Meromorphic Approximants to Complex Cauchy Transforms with Polar Singularities

نویسندگان

  • L. Baratchart
  • M. Yattselev
چکیده

We study AAK-type meromorphic approximants to functions of the form F (z) = Z dλ(t) z − t +R(z), where R is a rational function and λ is a complex measure with compact regular support included in (−1, 1), whose argument has bounded variation on the support. The approximation is understood in L-norm of the unit circle, p ≥ 2. We dwell on the fact that the denominators of such approximants satisfy certain non-Hermitian orthogonal relations with varying weights. They resemble the orthogonality relations that arise in the study of multipoint Padé approximants. However, the varying part of the weight implicitly depends on the orthogonal polynomials themselves, which constitutes the main novelty and the main difficulty of the undertaken analysis. We obtain that the counting measures of poles of the approximants converge to the Green equilibrium distribution on the support of λ relative to the unit disk, that the approximants themselves converge in capacity to F , and that the poles of R attract at least as many poles of the approximants as their multiplicity and not much more. Mathematics Subject Classification (2000). primary 41A20, 41A30, 42C05; secondary 30D50, 30D55, 30E10, 31A15.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Meromorphic Approximants to Complex Cauchy Transforms with Polar Singularities

We study AAK-type meromorphic approximants to functions of the form F (z) = ∫ dλ(t ) z − t +R(z), where R is a rational function and λ is a complex measure with compact regular support included in (−1,1), whose argument has bounded variation on the support. The approximation is understood in Lp -norm of the unit circle, p ≥ 2. We dwell on the fact that the denominators of such approximants sati...

متن کامل

Multipoint Padé approximants to complex Cauchy transforms with polar singularities

We study diagonal multipoint Padé approximants to functions of the form F (z) = Z dλ(t) z − t +R(z), where R is a rational function and λ is a complex measure with compact regular support included in R, whose argument has bounded variation on the support. Assuming that interpolation sets are such that their normalized counting measures converge sufficiently fast in the weak-star sense to some c...

متن کامل

Strong Asymptotics of Hermite-padé Approximants for Angelesco Systems

In this work type II Hermite-Padé approximants for a vector of Cauchy transforms of smooth Jacobi-type densities are considered. It is assumed that densities are supported on mutually disjoint intervals (an Angelesco system with complex weights). The formulae of strong asymptotics are derived for any ray sequence of multi-indices.

متن کامل

Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in L of the Circle

For all n large enough, we show uniqueness of a critical point in best rational approximation of degree n, in the L-sense on the unit circle, to functions of the form

متن کامل

The application of Padé approximants to Wiener–Hopf factorization

The key step in the solution of a Wiener–Hopf equation is the decomposition of the Fourier transform of the kernel, which is a function of a complex variable, α say, into a product of two terms. One is singularity and zero free in an upper region of the α-plane, and the other singularity and zero free in an overlapping lower region. Each product factor can be expressed in terms of a Cauchy-type...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2008