No abstract

Let fγ(x) = ∞ k=0 T k (x)/(γ) k , where (γ) k = γ(γ+1) · · · (γ+k−1) and T k (x) = cos (k arccos x) are Padé-Chebyshev polynomials. For such functions and their Padé-Chebyshev approximations π ch n,m (x; fγ), we find the asymptotics of decreasing the difference fγ(x) − π ch n, m (x; fγ) in the case where 0 m m(n), m(n) = o (n), as n → ∞ for all x ∈ [−1, 1]. Particularly, we determine that, under the same assumption, the Padé-Chebyshev approximations converge to fγ uniformly on the segment [−1, 1] with the asymptotically best rate.

New concepts are introduced in the present work. They are a quite normal index and a quite perfect system of functions. Using these concepts, the uniqueness criterion for solution of two Hermite – Pade problems is proved, the explicit determinant representations of type I and II Hermite – Padé polynomials for an arbitrary system of power series are obtained. The results obtained complement and generalize the well-known result in the theory of Hermite – Padé approximations.