[Paper Review] Zeros of polynomials orthogonal on several intervals

Franz Peherstorfer|ArXiv.org|Mar 28, 2002

Mathematical functions and polynomials12 references4 citations

TL;DR

This paper resolves long-standing questions about the distribution of zeros of orthogonal polynomials on multiple intervals by deriving exact formulas for the number of zeros in each interval and criteria for zeros to appear in gaps between intervals. Using harmonic measure and Jacobi inversion on a Riemann surface, it proves that if harmonic measures are rationally independent, every point in the gaps is an accumulation point of zeros; otherwise, only finitely many such points exist.

ABSTRACT

First a formula for the number of zeros of the orthogonal polynomial in the intervals is presented. Then a criteria about the appearance of a zero in a gap is given. Finally a necessary and sufficient condition is derived such that the zeros of the orthogonal polynomials have given accumulation points in the gaps. As a consequence it follows that every point from the gaps is an accumulation point of the zeros of the orthogonal polynomials, if the harmonic measures of the intervals are linearly independent over the rationals. If the harmonic measures are rational then there is a finite number of accumulation points only.

Motivation & Objective

• To determine the precise number of zeros of orthogonal polynomials in each subinterval of a union of disjoint intervals.
• To establish conditions under which zeros appear in the gaps between intervals, particularly whether they accumulate in these gaps.
• To characterize the set of accumulation points of zeros in the gaps based on the rational independence of harmonic measures.
• To extend classical results from single-interval orthogonal polynomials to the more complex case of multiple intervals using advanced tools from Riemann surface theory and potential theory.

Proposed method

• Derives a formula for the number of zeros in each interval $E_j$ using the harmonic measure $\omega_j(\infty)$ and the logarithmic mean value of the weight function.
• Applies the theory of Green's functions and Riemann surfaces defined by $y^2 = H(x)$ to analyze the asymptotic behavior of orthogonal polynomials.
• Uses the Jacobi inversion problem on the Riemann surface to encode information about zero locations and their accumulation.
• Employs approximation by Bernstein-Szegö weights to connect the general case to known asymptotic representations.
• Applies Kronecker's Lemma on simultaneous Diophantine approximation to control the convergence of solutions to the Jacobi inversion problem.
• Uses uniform bounds on orthonormal polynomials and their asymptotic expansions to prove the existence and location of accumulation points in gaps.

Experimental results

Research questions

• RQ1 How many zeros does the orthogonal polynomial $p_n$ have in each interval $E_j$ of the union $E = \bigcup_{j=1}^l E_j$?
• RQ2 Under what conditions does a zero of $p_n$ appear in a gap $(a_{2j}, a_{2j+1})$?
• RQ3 When is the set of accumulation points of the zeros of $p_n$ dense in a gap $[a_{2j}, a_{2j+1}]$?
• RQ4 What is the structure of the set of accumulation points of zeros in the gaps when harmonic measures are rationally dependent?
• RQ5 Can the zero distribution be characterized uniformly across all $n$ using potential-theoretic and algebraic-geometric tools?

Key findings

• The number of zeros of $p_n$ in each interval $E_j$ is given by $\left\lfloor n \omega_j(\infty) + \frac{1}{n} \log \left| \int_E \log |W| \, d\mu_e \right| \right\rfloor$ or similar expression involving harmonic measure and weight mean.
• A zero of $p_n$ appears in a gap $(a_{2j}, a_{2j+1})$ if and only if the solution to the associated Jacobi inversion problem lies in the corresponding branch of the Riemann surface.
• If the harmonic measures $\omega_1(\infty), \dots, \omega_{l-1}(\infty)$ are linearly independent over $\mathbb{Q}$, then every point in $[a_1, a_{2l}] \setminus E$ is an accumulation point of zeros of $p_n$.
• If the harmonic measures are rational, i.e., $\omega_j(\infty) = k_j/N$, then the number of accumulation points in each gap is at most $N + 2$.
• The asymptotic behavior of the orthonormal polynomials $P_n$ is governed by the Green's function and the solution to the Jacobi inversion problem on the Riemann surface $y^2 = H(x)$.
• For sequences $n_\kappa$ satisfying certain Diophantine conditions, the zeros of $P_{n_\kappa}$ can be made to accumulate precisely at any prescribed finite set of points in the gaps.