int_{-\infty}^\infty p_n(x)p_m(x)w^2(x)\,dx=\delta_{mn}, \] the Kronecker delta. The authors show that under certain conditions, a series of the form \(\sum_{k=0}^\infty b_k p_k(z)\) converges uniformly and absolutely on compact subsets of an open strip in the complex plane, and diverges outside the closure of this strip. To formulate the main result, we need some notations. The numbers \(a_{-t}<0<a_t\), \(t>0\), are uniquely determined by the following equations \[ t=\frac{1}{\pi}\int_{a_{-t}}^{a_t}\frac{x Q'(x)}{\sqrt{\left(x-a_{-t}\right)\left(a_t-x\right)}}\,dx; \] \[ 0=\frac{1}{\pi}\int_{a_{-t}}^{a_t}\frac{Q'(x)}{\sqrt{\left(x-a_{-t}\right)\left(a_t-x\right)}}\,dx. \] The density function for \(x\in\left[a_{-n},a_n\right]\) is defined by \[ \sigma_n\left(x\right)=\frac{1}{\pi^2}\sqrt{\left(x-a_{-n}\right)\left(a_n-x\right)} \int_{a_{-n}}^{a_n}\frac{Q'(s)-Q'(x)}{s-x}\frac{ds}{\sqrt{\left(s-a_{-n}\right)\left(a_n-s\right)}}. \] The main result of the article is given by the following statement. Theorem. Let \(w=\exp\left(-Q\right)\in{\mathcal F}\left(\text{Lip}\frac{1}{2}\right)\). For a sequence of complex numbers \(\left\{b_k\right\}\), let \[ B:=\liminf_{n\to\infty}\frac{1}{\pi\sigma_n\left(0\right)}\log\left(\frac{1}{\left|b_n\right|}\right). \] Then \(\sum b_k p_k(z)\) converges uniformly and absolutely on compact subsets of the strip \(\left|\text{Im}\,z\right|<B\) and diverges for every \(z\) outside the closed strip \(\left|\text{Im}\,z\right|\leq B\). The article should be interesting for specialists in approximation theory, harmonic analysis, complex analysis, and various applications, where orthogonal polynomials are used.
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