Zeros of non-Baxter paraorthogonal polynomials on the unit circle. (English) Zbl 1248.42024
This paper provides leading-order asymptotics for the size of the gap in the zeros around \(1\) of paraorthogonal polynomials on the unit circle whose Verblunsky coefficients satisfy a slow decay condition and are inside the interval \((0,1)\). Precisely it proves the following
Theorem. Let \(k\in \mathbb{N},\bar{\epsilon}>0\), and \(\xi\in (\frac \pi 2 +\bar{\epsilon}, \frac {3\pi} 2-\bar{\epsilon})\) be fixed, and let \(\beta=e^{i\xi}\). Let \(\{\alpha_n \}_{n\geq 0}\) be a sequence of real Verblunsky coefficients having slow decay controlled by \(f\). For each \(\delta>0\), there exists \(N_k=N_k(\delta)\in \mathbb{N}\) so that if \(M>N_k\), then the zero point \(\xi_k^{(M)}\) of the paraorthogonal polynomial \(\Phi_k^{\beta_k}(z)\) obeys \[ |\frac {\arg (\xi_k^{(M))})} {2|f(M)|} -1|<\delta, \] and \[ |\frac {2\pi-\arg (\xi_{M-k+1}^{(M))})} {2|f(M)|} -1|<\delta. \] This theorem proves a conjecture in the paper [Electron. Trans. Numer. Anal. 25 328-368 (2006; Zbl 1129.42011)] by B. Simon.
Theorem. Let \(k\in \mathbb{N},\bar{\epsilon}>0\), and \(\xi\in (\frac \pi 2 +\bar{\epsilon}, \frac {3\pi} 2-\bar{\epsilon})\) be fixed, and let \(\beta=e^{i\xi}\). Let \(\{\alpha_n \}_{n\geq 0}\) be a sequence of real Verblunsky coefficients having slow decay controlled by \(f\). For each \(\delta>0\), there exists \(N_k=N_k(\delta)\in \mathbb{N}\) so that if \(M>N_k\), then the zero point \(\xi_k^{(M)}\) of the paraorthogonal polynomial \(\Phi_k^{\beta_k}(z)\) obeys \[ |\frac {\arg (\xi_k^{(M))})} {2|f(M)|} -1|<\delta, \] and \[ |\frac {2\pi-\arg (\xi_{M-k+1}^{(M))})} {2|f(M)|} -1|<\delta. \] This theorem proves a conjecture in the paper [Electron. Trans. Numer. Anal. 25 328-368 (2006; Zbl 1129.42011)] by B. Simon.
Reviewer: Lizhong Peng (Beijing)
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 26C10 | Real polynomials: location of zeros |
| 47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
Citations:
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