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)

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)