Lethargy theorem

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In mathematics, a lethargy theorem is a statement about the distance of points in a metric space from members of a sequence of subspaces; one application in numerical analysis is to approximation theory, where such theorems quantify the difficulty of approximating general functions by functions of special form, such as polynomials. In more recent work, the convergence of a sequence of operators is studied: these operators generalise the projections of the earlier work.

Let ${\displaystyle V_{1}\subset V_{2}\subset \ldots }$ be a strictly ascending sequence of finite-dimensional linear subspaces of a Banach space X, and let ${\displaystyle \epsilon _{1}\geq \epsilon _{2}\geq \ldots }$ be a decreasing sequence of real numbers tending to zero. Then there exists a point x in X such that the distance of x to V_i is exactly ${\displaystyle \epsilon _{i}}$.

• Bernstein's theorem (approximation theory)

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• Elliott Ward Cheney (1982). Introduction to Approximation Theory (2nd ed.). American Mathematical Society. ISBN 978-0-8218-1374-4.
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• Frank Deutsch; Hein Hundal (2010). "Slow convergence of sequences of linear operators II: arbitrarily slow convergence". Journal of Approximation Theory. 162 (9): 1717–1738. doi:10.1016/j.jat.2010.05.002. MR 2718893.
• Badea, C.; Grivaux, S.; Müller, V. (2011). "The rate of convergence in the method of alternating projections". Algebra i Analiz. 23 (3): 1–30. arXiv:1006.2047. doi:10.1090/S1061-0022-2012-01202-1. MR 2896163.