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Section: New Results

Behavior of poles in rational and meromorphic approximation

Participants : Laurent Baratchart, Sylvain Chevillard, Juliette Leblond, Martine Olivi, Fabien Seyfert.

Rational approximation

The numerous experiments that we performed on synthetic data in the context of the MagLune project (see Sections 6.1.2 and 8.2.1) revealed an intriguing behavior of the local minima of the optimization problem underlying our method. In the context of that application, we are provided with sampled values on the unit circle $𝕋$ of a function $f$ which is known to be of the form $f\left(z\right)=p\left(z\right)/{\left(z-\beta \right)}^{5}$ where $p\left(z\right)\in {ℂ}_{4}\left[z\right]$ is a polynomial of degree at most 4 with complex coefficients and $\beta \in 𝔻$ belongs to the unit disk. A key problem consists in recovering $\beta$ from the values of $f$ on the unit circle. The same problem occurs in the core of FindSources3D (see 3.4.3 and 6.1.3) with $p$ being of degree at most 2 and a pole of order 3 rather than 5.

In order to estimate $\beta$, we seek for the global minimum on ${ℂ}_{4}\left[z\right]×𝔻$ of the function $\phi$ defined by

$\phi :\left(q,\phantom{\rule{0.166667em}{0ex}}\alpha \right)↦{∥\frac{q\left(z\right)}{{\left(z-\alpha \right)}^{5}}-f\left(z\right)∥}_{{L}^{2}\left(𝕋\right)}.$

When $f$ is actually a rational function of the considered form, $\phi$ obviously has a unique global minimum where it reaches the value 0. We experimentally observed that $\phi$ usually has several local minima, some of them achieving very small values, and these minima often have a complex argument close to the argument of $\beta$. This behavior is unusual and contrasts with the fact the function

$\psi :\left(q,\phantom{\rule{0.166667em}{0ex}}{\alpha }_{1},\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}{\alpha }_{5}\right)↦{∥\frac{q\left(z\right)}{{\prod }_{i=1}^{5}\left(z-{\alpha }_{i}\right)}-f\left(z\right)∥}_{{L}^{2}\left(𝕋\right)}$

is known to have a unique local minimum on ${ℂ}_{4}\left[z\right]×{𝔻}^{5}$ (which is global) when $f$ is a rational function of the same form.

In order to understand the reasons underlying our observations, we started studying the theoretical properties of the critical points of $\phi$, in the general case of a pole of order $n\in {ℕ}^{*}$ and with a polynomial of degree less or equal to $n-1$ at the numerator. Our results so far are the following.

We introduce the family ${\left({g}_{j}^{\left(\alpha \right)}\right)}_{j\in {ℕ}^{*}}$ where ${g}_{j}^{\left(\alpha \right)}\left(z\right)={\left(1-\overline{\alpha }z\right)}^{j-1}/{\left(z-\alpha \right)}^{j}$ which is an orthogonal basis (for the usual ${L}^{2}\left(𝕋\right)$ Hilbert product) of the space of rational functions with a single pole (of arbitrary order) in $\alpha$. Thanks to this family, we prove that $\left(q,\alpha \right)$ is a critical point of $\phi$ if and only if $f$ is orthogonal either to ${g}_{n}^{\left(\alpha \right)}$ or ${g}_{n+1}^{\left(\alpha \right)}$ and, for such a given $\alpha$, $q/{\left(z-\alpha \right)}^{n}$ is the orthogonal projection of $f$ onto the rational functions of that form. The case when $f$ is orthogonal to ${g}_{n}^{\left(\alpha \right)}$ combined with the fact that $q/{\left(z-\alpha \right)}^{n}$ is the orthogonal projection of $f$ implies a pole-zero simplification of $q/{\left(z-\alpha \right)}^{n}$ at $z=\alpha$ and we conjecture that it exactly corresponds to local maxima of $\phi$ with respect to variable $\alpha$. We also conjecture that the other case exactly corresponds to local minima of $\phi$. We are currently working on proving these conjectures, which should not be too hard.

We also obtained an explicit algebraic equation characterizing $\alpha$, and we know how to solve it when $f$ is of the form $1/{\left(z-\beta \right)}^{k}$ ($1\le k\le n$). For small values of $n$, we proved (and conjecture that it holds for any $n$) that there are $2k-1$ solutions in the unit disk, all lying on the diameter passing though $\beta$. This is a remarkable result that somehow theoretically confirms the kind of experimental observations we got. The theoretical case of a function $f$ with a non trivial numerator seems currently out of reach, though.

Meromorphic approximation

We showed that best meromorphic approximation on a contour, in the uniform norm, to functions with countably many branched singularities with polar closure inside the contour produces poles whose counting measure accumulate weak-* to the Green equilibrium distribution on the cut of minimal capacity outside of which the function is single-valued. This is joint work with M. Yattselev (University of Indianapolis, Purdue University at Indianapolis). An article is currently being written on this topic.