Best approximations, sequence of

A sequence $ \{ E (x, F _ {n} ) \} $, $ n = 1, 2 \dots $ of numbers, where $ E (x, F _ {n} ) $ is the best approximation of an element $ x $ of a normed linear space $ X $ by elements of an $ n $- dimensional subspace $ F _ {n} \subset X $, with $ F _ {1} \subset F _ {2} \subset \dots $, so that $ E (x, F _ {1} ) \geq E (x, F _ {2} ) \geq \dots $. Usually, $ F _ {n} $ is the linear span of the first $ n $ elements of some fixed system $ \{ u _ {1} , u _ {2} , . . . \} $ of linearly independent elements of $ X $.

In the case $ X = C [a, b] $ and $ F _ {n} = F _ {n} ^ {A} $ is the subspace of algebraic polynomials of degree $ n - 1 $, sequences of best approximations were first considered in the 1850s by P.L. Chebyshev; the fact that

$$ E (x, F _ {n} ^ {A} ) \rightarrow 0 \ \textrm{ as } n \rightarrow \infty $$

for any function $ x (t) \in C [a, b] $ was established in 1885 by K. Weierstrass. In the general case, the relation

$$ \lim\limits _ {n \rightarrow \infty } E (x, F _ {n} ) = 0 \ \ \textrm{ for } \textrm{ all } x \in X $$

is always satisfied when the union of the subspaces $ F _ {n} $, $ n = 1, 2 \dots $ is everywhere dense in $ X $,

$$ \overline{ {\cup F _ {n} }}\; = X $$

(essentially, this is an equivalent statement). However, the sequence $ \{ E (x, F _ {n} ) \} $ may converge to zero arbitrarily slowly. This follows from a theorem of Bernstein: If $ \{ F _ {n} \} $ is a sequence of subspaces of dimension $ n = 1, 2 \dots $ of a normed linear space $ X $, such that $ F _ {1} \subset F _ {2} \subset \dots $ and $ \overline{ {\cup F _ {n} }}\; = X $, then, for any monotone decreasing null sequence $ \{ \mu _ {n} \} $ of non-negative real numbers, there exists an $ x \in X $ such that $ E (x, F _ {n} ) = \mu _ {n} $, $ n = 1, 2 ,\dots $. In the function spaces $ C $ and $ L _ {p} $, the rate at which a sequence of best approximations tends to zero depends both on the system of subspaces $ F _ {n} $ and on the smoothness characteristics of the approximated function $ x $( the modulus of continuity, the existence of derivatives up to a specific order, etc.), and it can be estimated in terms of these characteristics. Conversely, knowing the rate of convergence to zero of the sequence $ \{ E (x, F _ {n} ) \} $, one can draw conclusions with respect to the smoothness of $ x (t) $( see Approximation of functions, direct and inverse theorems).

References

[1]	S.N. Bernshtein, "Collected works" , 2 , Moscow (1954) (In Russian)
[2]	V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian)
[3]	A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)

Comments

Theorems inferring smoothness characteristics of a function $ x \in C $ or $ L _ {p} $ from properties of $ E (x, F _ {n} ) $ were first given by D. Jackson in 1911 for algebraic or trigonometric approximation, see Jackson theorem. Theorems converse to these, i.e. inferring properties of $ E (x, F _ {n} ) $ from smoothness characteristics of the function $ x $, have been proved by S.N. Bernstein [S.N. Bernshtein] and A. Zygmund, cf. Bernstein theorem. See also [a2], Chapt. 4, Sect. 6 and Chapt. 6, Sect. 3.

References

[a1]	I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian)
[a2]	E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff