تقرير
Zeros of random linear combinations of OPUC with complex Gaussian coefficients
| العنوان: | Zeros of random linear combinations of OPUC with complex Gaussian coefficients |
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| المؤلفون: | Yeager, Aaron M. |
| سنة النشر: | 2016 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Probability, Mathematics - Complex Variables |
| الوصف: | We study zero distribution of random linear combinations of the form $$P_n(z)=\sum_{j=0}^n\eta_j\phi_j(z),$$ in any Jordan region $\Omega \subset \mathbb C$. The basis functions $\phi_j$ are orthogonal polynomials on the unit circle (OPUC) that are real-valued on the real line, and $\eta_0,\dots,\eta_n$ are complex-valued iid Gaussian random variables. We derive an explicit intensity function for the number of zeros of $P_n$ in $\Omega$ for each fixed $n$. Using the Christoffel-Darboux formula, the intensity function takes a very simple shape. Moreover, we give the limiting value of the intensity function when the orthogonal polynomials are associated to Szeg\H{o} weights. Comment: This article relies heavily on the results of section 2 from arXiv:1605.06836 to give the analogues of the applications in sections 3 and 4 of arXiv:1605.06836 for OPUC |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1608.02805 |
| رقم الأكسشن: | edsarx.1608.02805 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |