Section: New Results

Rational and meromorphic approximation

Participants : Laurent Baratchart, Herbert Stahl [ TFH Berlin ] , Maxim Yattselev.

The results of [4] and [6] were extensively used over the last years to prove the convergence in capacity of L^p -best meromorphic approximants on the circle (i.e. solutions to problem (P_N ) of section 3.1.2 ) when p2 , for those functions f that can be written as Cauchy transforms of complex measures supported on a hyperbolic geodesic arc [18] , [17] , [25] , [26] . A rational function can also be added to f without modifying the results, which is useful for applications to inverse sources problems. Some mild conditions (bounded variation of the argument and power-thickness of the total variation) were required on the measure. Here, we recall that convergence in capacity means that the (logarithmic) capacity of the set where the error is greater than goes to 0 for each fixed >0 . This convergence can be quantified, namely it is geometric with pointwise rate exp{-1/C-G} where C is the capacity of the condensor and G the Green potential of the equilibrium measure. The results can be adapted to somewhat general interpolation schemes [18] , [17] . From this work it follows that the counting measures of the poles of the approximants converge, in the weak-* sense, to the Green equilibrium distribution on . In particular the poles cluster to the endpoints of the arc, which is of fundamental use in the team's approach to source detection (see section  6.3.2 ).

This year the weak-* convergence of the poles of best H² rational approximants to Cauchy integrals over general symmetric contours for the Green Potential, and not merely geodesic arcs, were established using the reflected symmetry of the poles and the interpolation nodes of such approximants across the circle, and analyzing the location of continua of minima weighted through a discretization of the weight and a limiting process. This warrants the use of rational approximation to functions with arbitrarily many branchpoints in source detection. A paper is currently being written on this topic.

The technique we just described only yields convergence in capacity and n-th root asymptotics. To obtain strong asymptotics, additional assumptions must be made on the approximated function. Last year, we proved strong asymptotics of multipoint Padé interpolants, in appropriate interpolation nodes, to Cauchy integrals over arbitrary analytic arcs, when the density of the measure with respect to a positive power of the equilibrium distribution on the arc is Dini-smooth. In addition, the density may in fact vanish in finitely many points like a small fractional power of the distance to these point [16] . Moreover, the polar singularities of the function, if any, are asymptotically reproduced by the approximants with their multiplicities. This is important for inverse problem of mixed type, like those mentioned in section 6.3.2 , where monopolar and dipolar sources are handled simultaneously.

This year we proved under appropriate smoothness assumptions that the result still holds without restrictions on the density, that is, the power of the equilibrium distribution with respect to which we compute its derivative needs no longer be positive (in the language of orthogonal polynomials, this means we can handle arbitrary Jacobi weights). The lower the power the smoother the density should be. Typically, if the power is zero (so that we only consider the density with respect to arclength on the arc), a fraction of a derivative is sufficient (i.e. the density should belong to a W^{1-1/p, p} -class on the arc for some p>2 . When the power gets negative, -classes of Hölder-smoothness for the k -th derivative are required, where k is related to the integer part of the Jacobi exponents and to their fractional part. This time however, the density is not allowed to vanish.

This result more or less settles the issue of convergence of multipoint Padé approximants to Cauchy integrals over arcs, because it asserts that uniform convergence holds, under mild assumptions on the density, when the interpolation points are chosen in some appropriate manner (symmetric with respect to a weighted equilibrium potential adapted to the contour), and because we also proved that whenever a convergent interpolation scheme exists to a Cauchy integral with smooth density on an arc, with interpolation keeping off the arc, then the arc must be analytic.

The technique of proof uses a -generalization, over varying contours, of the Riemann-Hilbert approach to the asymptotics of orthogonal polynomials as adapted to the segment in [77] This provides us with precise (Plancherel-Rotach type) asymptotics for the non-Hermitian orthogonal polynomials which is the denominator of the approximant. Asymptotics for the latter are even obtained on the arc where the measure of orthogonality is supported. In the case of non-positive Jacobi powers, -estimates and Muckenhoupt weights are also needed. A paper has been written and submitted to report on this research [50] .

We have pursued this line of research for functions defined as Cauchy integrals over union of (possibly intersecting) arcs, and obtained convergence results over regular threefolds. The current goal is to understand which systems of arcs can be construed as critical configurations for weighted potential problems, and whether the above analysis can be extended beyond arcs to 2-D singular sets.

In another collection, the results of [9] have been carried over for analytic approximation to the matrix case in [14] . The surprising fact was that not every matrix valued function generates a vectorial Hankel operator meeting the AAK theorem when p< . This led us to the generalization of the latter based on Hankel operators with matrix argument.