Semi-iterative methods for the approximate solution of Ill-posed problems
Numerische Mathematik
Two generalizations of Dykstra's cyclic projections algorithm
Mathematical Programming: Series A and B
A dual approach to constrained interpolation from a convex subset of Hilbert space
Journal of Approximation Theory
Bayesian inference for nonlinear multivariate diffusion models observed with error
Computational Statistics & Data Analysis
Slow convergence of sequences of linear operators I: Almost arbitrarily slow convergence
Journal of Approximation Theory
Slow convergence of sequences of linear operators I: Almost arbitrarily slow convergence
Journal of Approximation Theory
Full length article: Approximation schemes satisfying Shapiro's Theorem
Journal of Approximation Theory
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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. This is a continuation of the paper Deutsch and Hundal (2010) [14]. The main result is a ''lethargy'' theorem (Theorem 3.3) which gives useful conditions that guarantee arbitrarily slow convergence. In the particular case when the sequence of linear operators is generated by the powers of a single linear operator, we obtain a ''dichotomy'' theorem, which states the surprising result that either there is linear (fast) convergence or arbitrarily slow convergence; no other type of convergence is possible. The dichotomy theorem is applied to generalize and sharpen: (1) the von Neumann-Halperin cyclic projections theorem, (2) the rate of convergence for intermittently (i.e., ''almost'' randomly) ordered projections, and (3) a theorem of Xu and Zikatanov.