Approximation theory

The branch of mathematical analysis studying methods for approximating some mathematical objects by others and also studying questions related to the research and estimation of the error that arises here.

The main contents of approximation theory concerns the approximation of functions. Its foundations are laid by the work of P.L. Chebyshev (1854–1859) on best uniform approximation of functions by polynomials and by K. Weierstrass, who in 1885 established that in principle it is possible to approximate a continuous function on a finite interval by polynomials with arbitrary pre-given error. The development of approximation theory was to a large extent determined by the fundamental work of H. Lebesgue, Ch.J. de la Vallée-Poussin, S.N. Bernstein [S.N. Bernshtein], D. Jackson, J. Favard, A.N. Kolmogorov and S.M. Nikol'skii on the approximation of functions and function classes.

With the development of functional analysis, many problems in approximation theory were considered in the most general setting, e.g. as the approximation of elements of an arbitrary linear normed space $ X $. Three classes of problems arose here, corresponding more or less to the main chronological stages of development of research in approximation theory.

1) The approximation of a fixed element $ x \in X $ by elements of a fixed set $ \mathfrak N \subset X $. If

$$ E (x, \mathfrak N ) = \ \inf _ {u \in \mathfrak N } \ \| x - u \| $$

is taken as the approximation measure, i.e. the best approximation of $ x $ by $ \mathfrak N $, then, along with the investigation and estimation of $ E (x, \mathfrak N ) $, questions on the existence of an element of best approximation $ u _ {0} \in \mathfrak N $( for which $ \| x - u _ {0} \| = E (x, \mathfrak N ) $), its uniqueness and characteristic features arise. Any operator $ A $ mapping $ X $ into $ \mathfrak N $ gives rise to an approximation method with error $ \| x - Ax \| $. If $ \mathfrak N $ is a linear manifold, linear operators are of particular importance. For sequences $ \{ A _ {n} \} $ of such operators, the question of the conditions of convergence $ A _ {n} x \rightarrow x $ for any $ x \in X $ arises.

2) The approximation of a fixed set $ \mathfrak M \subset X $ by elements of another fixed set $ \mathfrak N \subset X $. The best approximation here is expressed by

$$ E ( \mathfrak M , \mathfrak N ) = \ \sup _ {x \in \mathfrak M } \ E (x, \mathfrak N ), $$

which gives the minimal possible error estimate when approximating an arbitrary $ x \in X $ by elements from $ \mathfrak N $. In concrete cases, the problem consists of estimating, or expressing exactly, $ E ( \mathfrak M , \mathfrak N ) $ by characteristic properties of the given sets $ \mathfrak M $ and $ \mathfrak N $. If the approximation is established by an operator $ A $, the supremum

$$ \sup _ {x \in \mathfrak M } \ \| x - Ax \| $$

is investigated, as well as (if $ \mathfrak N $ is a linear manifold)

$$ {\mathcal E} ( \mathfrak M , \mathfrak N ) = \ \inf _ {AX \subset \mathfrak N } \ \sup _ {x \in \mathfrak M } \ \| x - Ax \| , $$

where the infimum is taken over all linear operators $ A $ mapping $ X $ into $ \mathfrak N $. A linear operator realizing this infimum (if it exists) gives rise to a best linear method of approximation. The case $ {\mathcal E} ( \mathfrak M , \mathfrak N ) = E ( \mathfrak M , \mathfrak N ) $ is of particular interest.

3) Best approximation of a fixed set $ \mathfrak M \subset X $ by a given class $ \{ \mathfrak N \} $ of approximating sets in $ X $. It is assumed that in $ \{ \mathfrak N \} $ there are, in a definite sense, "equally-expensive" classes, e.g. containing the same amount of elements or having the same dimension. The first case leads to the $ \epsilon $- entropy of $ \mathfrak M $( relative to $ X $), the second — to the problem of calculating the width of $ \mathfrak M $( in $ X $), in particular,

$$ \tag{1 } d _ {N} ( \mathfrak M , X) = \ \inf _ {\mathfrak N _ {N} } \ E ( \mathfrak M , \mathfrak N _ {N} ), $$

and

$$ \tag{2 } d _ {N} ^ { \prime } ( \mathfrak M , X) = \ \inf _ {\mathfrak N _ {N} } \ {\mathcal E} ( \mathfrak M , \mathfrak N _ {N} ), $$

where the infimum is taken over all subspaces $ \mathfrak N _ {N} $ in $ X $ of fixed dimension $ N $( or over all possible translations $ \mathfrak N _ {N} + a $ of it). Thus in (1), (2) the problem is to determine the best (respectively the best linear) approximation tool of dimension $ N $ for $ \mathfrak M $.

References

[1]	N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[2]	V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian)
[3]	V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian)
[4]	S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[5]	N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian)
[6]	V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)
[7]	A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)
[8]	J.R. Rice, "The approximation of functions" , 1–2 , Addison-Wesley (1964–1968)
[9]	P.J. Davis, "Interpolation and approximation" , Blaisdell (1965)
[10]	G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966)

Comments

There are many good books on approximation (and interpolation) theory. Below some of them are mentioned. [a1] contains extensive notes (related to the history of theorems and methods), as well as a useful bibliography. The more recent book [a2] also contains an extensive bibliography.

References

[a1]	E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982)
[a2]	A.S.B. Holland, B.N. Sahney, "The general problem of approximation and spline functions" , R.E. Krieger (1979)
[a3]	I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian)
[a4]	I.M. Singer, "Best approximation in normed linear spaces by elements of linear subspaces" , Springer (1970) (Translated from Rumanian)
[a5]	A. Pinkus, "$n$-widths in approximation theory" , Springer (1985) (Translated from Russian)