Best uniform rational approximations of functions by orthoprojections

Let C[-1, 1] be the Banach space of continuous complex functions f on the interval [-1, 1] equipped with the standard maximum norm parallel tofparallel to ; let omega((.)) = omega((.), f) be the modulus of continuity of f; and let R-n = R-n(f) be the best uniform approximation of f by rational functions (r.f.) whose degrees do not exceed n = 1, 2,.... The space C[-1, 1] is also regarded as a pre-Hilbert space with respect to the inner product given by (f, g) = (1/pi) integral(-1)(1) f (x)/g(x)(1 - x(2))(-1/2) dx. Let z(n) = {z(1), z(2),..., z(n)} be a set of points located outside the interval [-1, 1]. By F( (.), f, z(n)) we denote an orthoprojection operator acting from the pre-Hilbert space C[-1, 1] onto its (n + 1)-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set z(n). In this paper, we show that if f is not a rational function of degree less than or equal to n, then we can find a set of points z(n) = z(n) (f) such that