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An introduction to the approximation of functions

Theodore J. Rivlin

Chapter 4

POLYNOMIAL AND SPLINE INTERPOLATION - all with Video Answers

Educators

Chapter Questions

Problem 1

Given any $X$, show that
$$
\sum_{j=1}^{k+1} l_f^{(k)}(x) \equiv 1, \quad k=0,1, \ldots,
$$
and hence that
$$
\lambda_k(X ; x) \geq 1, \quad k=0,1, \ldots, x \in I .
$$

When does equality hold in (4.4.27)?

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01:05

Problem 2

Suppose that $f$ has an $n$th derivative in $I$ that satisfies $\left\|f^{(n)}\right\|=M$. Show that, for each $x \in I$,
$$
f(x)-L_{n-1}(x)=\frac{\left(x-x_1^{(n)}\right) \cdots\left(x-x_n^{(n)}\right)}{n !} f^{(n)}(\xi)
$$
for some $\xi(x) \in I$ and, hence, that
$$
G_{n-1}=\left\|f-L_{n-1}\right\| \leq\left\|\left(x-x_1^{(n)}\right) \cdots\left(x-x_n^{(n)}\right)\right\| \frac{M}{n !} .
$$

Carson Merrill
Carson Merrill
Numerade Educator
02:34

Problem 3

Show that, if $M$ is as given in Exercise 4.2, the right-hand side of (4.4.29) is least when $x_1^{(n)}, \ldots, x_n^{(n)}$ are the zeros of the Chebyshev polynomial, $T_n(x)$. What is the right-hand side of (4.4.29) in this case?

WM
William Mead
Numerade Educator
01:02

Problem 4

Let $T$ be the matrix of nodes whose $k$ th row consists of the zeros of $T_k(x), k=1,2, \ldots$; that is

Tyler Moulton
Tyler Moulton
Numerade Educator
03:54

Problem 5

If we write $c_f^{+}$for $\cot \left(\theta+\theta_f\right) / 2$ and $c_f^{-}$for $\cot \left(\theta-\theta_f\right) / 2$, show that, if $\theta$ is replaced by $\theta-k \pi /(n+1), k=1,2, \ldots, n+1$, then
$$
\begin{array}{ll}
c_j^{+} \rightarrow c_{j-k}^{+}, & j=k+1, \ldots, n+1, \\
c_j^{+} \rightarrow c_{k+1-f}^{-}, & j=1, \ldots, k, \\
c_j^{-} \rightarrow c_{j+k}^{-}, & j=1, \ldots, n+1-k, \\
c_j^{-} \rightarrow c_{2 n+3-j-k}^{+}, & j=n+2-k, \ldots, n+1,
\end{array}
$$

Aman Gupta
Aman Gupta
Numerade Educator
01:00

Problem 6

Show that
$$
\lim _{n \rightarrow \infty}\left[\Lambda_n(T)-\left(\frac{2}{\pi} \log n+\frac{2}{\pi}\left(\log \frac{8}{\pi}+\gamma\right)\right)\right]=0,
$$
where $\gamma$ is Euler's constant, $\gamma=0.5772 \ldots$; that is,
$$
\gamma=\lim _{k \rightarrow \infty}\left[\sum_{j=1}^{k-1} \frac{1}{j}-\log k\right] .
$$

Wendi Zhao
Wendi Zhao
Numerade Educator

Problem 7

Show that, if $f$ defined on $I$ satisfies (the Dini-Lipschitz condition)
$$
\lim _{\delta \rightarrow 0} \omega(f ; \delta) \log \delta=0,
$$
then $L_n(f, T)$ converges uniformly to $f$ on $I$. In particular, (4.4.30) surely holds if $f$ has a bounded derivative on $I$.

