اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدDo orthogonal polynomials dream of symmetric curves?
85
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
The complex or non-hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the freedom in the choice of the integration contour for such polynomials, the location of their zeros is a priori not clear. Nevertheless, numerical experiments, such as those presented in this paper, show that the zeros not simply cluster somewhere on the plane, but persistently choose to align on certain curves, and in a very regular fashion. The problem of the limit zero distribution for the non-hermitian orthogonal polynomials is one of the central aspects of their theory. Several important results in this direction have been obtained, especially in the last 30 years, and describing them is one of the goals of the first parts of this paper. However, the general theory is far from being complete, and many natural questions remain unanswered or have only a partial explanation. Thus, the second motivation of this paper is to discuss some mysterious configurations of zeros of polynomials, defined by an orthogonality condition with respect to a sum of exponential functions on the plane, that appeared as a results of our numerical experiments. In this apparently simple situation the zeros of these orthogonal polynomials may exhibit different behaviors: for some of them we state the rigorous results, while other are presented as conjectures (apparently, within a reach of modern techniques). Finally, there are cases for which it is not yet clear how to explain our numerical results, and where we cannot go beyond an empirical discussion.
قيم البحث
اقرأ أيضاً
We use Turan type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the three term
recurrence $p_{k+1}=x p_k-c_k p_{k-1},$ with a nondecreasing sequence ${c_k}$. As a special case they include a non-asymptotic version of Mate, Nevai and Totik result on the largest zeros of orthogonal polynomials with $c_k=k^{delta} (1+ o(k^{-2/3})).$
We consider a problem of bounding the maximal possible multiplicity of a zero at of some expansions $sum a_i F_i(x)$, at a certain point $c,$ depending on the chosen family ${F_i }$. The most important example is a polynomial with $c=1.$ It is shown
that this question naturally leads to discrete orthogonal polynomials. Using this connection we derive some new bounds, in particular on the multiplicity of the zero at one of a polynomial with a prescribed norm.
Let $x_1$ and $x_k$ be the least and the largest zeros of the Laguerre or Jacobi polynomial of degree $k.$ We shall establish sharp inequalities of the form $x_1 <A, x_k >B,$ which are uniform in all the parameters involved. Together with inequalitie
s in the opposite direction, recently obtained by the author, this locates the extreme zeros of classical orthogonal polynomials with the relative precision, roughly speaking, $O(k^{-2/3}).$
In this contribution we deal with sequences of monic polynomials orthogonal with respect to the Freud Sobolev-type inner product begin{equation*} leftlangle p,qrightrangle _{s}=int_{mathbb{R}}p(x)q(x)e^{-x^{4}}dx+M_{0}p(0)q(0)+M_{1}p^{prime }(0)q^{pr
ime }(0), end{equation*}% where $p,q$ are polynomials, $M_{0}$ and $M_{1}$ are nonnegative real numbers. Connection formulas between these polynomials and Freud polynomials are deduced and, as a consequence, a five term recurrence relation for such polynomials is obtained. The location of their zeros as well as their asymptotic behavior is studied. Finally, an electrostatic interpretation of them in terms of a logarithmic interaction in the presence of an external field is given.
Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ($q=1$) classical orthogonal polynomials, and study those cases in which the exponential generati
ng function has a nice form. In the opposite direction, we show that the generalized Dumont-Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
