Bernstein's Lethargy Theorem
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
is to
approximation theory In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...
, where such theorems quantify the difficulty of approximating general functions by functions of special form, such as
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s. 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 In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
''X'', and let \epsilon_1 \ge \epsilon_2 \ge \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) In approximation theory, Bernstein's theorem is a converse to Jackson's theorem. The first results of this type were proved by Sergei Bernstein in 1912. For approximation by trigonometric polynomials, the result is as follows: Let ''f'': , 2 ...


References

* * * * * * {{cite journal , title=The Rate of Convergence in the Method of Alternating Projections , author1=Catalin Badea, author2=Sophie Grivaux, author3=Vladimir MÄuller, year=2011 *
Preprint
Theorems in approximation theory