Best linear method
With respect to the approximation of elements in a given set 
, the linear method that yields the smallest error among all linear methods. In a normed linear space 
, a linear method for the approximation of elements 
by elements of a fixed subspace 
is represented by a linear operator that maps the entire space 
, or some linear manifold containing 
, into 
. If 
is the set of all such operators, a best linear method for 
(if it exists) is defined by an operator 
for which
 |
The method defined by an operator 
in 
will certainly be a best linear method for 
relative to the approximating set 
if, for all 
,
 |
(
is the best approximation of 
by 
) and if, moreover, for all 
,
 |
The latter is certainly true if 
is a Hilbert space, 
is an 
-dimensional subspace of 
, 
and 
is the orthogonal projection onto 
, i.e.
 |
where 
is an orthonormal basis in 
.
Let 
be a Banach space of functions defined on the entire real line, with a translation-invariant norm: 
(this condition holds, e.g. for the norms of the spaces 
and 
, 
, of 
-periodic functions); let 
be the subspace of trigonometric polynomials of order 
. There exist best linear methods (relative to 
) for a class 
of functions 
that contains 
for any 
whenever it contains 
. An example is the linear method
 | (*) |
 |
where 
and 
are the Fourier coefficients of 
relative to the trigonometric system, and 
and 
are numbers.
Now consider the classes 
(and 
), 
of 
-periodic functions 
whose derivatives 
are locally absolutely continuous and whose derivatives 
are bounded in norm in 
(respectively, in 
) by a number 
. For these classes, best linear methods of the type (*) yield the same error (over the entire class) in the metric of 
(respectively, 
) as the best approximation by a subspace 
; the analogous assertion is true for these classes with any rational number 
(interpreting the derivatives 
in the sense of Weyl). For integers 
best linear methods of type (*) have been constructed using only the coefficients 
(all 
).
If 
is the subspace of 
-periodic polynomial splines of order 
and defect 1 with respect to the partition 
, 
then a best linear method for the classes 
(and 
), 
is achieved in 
, 
(resp. in 
) by splines in 
interpolating the function 
at the points 
.
References
| [1] | N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian) |
| [2] | N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian) |
| [3] | V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian) |
Comments
References
| [a1] | H. Kiesewetter, "Vorlesungen über lineare Approximation" , Deutsch. Verlag Wissenschaft. (1973) |
| [a2] | J.R. Rice, "The approximation of functions" , 1. Linear theory , Addison-Wesley (1964) |
Best linear method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Best_linear_method&oldid=46042