The interaction of alternation points and poles in rational approximation

Abstract

The interrelation of alternation points for the minimal error function and poles of best Chebyshev approximants is investigated if uniform approximation on the interval [ - 1 , 1 ] by rational functions of degree ( n ( s ) , m ( s ) ) is considered, s ∈ N . In general, the alternation points need not to be uniformly distributed with respect to the equilibrium measure on [ - 1 , 1 ] , even not to be dense on the interval. We show that, at least for a subsequence Λ ⊂ N , the asymptotic behaviour of the alternation points to the degrees ( n ( s ) , m ( s ) ) , s ∈ Λ , is completely determined by the location of the poles of the best approximants, and vice versa, if m ( s ) ⩽ n ( s ) or m ( s ) - n ( s ) = o ( s / log s ) as s → ∞ .