Exercise 8.5.11

Question

\begin{enumerate}[label=(\alph*)] 
\item Use the fact that the Taylor series for $\sin(x)$ and $\cos(x)$ converge uniformly on any compact set to prove WAT under the added assumption that $[a,b]$ is $[0,\pi]$.
\item Show how the case for an arbitrary interval $[a, b]$ follows from this one.
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item For any function $f$, Fejér's Theorem implies we can construct a function of the form
$$g(x) = k_0 + \sum^{N_1}_{i=1} k_i \sin(c_i x) + \sum^{N}_{i=N_1 + 1} k_i \cos(c_i x)$$
satisfying $\left|g(x) - f(x)\right| < \epsilon/2$ over $x \in [0, \pi]$, and uniform convergence of the Taylor series of $\sin x$ and $\cos x$ imply we can find a set of polynomials $P_i$ satisfying
$$\left|P_i(x) - k_i \sin(c_i x)\right| < \frac{\epsilon}{2N}$$
for $1 \leq i \leq N_1$ and
$$\left|P_i(x) - k_i \cos(c_i x)\right| < \frac{\epsilon}{2N}$$
for $N_1 + 1 \leq i \leq N$
Then the polynomial $P(x) = k_0 + \sum^N_{i=1} P_i(x)$ satisfies
$$\left|P(x) - g(x)\right| < \frac{\epsilon}{2} \text{\quad and \quad} \left|P(x) - f(x) \right|< \epsilon$$
\item For $x \in [a,b]$, apply the variable substitution $y = \frac{\pi}{b-a}(x-a) $ and use part (a) on $y$.
 \end{enumerate}

Exercise 8.5.11

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Exercise 8.5.11

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