Slow convergence of sequences of linear operators I: Almost arbitrarily slow convergence Original Research Article

We study the rate of convergence of a sequence of linear operators that converges pointwise to a linear operator. Our main interest is in characterizing the slowest type of pointwise convergence possible. A sequence of linear operators (Ln)(Ln) is said to converge to a linear operator LLarbitrarily slowly (resp., almost arbitrarily slowly) provided that (Ln)(Ln) converges to LL pointwise, and for each sequence of real numbers (ϕ(n))(ϕ(n)) converging to 0, there exists a point x=xϕx=xϕ such that ‖Ln(x)−L(x)‖≥ϕ(n)‖Ln(x)−L(x)‖≥ϕ(n) for all nn (resp., for infinitely many nn). The main result in this paper is a “lethargy” theorem that characterizes almost arbitrarily slow convergence. It states (Theorem 3.1) that a sequence of linear operators converges almost arbitrarily slowly if and only if it converges pointwise, but not in norm. The Lethargy Theorem is then applied to show that a large class of polynomial operators (e.g., Bernstein, Hermite–Fejer, Landau, Fejer, and Jackson operators) all converge almost arbitrarily slowly to the identity operator. It is also shown that all the classical quadrature rules (e.g., the composite Trapezoidal Rule, composite Simpson’s Rule, and Gaussian quadrature) converge almost arbitrarily slowly to the integration functional. In the second part of this paper, Deutsch and Hundal (2010) [5], we make a similar study of arbitrarily slow convergence.