Asymptotic uniqueness of best rational approximants to complex Cauchy transforms in 𝐿² of the circle

Abstract

For all n large enough, we show uniqueness of a critical point in best rational approximation of degree n, in the L 2 -sense on the unit circle, to functions of the formwith r a rational function andμ a complex-valued Dini-continuous function on a real segment [a, b] ⊂ (−1, 1) which does not vanish, and whose argument is of bounded variation.Here ω [a,b] stands the normalized arcsine distribution on [a, b].Mathematics Subject Classification (2000). 41A52, 41A20, 30E10.

Cyclic AFD algorithm for the best rational approximation

“…The expression (12) was first used in [15] and [14], in which AFD and unwending AFD are proposed. The AFD algorithm depends on a maximal selection principle: For any f 2 H 2 , b D arg maxfjhf , e a ij : a 2 Dg is attainable inside D (see [15] or [14]).…”

“…It is a consecutive selection process of the parameters a 1 , : : : , a n . To solve the best n-Blaschke approximation problem, or, equivalently, the best n-rational approximation problem, one is reduced to select all the parameters a 1 , : : : , a n inside D at one time to give rise to the global minimum value of (12). As mentioned, [3] and [4] show that such simultaneous selections exist.…”

“…The criterion developed in is further refined to show that the asymptotic uniqueness holds for Markov functions whose defining measure satisfies precisely the Szegö condition . Further, development along this line is given in in which Markov functions defined by the Cauchy transformation are generalized to Cauchy transforms of complex measures that are absolutely continuous with respect to the equilibrium distributions on a real segment [ a , b ] ⊂ ( − 1,1) with the density function being Dini‐smooth and non‐vanishing. Another type of the uniqueness conditions is given in.…”

Optimal approximation by Blaschke forms

“…The problem of best rational approximation to H 2 -functions is a classical issue. Some early references include [5][6][7][8][9][10]. Although the existence of minimizers has been proved [10], from the constructive point of view no practical algorithms have been established.…”

“…In the transformations we utilized (6) and the fact that the Blaschke product e B kÀ1 is unimodular on the unit circle. By mathematical induction (12) holds for all k. Finally we have…”