On the Distribution of Zeros and Poles of Rational Approximants on Intervals

The distribution of zeros and poles of best rational approximants is well understood for the functions ð ‘“ ( ð ‘¥ ) = | ð ‘¥ | ð ›¼ , ð ›¼ > 0 . If ð ‘“ ∈ ð ¶ [ − 1 , 1 ] is not holomorphic on [ − 1 , 1 ] , the distribution of the zeros of best rational approximants is governed by the equilibrium measure of [ − 1 , 1 ] under the additional assumption that the rational approximants are restricted to a bounded degree of the denominator. This phenomenon was discovered first for polynomial approximation. In this paper, we investigate the asymptotic distribution of zeros, respectively, ð ‘Ž -values, and poles of best real rational approximants of degree at most ð ‘› to a function ð ‘“ ∈ ð ¶ [ − 1 , 1 ] that is real-valued, but not holomorphic on [ − 1 , 1 ] . Generalizations to the lower half of the Walsh table are indicated.