Asymptotics of Polynomials Orthogonal on a Cross with a Jacobi-Type Weight
This post discusses some of the results obtained by myself and my advisor, Maxim Yattselev, in the paper "Asymptotics of Polynomials Orthogonal on a Cross with a Jacobi-Type Weight" (arxiv version). In this work we obtain asymptotic formulas for non-Hermitian orthogonal polynomials (no conjugation in orthogonality condition) on a cross, i.e. polynomials $Q_n(z)$ satisfying \[ \int_{\Delta} z^k Q_n(z) \rho(z) \ dz = 0 , \quad \text{ for } k = 0, 1, ..., n-1 \] where $\Delta = [-a, a] \cup [-ib, ib] \subset \mathbb{C}$. This is done via the Riemann-Hilbert problem for orthogonal polynomials (see this blog post for more on this). It will be convenient to denote the "legs" of the cross by $\Delta_i, \ i = 1, 2, 3, 4$, and the corresponding nonzero endpoints by $a_i$, as shown. One unique feature of such non-Hermitian orthogonal polynomials is that, due to the lack of positivity, one cannot guarantee anymore than $\deg Q_n \leq n$. In fact, one reason to consider this specific family of polynomials is that they experience a strange degeneration pattern. These degenerations turn out to be controlled by a secret parameter (in the paper, this is $\nu$).
Example Consider the weight function $\rho|_{\Delta_k}(z) = -\frac{i^{4 - k}}{|z^4 - 1|^{-1/4}}$, then a short calculation yields \[Q_{4n}(z) = P_{n, 1}(z^4), \quad Q_{4n + 1}(z) = Q_{4n + 2}(z) = Q_{4n + 3}(z) = z P_{n, 2}(z) \] where $P_{n,1}$ are polynomials orthogonal with respect to $x^{-3/4}(1 - x)^{-1/4}, \ x \in [0, 1]$ and $P_{n, 2}$ are orthogonal with respect to $x^{1/4}(1 - x)^{-1/4}, \ x \in [0, 1]$.
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| Our cross |
The function $\rho(z)$ is, in some sense, an analytic perturbation of the power functions, and in the simplest case takes the form \[ \rho|_{\Delta_k}(z) = c_k \prod_{i = 1}^{4} (z - a_i)^{\alpha_i}, \quad \alpha_i > -1\]where $\sum c_k = 0$ and the $(z - a_i)^{\alpha_i}$ is a branch holomorphic across $\Delta \setminus a_i$. More generally, we may consider weights in a nested sequence of classes $\mathcal{W}_{k}, \ k \in \mathbb{N}$, and for $\rho$ as described above, we say $\rho \in \mathcal{W}_{\infty}$. The details of this are technical, so I won't say much more on this (I have to give you *some* reason to read the paper!).
To motivate the discussion, let me start with the following classical theorem
Theorem Let $w(x) >0$ be a positive, continuous function on $[a, b]$ and consider the polynomials $P_n(x)$ satisfying \[ \int_a^b x^k P_n(x) w(x) \ dx\ = 0, \quad \text{ for } \quad k = 0, 1, ..., n-1. \] Then, $P_n(x)$ has $n$ simple zeros all of which belong to $[a, b]$.
More sophisticated versions of this theorem exist (see Theorem 2.1.1 in General Orthogonal Polynomials by H. Stahl and V. Totik, for example), but all heavily depend on working with an inner product. Dropping the conjugation in the definition of our orthogonal polynomials and considering complex-valued densities means that $\deg (Q_n) \leq n$ is the best we can say in general. In fact, since the expression $z^k Q_n(z) \rho(z)$ is holomorphic in a small enough neighborhood of any point in $\Delta \setminus \left(\cup \{a_i \} \right)$, one could deform the contour of integration to any cross-like contour. So, which contour will attract the zeros of $Q_n(z)$, and how many are there?
In a series of papers, Herbert Stahl answered the question regarding the zero-attracting curve: it is characterized as being the curve of smallest capacity connecting the points $\{ a_1, a_2, a_3, a_4 \}$. The problem of finding a continuum of minimal capacity connecting a finite set of points $\{a_1, a_2, ..., a_n \}$ had already been studied by many in the context of geometric function theory, and is commonly known as the Chebotarev problem. It is also known from works of Grötzsch and Lavrentiev that this set can be described as a union of critical trajectories of certain quadratic differentials.
