2,174 research outputs found
Slow convergence of sequences of linear operators II: Arbitrarily slow convergence
AbstractWe 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
Functions with Prescribed Best Linear Approximations
A common problem in applied mathematics is to find a function in a Hilbert
space with prescribed best approximations from a finite number of closed vector
subspaces. In the present paper we study the question of the existence of
solutions to such problems. A finite family of subspaces is said to satisfy the
\emph{Inverse Best Approximation Property (IBAP)} if there exists a point that
admits any selection of points from these subspaces as best approximations. We
provide various characterizations of the IBAP in terms of the geometry of the
subspaces. Connections between the IBAP and the linear convergence rate of the
periodic projection algorithm for solving the underlying affine feasibility
problem are also established. The results are applied to problems in harmonic
analysis, integral equations, signal theory, and wavelet frames
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