Best approximation
of a function by functions
from a fixed set
The quantity
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where is the error of approximation (see Approximation of functions, measure of). The concept of a best approximation is meaningful in an arbitrary metric space
when
is defined by the distance between
and
; in this case
is the distance from
to the set
. If
is a normed linear space, then for a fixed
the best approximation
![]() | (1) |
may be regarded as a functional defined on (the functional of best approximation).
The functional of best approximation is continuous, whatever the set . If
is a subspace, the functional of best approximation is a semi-norm, i.e.
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and
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for any . If
is a finite-dimensional subspace, then for any
there exists an element
(an element of best approximation) at which the infimum in (1) is attained:
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In a space with a strictly convex norm, the element of best approximation is unique.
Through the use of duality theorems, the best approximation in a normed linear space can be expressed in terms of the supremum of the values of certain functionals from the adjoint space
(see, e.g. [5], [8]). If
is a closed convex subset of
, then for any
![]() | (2) |
in particular, if is a subspace, then
![]() | (3) |
where is the set of functionals
in
such that
for any
. In the function spaces
or
, the right-hand sides of (2) and (3) take explicit forms depending on the form of the linear functional. In a Hilbert space
, the best approximation of an element
by an
-dimensional subspace
is obtained by orthogonal projection on
and can be calculated; one has:
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where form a basis of
and
is the Gram determinant, the elements of which are the scalar products
,
. If
is an orthonormal basis, then
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In the space one has the following estimate for the best uniform approximation of a function
by an
-dimensional Chebyshev subspace
(the de la Vallée-Poussin theorem): If for some function
there exist
points
,
, for which the difference
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takes values with alternating signs, then
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For best approximations in see Markov criterion. In several important cases, the best approximations of functions by finite-dimensional subspaces can be bounded from above in terms of differential-difference characteristics (e.g. the modulus of continuity) of the approximated function or its derivatives.
The concept of a best uniform approximation of continuous functions by polynomials is due to P.L. Chebyshev (1854), who developed the theoretical foundations of the concept and established a criterion for polynomials of best approximation in the metric space (see Polynomial of best approximation).
The best approximation of a class of functions is the supremum of the best approximations of the functions in the given class
by a fixed set of functions
, i.e. the quantity
![]() |
The number characterizes the maximum deviation (in the specific metric chosen) of the class
from the approximating set
and indicates the minimal possible error to be expected when approximating an arbitrary function
by functions of
.
Let be a subset of a normed linear function space
, let
be a linearly independent system of functions in
and let
,
be the subspaces generated by the first
elements of this system. By investigating the sequence
,
one can draw conclusions regarding both the structural and smoothness properties of the functions in
and the approximation properties of the system
relative to
. If
is a Banach function space and
is closed in
, i.e.
, then
as
if and only if
is a compact subset of
.
In various important cases, e.g. when the are subspaces of trigonometric polynomials or periodic splines, and the class
is defined by conditions imposed on the norm or on the modulus of continuity of some derivative
, the numbers
can be calculated explicitly [5]. In the non-periodic case, results are available concerning the asymptotic behaviour of
as
.
References
[1] | P.L. Chebyshev, "Complete collected works" , 2 , Moscow (1947) (In Russian) |
[2] | N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian) |
[3] | V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian) |
[4] | V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian) |
[5] | N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian) |
[6] | S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian) |
[7] | A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian) |
[8] | V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian) |
[9] | P.J. Laurent, "Approximation et optimisation" , Hermann (1972) |
Comments
In Western literature an element, a functional or a polynomial of best approximation is also called a best approximation.
References
[a1] | G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966) |
[a2] | E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff |
[a3] | J.R. Rice, "The approximation of functions" , 1. Linear theory , Addison-Wesley (1964) |
[a4] | A. Pinkus, "![]() |
Best approximation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Best_approximation&oldid=16361