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 , . If is not holomorphic on , the distribution of the zeros of best rational approximants is governed by the equilibrium measure of 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, -values, and poles of best real rational approximants of degree at most to a function that is real-valued, but not holomorphic on . Generalizations to the lower half of the Walsh table are indicated.

Ray Sequences of Best Rational Approximants For |x|α

The convergence behavior of best uniform rational approximations with numerator degree m and denominator degree n to the function |x|α, α > 0, on [-1, 1] is investigated. It is assumed that the indices (m, n) progress along a ray sequence in the lower triangle of the Walsh table, i.e. the sequence of indices {(m, n)} satisfiesIn addition to the convergence behavior, the asymptotic distribution of poles and zeros of the approximants and the distribution of the extreme points of the error function on [-1, 1] will be studied. The results will be compared with those for paradiagonal sequences (m = n + 2[α/2]) and for sequences of best polynomial approximants.