# 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)