Lethargy theorem

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.

Bernstein's lethargy theorem

Let $V_{1}\subset V_{2}\subset \ldots$ be a strictly ascending sequence of finite-dimensional linear subspaces of a Banach space X, and let $\epsilon _{1}\geq \epsilon _{2}\geq \ldots$ 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 $\epsilon _{i}$.

See also

• Bernstein's theorem (approximation theory)

References

• S.N. Bernstein (1938). "On the inverse problem of the theory of the best approximation of continuous functions". Sochinenya. II: 292–294.
• Elliott Ward Cheney (1982). Introduction to Approximation Theory (2nd ed.). American Mathematical Society. ISBN 978-0-8218-1374-4.
• Bauschke, Heinz H.; Burachik, Regina S.; Combettes, Patrick L.; Elser, Veit; Luke, D. Russell; Wolkowicz, Henry, eds. (2011). Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications. doi:10.1007/978-1-4419-9569-8. ISBN 9781441995681.
• Frank Deutsch; Hein Hundal (2010). "Slow convergence of sequences of linear opertors I: almost arbitrarily slow convergence". Journal of Approximation Theory. 162 (9): 1701–1716. doi:10.1016/j.jat.2010.05.001. MR 2011h:41020.
• Frank Deutsch; Hein Hundal (2010). "Slow convergence of sequences of linear opertors II: arbitrarily slow convergence". Journal of Approximation Theory. 162 (9): 1717–1738. doi:10.1016/j.jat.2010.05.002. MR 2011h:41021.
• Catalin Badea; Sophie Grivaux; Vladimir MÄuller (2011). "The Rate of Convergence in the Method of Alternating Projections".
  • Preprint