Best linear method

With respect to the approximation of elements in a given set $ \mathfrak M $, the linear method that yields the smallest error among all linear methods. In a normed linear space $ X $, a linear method for the approximation of elements $ x \in \mathfrak M \subset X $ by elements of a fixed subspace $ F \subset X $ is represented by a linear operator that maps the entire space $ X $, or some linear manifold containing $ \mathfrak M $, into $ F $. If $ {\mathcal L} $ is the set of all such operators, a best linear method for $ \mathfrak M $( if it exists) is defined by an operator $ \widetilde{A} \in {\mathcal L} $ for which

$$ \sup _ {x \in \mathfrak M } \ \| x - \widetilde{A} x \| = \ \inf _ {A \in {\mathcal L} } \ \sup _ {x \in \mathfrak M } \ \| x - Ax \| . $$

The method defined by an operator $ A $ in $ {\mathcal L} $ will certainly be a best linear method for $ \mathfrak M $ relative to the approximating set $ F $ if, for all $ x \in \mathfrak M $,

$$ \| x - Ax \| \leq \ \sup _ {x \in \mathfrak M } E (x, F) $$

( $ E (x, F) $ is the best approximation of $ x $ by $ F $) and if, moreover, for all $ x \in X $,

$$ \| x - Ax \| = E (x, F). $$

The latter is certainly true if $ X $ is a Hilbert space, $ F = F _ {n} $ is an $ n $- dimensional subspace of $ X $, $ n = 1, 2 \dots $ and $ A $ is the orthogonal projection onto $ F _ {n} $, i.e.

$$ Ax = \ \sum _ {k = 1 } ^ { n } (x, e _ {k} ) e _ {k} , $$

where $ \{ e _ {1} \dots e _ {n} \} $ is an orthonormal basis in $ F _ {n} $.

Let $ X $ be a Banach space of functions defined on the entire real line, with a translation-invariant norm: $ \| x ( \cdot + \tau ) \| = \| x ( \cdot ) \| $( this condition holds, e.g. for the norms of the spaces $ C = C [0, 2 \pi ] $ and $ L _ {p} = L _ {p} (0, 2 \pi ) $, $ 1 \leq p \leq \infty $, of $ 2 \pi $- periodic functions); let $ F = T _ {n} $ be the subspace of trigonometric polynomials of order $ n $. There exist best linear methods (relative to $ T _ {n} $) for a class $ \mathfrak M $ of functions $ x (t) \in X $ that contains $ x (t + \alpha ) $ for any $ \alpha \in \mathbf R $ whenever it contains $ x (t) $. An example is the linear method

$$ \tag{* } A (x; t; \mu , \nu ) = \frac{\mu _ {0} a _ {0} }{2} + $$

$$ + \sum _ {k = 1 } ^ { n } \{ \mu _ {k} (a _ {k} \cos kt + b _ {k} \ \sin kt) + \nu _ {k} (a _ {k} \sin kt - b _ {k} \cos kt) \} , $$

where $ a _ {k} $ and $ b _ {k} $ are the Fourier coefficients of $ x (t) $ relative to the trigonometric system, and $ \mu _ {k} $ and $ \nu _ {k} $ are numbers.

Now consider the classes $ W _ \infty ^ {r} M $( and $ W _ {1} ^ {r} M $), $ r = 1, 2 \dots $ of $ 2 \pi $- periodic functions $ x (t) $ whose derivatives $ x ^ {(r - 1) } (t) $ are locally absolutely continuous and whose derivatives $ x ^ {(r)} (t) $ are bounded in norm in $ L _ \infty $( respectively, in $ L _ {1} $) by a number $ M $. For these classes, best linear methods of the type (*) yield the same error (over the entire class) in the metric of $ C $( respectively, $ L _ {1} $) as the best approximation by a subspace $ T _ {n} $; the analogous assertion is true for these classes with any rational number $ r > 0 $( interpreting the derivatives $ x ^ {(r)} (t) $ in the sense of Weyl). For integers $ r = 1, 2 \dots $ best linear methods of type (*) have been constructed using only the coefficients $ \mu _ {k} $( all $ \nu _ {k} = 0 $).

If $ F = S _ {n} ^ {r} $ is the subspace of $ 2 \pi $- periodic polynomial splines of order $ r $ and defect 1 with respect to the partition $ k \pi /n $, $ k = 0, \pm 1 \dots $ then a best linear method for the classes $ W _ \infty ^ {r + 1 } M $( and $ W _ {1} ^ {r + 1 } $), $ r = 1, 2 \dots $ is achieved in $ L _ {p} $, $ 1 \leq p \leq \infty $( resp. in $ L _ {1} $) by splines in $ S _ {n} ^ {r} $ interpolating the function $ x (t) $ at the points $ k \pi /n + [1 + (-1) ^ {r} ] \pi /4n $.

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)