Approximation Theory and Method part 2

Approximation operators

在前面的讨论中，我们得到了 best approximation 的一些性质. 但是实际上我们并不总是能有 best approximation 这么好的结果。那么我们能不能退而求其次，研究一下更具一般性的 approximation operators 呢？

We note that the operator \(\boldsymbol X\) is defined to be a projection if the equation

\[\boldsymbol{X}[\boldsymbol{X}(f)]=\boldsymbol{X}(f), \quad f \in \mathscr{B}, \]

is satisfied. Hence a sufficient condition for \(X\) to be a projection is the equation

\[X(a)=a, \quad a \in \mathscr{A} \]

Most of the approximation methods that are considered in this book do satisfy condition (3.2), but an important exception is the Bernstein operator, which is discussed in Chapter 6. Sometimes \(X(f)\) is written as \(X f\).

The idea of a linear operator is also well known; namely, we define \(X\) to be linear if the equation

\[X(\lambda f)=\lambda \boldsymbol{X}(f) \]

holds for all \(f \in \mathscr{B}\), where \(\lambda\) is any real number, and if the equation

\[\boldsymbol{X}(f+g)=\boldsymbol{X}(f)+\boldsymbol{X}(g) \]

is obtained for all \(f \in \mathscr{B}\) and for all \(g \in \mathscr{B}\).

最好还是定义一下这些 approximation operator 的 norm.

Also we make frequent use of the norm of an approximation operator. The norm of \(X\) is written as \(\|X\|\), and it is the smallest real number such that the inequality

\[\|\boldsymbol{X}(f)\| \leqslant\|\boldsymbol{X}\|\|f\| \]

或者可以这样写

\[\|\boldsymbol{X} \| = \sup _{ \|f\| \neq 0} \frac{\|\boldsymbol{X}(f)\| }{\|f\|} \]

这和我们定义矩阵时是类似的. 比如矩阵\(A\) 的 norm:

\[\|A \| = \sup _{ \|x\| \neq 0} \frac{\|Ax\| }{\|x\|} \]

关于 norm of approximation operator 的取值问题，可以考虑一个例子.

An example of an approximation operator that is useful because it is easy to apply is as follows. Let \(\mathscr{B}\) be the space \(\mathscr{C}[0,1]\) of real-valued functions that are continuous on \([0,1]\), and let \(\mathscr{A}\) be the linear space \(\mathscr{P}_1\) of all real polynomials of degree at most one. Then, in order that the calculation of an approximation to a function \(f\) in \(\mathscr{R}\) depends on only two function evaluations, we let \(p\) be the polynomial in \(\mathscr{A}\) that satisfies the interpolation conditions

\[\left.\begin{array}{l} p(0)=f(0) \\ p(1)=f(1) \end{array}\right\} \]

Thus \(p=X(f)\), where \(X\) is a linear projection operator from \(\mathscr{B}\) to \(\mathscr{A}\).
In order to define the norm of this operator we choose a norm for the space \(\mathscr{C}[0,1]\). However, if the 2-norm

\[\|f\|_2=\left\{\int_0^1[f(x)]^2 \mathrm{~d} x\right\}^{\frac{1}{2}}, \quad f \in \mathscr{C}[0,1], \]

is used, we find that the operator \(X\) is unbounded, because it is possible for \(\|X f\|_2\) to be one when \(\|f\|_2\) is arbitrarily small. It is therefore necessary to prefer the \(\infty\)-norm

\[\|f\|_{\infty}=\max _{0 \leqslant x \leqslant 1}|f(x)|, \quad f \in \mathscr{C}[0,1] \]

when considering approximation operators that are defined by interpolation conditions. In this case, because \(p\) is in \(\mathscr{P}_1\), equation (3.6) implies the inequality

\[\begin{aligned} \|X(f)\| & =\|p\| \\ & =\max [|p(0)|,|p(1)|] \\ & =\max [|f(0)|,|f(1)|] \\ & \leqslant\|f\|, \quad f \in \mathscr{C}[0,1] . \end{aligned} \]

在这个例子中，我们的 approximation operator 可以理解为把 0 和 1 处，也就是函数的两端进行连线. 可以发现此时 2-norm 可以去到很大，而 inf-norm 是有上确界的.