# 数学代写|数值分析代写numerical analysis代考|MATHS7104

## 数学代写|数值分析代写numerical analysis代考|Oscillation Theorem

The previous example hints that the points at which the error $f-p_*$ attains its maximum magnitude play a central role in the theory of minimax approximation. The Theorem of de la Vallée Poussin is a first step toward such a result. We include its proof to give a flavor of how such results are established.

Before proving this result, look at Figure $2.2$ for an illustration of the theorem. Suppose we wish to approximate $f(x)=\mathrm{e}^x$ with some quintic polynomial, $r \in \mathcal{P}5$ (i.e., $n=5$ ). This polynomial is not necessarily the minimax approximation to $f$ over the interval $[0,1]$. However, in the figure it is clear that for this $r$, we can find $n+2=7$ points at which the sign of the error $f(x)-r(x)$ oscillates. The red curve shows the error for the optimal minimax polynomial $p$ (whose computation is discussed below). This is the point of de la Vallée Poussin’s theorem: Since the error $f(x)-r(x)$ oscillates sign $n+2$ times, the minimax error $\pm\left|f-p_\right|_{\infty}$ exceeds $\left|f\left(x_j\right)-r\left(x_j\right)\right|$ at one of the points $x_j$ that give the oscillating sign. In other words, de la Vallée Poussin’s theorem gives a nice mechanism for developing lower bounds on $\left|f-p_*\right|_{\infty}$.

Proof. Suppose we have $n+2$ ordered points, $\left{x_j\right}_{j=0}^{n+1} \subset[a, b]$, such that $f\left(x_j\right)-r\left(x_j\right)$ alternates sign at consecutive points, and let $p_$ denote the minimax polynomial, $$\left|f-p_\right|_{\infty}=\min {p \in \mathfrak{P}_n}|f-p|{\infty .}$$
We will prove the result by contradiction. Thus suppose
(2.3) $\quad\left|f-p_*\right|_{\infty}<\left|f\left(x_j\right)-r\left(x_j\right)\right|, \quad$ for all $j=0, \ldots, n+1$.

## 数学代写|数值分析代写numerical analysis代考|Optimal Interpolation Points via Chebyshev Polynomials

As an application of the minimax approximation procedure, we consider how best to choose interpolation points $\left{x_j\right}_{j=0}^n$ to minimize
$$\left|f-p_n\right|_{\infty},$$
where $p_n \in \mathcal{P}n$ is the interpolant to $f$ at the specified points. Recall the interpolation error bound developed in Section 1.6: If $f \in C^{n+1}[a, b]$, then for any $x \in[a, b]$ there exists some $\xi \in[a, b]$ such that $$f(x)-p_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1) !} \prod{j=0}^n\left(x-x_j\right) .$$
Taking absolute values and maximizing over $[a, b]$ yields the bound
$$\left|f-p_n\right|_{\infty}=\max {\delta \in[a, b]} \frac{\left|f^{(n+1)}(\xi)\right|}{(n+1) !} \max {\mathrm{r} \in[a, h]}\left|\prod_{j=0}^n\left(x-x_j\right)\right| .$$
For Runge’s example, $f(x)=1 /\left(1+x^2\right)$ for $x \in[-5,5]$, we observed that $\left|f-p_n\right|_{\infty} \rightarrow \infty$ as $n \rightarrow \infty$ if the interpolation points $\left{x_j\right}$ are uniformly spaced over $[-5,5]$. However, Marcinkiewicz’s theorem (Section 1.6) guarantees there is always some scheme for assigning the interpolation points such that $\left|f-p_n\right|_{\infty} \rightarrow 0$ as $n \rightarrow \infty$. While there is no fail-safe a priori system for picking interpolations points that will yield uniform convergence for all $f \in C[a, b]$, there is a distinguished choice that works exceptionally well for just about every function you will encounter in practice. We determine this set of interpolation points by choosing those $\left{x_j\right}_{j=0}^n$ that minimize the error bound (which is distinct from – but hopefully akin to – minimizing the error itself, $\left|f-p_n\right|_{\infty}$ ). That is, we want to solve
$$\min {x_0, \ldots, x_n} \max {x \in[a, b]}\left|\prod_{j=0}^n\left(x-x_j\right)\right| .$$

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Oscillation Theorem

(2.3) $\left|f-p_*\right|_{\infty}<\left|f\left(x_j\right)-r\left(x_j\right)\right|$, 对所有人 $j=0, \ldots, n+1$.

## 数学代写|数值分析代写numerical analysis代考|Optimal Interpolation Points via Chebyshev Polynomials

$$f(x)-p_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1) !} \prod j=0^n\left(x-x_j\right) .$$

$$\left|f-p_n\right|{\infty}=\max \delta \in[a, b] \frac{\left|f^{(n+1)}(\xi)\right|}{(n+1) !} \operatorname{maxr} \in[a, h]\left|\prod{j=0}^n\left(x-x_j\right)\right| .$$

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