Meeting Details

Title: Arbitrarily slow convergence of sequences of linear operators Computational and Applied Mathematics Colloquium Frank Deutsch, Department of Mathematics, Penn State Our objective is to 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) is said to converge to a linear operator L arbitrarily slowly (resp., almost arbitrarily slowly) provided that (Ln) converges to L pointwise, and for each sequence of real numbers (φ(n)) converging to 0, there exists a point x = xφ such that ||Ln (x) − L(x|| ≥ φ(n) for all n (resp., for inﬁnitely many n). The main results in this paper are two “lethargy” theorems. The ﬁrst one characterizes almost arbitrarily slow convergence. The second one gives a useful sufficient condition that guarantees arbitrarily slow convergence. In the particular case when the sequence of linear operators is generated by the powers of a single linear operator, it turns out that almost arbitrarily slow convergence and arbitrarily slow convergence are equivalent. In this case we obtain a “dichotomy” theorem that states the perhaps surprising result that either there is linear (fast) convergence or arbitrarily slow convergence; no other type of convergence is possible. The ﬁrst lethargy theorem is 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. Finally, the dichotomy theorem is applied in Hilbert space to generalize and sharpen: (1) the von Neumann-Halperin cyclic pro jections theorem, (2) the rate of convergence for randomly ordered pro jections, and (3) a theorem of Xu and Zikatanov.

Room Reservation Information

Room Number: MB106 10 / 16 / 2009 03:35pm - 04:25pm