Rational approximation of functions with logarithmic singularities
Also appears in collection : Annual conference of the functional analysis, harmonic analysis and probability Gdr research group / Journées du Gdr analyse Fonctionnelle, harmonique et probabilités
I will report on the results of my recent work with Dmitri Yafaev (Rennes I). We consider functions $\omega$ on the unit circle with a finite number of logarithmic singularities. We study the approximation of $\omega$ by rational functions in the BMO norm. We find the leading term of the asymptotics of the distance in the BMO norm between $\omega$ and the set of rational functions of degree $n$ as $n$ goes to infinity. Our approach relies on the Adamyan-Arov-Krein theorem and on the study of the asymptotic behaviour of singular values of Hankel operators. In particular, we make use of the localisation principle, which allows us to combine the contributions of several singularities in one asymptotic formula.
![$k$-sum free sets in $[0,1]$](/media/cache/video_light/uploads/video/2015-09-10_de_Roton-video--19cf70874d3c5af7378c268cab865b06.jpg)