To say more, let's first put this work in context. Stahl's work was actually concerned with the convergence of Padé approximants. These are approximants that are, in some sense, the best rational approximants to a function given by a Taylor (or Laurent) series (more on these can be find here). Roughly speaking, Stahl proved the following: given a function $f$ holomorphic and single-values outside of a small (capacity zero) compact set $K$. Then, the Padé approximants of $f$ converge to $f$ in capacity. Here, convergence in capacity is defined in the same way as convergence of measure (except...with capacity instead of measure). As a general result, this is actually the best possible due to the existence of "spurious" poles of the Padé approximants.
To say more, let's first put this work in context. Stahl's work was actually concerned with the convergence of Padé approximants. These are approximants that are, in some sense, the best rational approximants to a function given by a Taylor (or Laurent) series (more on these can be find here). Roughly speaking, Stahl proved the following: given a function $f$ holomorphic and single-values outside of a small (capacity zero) compact set $K$. Then, the Padé approximants of $f$ converge to $f$ in capacity. Here, convergence in capacity is defined in the same way as convergence of measure (except...with capacity instead of measure). As a general result, this is actually the best possible due to the existence of "spurious" poles of the Padé approximants.
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| peep the spurious poles of the "diagonal" approximant of degree 100 of $\sqrt[4]{1 - \frac{2}{z^2} + \frac{9}{z^4}}$ |
Let's now bring this back to our cross setting. The orthogonal polynomials above appear as denominators of the Padé approximants for functions of the form \[ f(z) = \int_{\Delta} \dfrac{\rho(x) dx}{z - x}, \quad z \in \mathbb{C} \setminus \Delta, \]knowns as Markov-type functions. These include functions like \[\log \left(\dfrac{z^2 - 1}{z^2 + 1} \right) \quad \text{and} \quad \sqrt{z^4 - 1},\]for example. Going through the Padé approximants makes it clear that while most of the zeros of $Q_n$ will be attracted to $\Delta$, some might not. This displays itself in the theorem below
Theorem (Asymptotics of $Q_n$ in the bulk for $\rho \in \mathcal{W}_{\infty}$) Given $\epsilon > 0$, it holds for all $n \in \mathbb{N}_{\rho, \epsilon}$ large enough that \[ Q_n(z) = \gamma_{n} (1 + \upsilon_{n,1}(z))\Psi_{n}(z) + \gamma_{n} \upsilon_{n,2}(z) \Psi_{n-1}(z)\]locally uniformly in $\mathbb{C} \setminus \Delta$ with \[\upsilon_{n,i} = \dfrac{L_{n,i}}{z} + \mathcal{o}\left(1 \right), \quad L_{n,i} = \mathcal{O} \left( n^{|\mathsf{Re}(\nu)| - 1/2} \right)\]and $\gamma_{n} = \lim_{z \to \infty}z^{-n} \Psi_n(z) < \infty$ whenever $n \in \mathbb{N}_{\rho, \epsilon}$.
The functions $\Psi_n$ ar known as Baker-Akhiezer functions in the integrable systems circles, and are defined on an associated Riemann surface. $\nu$ is the parameter we mentioned earlier, chosen with $\mathsf{Re}(\nu) \in \left(-\frac{1}{2}, \frac{1}{2} \right]$. It holds that $\mathbb{N}_{\rho, \epsilon}$ is either all even naturals, odd, naturals, or all natural numbers (large enough) whenever $\mathsf{Re}(\nu) \neq 1/2$. In particular, the dependence on $\epsilon$ and the excessive degeneration occur only when $\mathsf{Re}(\nu) = 1/2$.
The Riemann-Hilbert analysis also yields asymptotics of the error of approximation of Markov functions via Padé approximants (away from spurious poles!), upgrading the mode of convergence from Stahl's result for this subclass of functions. Once again here one needs to consider limits along subsequences to avoid the degenerations from above. With this the case of approximants of a Jacobi-type function with four branch points is almost done, but more on this later!