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01:31

Problem 8

Show that
$$
\lim _{n \rightarrow \infty} L_{2 n-1}(|x|, E ; 0)=0
$$
and, hence, that
$$
\lim _{n \rightarrow \infty} L_n(|x|, E ; 0)=0 .
$$
approximate $D^k f$ by $D^k L_n(f, X)$, where $X$ is an array of nodes as defined in (4.1.4). Thus
$$
D^k L_n(f, X)=\sum_{i=1}^{n+1} f\left(x_i\right) D^k l_i(x) .
$$

Khushbu Rani
Khushbu Rani
Numerade Educator

Problem 9

If $W$ is given, show that
$$
\left|I-I_n(W)\right| \leq 2 E_n\left[\int_{-1}^1 w(x) d x \cdot \int_{-1}^1 \frac{1}{w(x)} d x\right]^{1 / 2},
$$
so that, for Gaussian Quadrature, $W=W_0$,
$$
\left|I-I_{\mathrm{n}}\left(W_0\right)\right| \leq 4 E_{\mathrm{n}} .
$$

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01:19

Problem 10

If $T$ is the array of Chebyshev nodes, show that
$$
\left|I-I_n(T)\right| \leq \sqrt{2} \pi E_n .
$$

Just as interpolating polynomials provide a method for numerical integration, they can be used for numerical differentiation, that is, approximating the derivatives of a given function. Suppose that $f \in C^k[-1,1]$; the idea is to
approximate $D^k f$ by $D^k L_n(f, X)$, where $X$ is an array of nodes as defined in (4.1.4). Thus
$$
D^k L_n(f, X)=\sum_{i=1}^{n+1} f\left(x_i\right) D^k l_i(x) .
$$

Let us put
$$
\Lambda_n^{(k)}(X)=\max _{-1 \leq x \leq 1} \sum_{i=1}^{n+1}\left|D^k l_i(x)\right|,
$$
and
$$
\mathscr{L}_n^{(k)}=\inf _x \Lambda_n^{(k)}(X) .
$$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 11

Show that, if $k$ does not exceed $n$,
$$
\left\|D^k f-D^k L_n(f, X)\right\| \leq \Lambda_n^{(k)}(X) E_n(f ;[-1,1])+\left\|D^k f-D^k p\right\|,
$$
where $p$ is the best uniform approximation to $f$ on $[-1,1]$ out of $P_n$ and $\|\cdot\|$ is the uniform norm.

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01:56

Problem 12

Show that Exercise 4.11 implies that, for $0 \leq k \leq n$,
$$
\left\|D^k f-D^k L_n(f, X)\right\| \leq E_{n-k}\left(D^k f\right)\left[1+\frac{C}{n^k} \Lambda_n^{(k)}(X)\right],
$$
where $C=6^k e^k(1+k)^{-1}$.

Anna Jones
Anna Jones
Numerade Educator
01:56

Problem 13

Show that, if $1 \leq k \leq n$,
$$
\mathscr{L}_n^{(k)} \geq D^k T_n(1) .
$$

The value of $D^k T_n(1)=T_n^{(k)}(1)$ is given in (1.2.22).

Anna Jones
Anna Jones
Numerade Educator
05:46

Problem 14

Show that, if $1 \leq k \leq n$,
$$
\mathscr{L}_n^{(k)}=\Lambda_n^{(k)}(T)=D^k T_n(1) .
$$

This result may be interpreted as saying that the Chebyshev nodes, $T$ (defined on p. 93), are an optimal set for numerical differentiation. Compare the discussion on p. 96.

Mengchun Cai
Mengchun Cai
Numerade Educator

Problem 15

Show that there is a unique cubic spline satisfying
$$
s\left(x_i\right)=f_i, \quad i=0, \ldots, n \quad \text { and } \quad s^{\prime \prime}\left(x_i\right)=0, \quad i=0, n .
$$

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Problem 16

Suppose that $n>3$ and $s(x)=0$ for $x$ outside $\left(x_1, x_{i+3}\right)$ for some $i$, $0 \leq i \leq n-3$. Show that $s(x) \equiv 0$.

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01:51

Problem 17

Show that, if $n>3$, there exists $s(x) \in S$, not identically zero, which is zero outside $\left(x_i, x_{1+4}\right)$ for each $i=0, \ldots, n-4$.

James Chok
James Chok
Numerade Educator

Problem 18

If
$$
s(x) \in C^3[a, b],
$$
show that $s \in P_3$.

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Problem 19

Show that every $s \in S\left(X_n\right)$ has the representation
$$
s(x)=p(x)+\sum_{i=1}^{n-1} c_i\left(x-x_i\right)_{+}^3
$$
for some choice of $c_6$ and $p \in P_3$.

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