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)

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.