Chebyshev’s alternance in the approximation of constants by simple partial fractions

Best approximation rate of constants by simple partial fractions and Chebyshev alternance

“…It is well known that a simple partial fraction of order ≤ n interpolating an n-node table may be nonunique or even fail to exist [3], [4]. However, the former situation cannot happen in the case of constants; namely, it was shown in [3] and [5] that if there exists a simple partial fraction of order ≤ n interpolating a constant function f (x) = c on a set…”

“…For example, the nonuniqueness of ρ * n even for the case in which there exists an alternance of n + 1 points, as well as the fact that the alternance is not necessary for the best approximation, was originally shown in [5] in an example with n = 2. This example was completely generalized to n = 3 in [6] and to an arbitrary n ≥ 2 in [7].…”

“…It was shown in [5] that, for sufficiently large n > n 0 (c), the existence of an (n + 1)-point alternance is necessary for the best approximation of a constant c ∈ (0, ln √ 2) on the interval [−1, 1] by a simple partial fraction of order ≤ n and that the simple partial fraction ρ * n (c; · ) is unique. In particular, since the (n + 1)-point alternance ensures the coincidence of the simple partial fraction ρ * n with the constant c at n nodes on [−1, 1], it follows that this fraction is the simple partial fraction of the best approximation and has order n.…”

An Analog of the Haar Condition for Simple Partial Fractions

“…We are interested in a similar condition of the uniqueness of a simple partial fraction of least deviation (unlike approximation by algebraic polynomials, such a fraction is not necessarily unique [2,3]). We recall that a (real-valued) simple partial fraction of degree n ∈ N is a rational function of the form…”

“…A detailed review of results on approximation by simple partial fractions can be found in [2]- [7]. Throughout the paper, we deal with approximation of a continuous real-valued function f on the segment [−1, 1].…”