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, the difficulty of approximating general functions by functions of special form, such as polynomials.

Bernstein's lethargy theorem

Let $V_{1}\subset V_{2}\subset \ldots$ be a strictly ascending sequence 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 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.
• Heinz H. Bauschke; Regina S. Burachik; Patrick L. Combettes; Veit Elser; D. Russell Luke, 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 978-1-4419-9568-11. {{cite book}}: Check |isbn= value: length (help); Unknown parameter |edior6= ignored (help)