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Problem 405

Find the Fourier series of the function \(\mathrm{f}(\mathrm{x})=|\mathrm{x}|,-\pi<\mathrm{x} \leq \pi\).

Expert verified

The Fourier series of the function \(f(x) = |x|\), within the interval \(-\pi < x \leq \pi\), is given by:
\[f(x) = \frac{\pi}{4} + \sum_{n=1}^{\infty} \frac{2}{n\pi} (\frac{-1}{n^2}[(-1)^n - 1])\cos(nx)\]

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Chapter 14

The Fourier series for $\mathrm{f}(\mathrm{x})=|\mathrm{x}|
\quad-\pi<\mathrm{x} \leq \pi$ and
$$
\begin{aligned}
&\mathrm{f}(\mathrm{x}+2 \pi)=\mathrm{f}(\mathrm{x}) \text { is } \\
&\qquad(\pi / 2)-(4 / \pi)^{\infty} \sum_{\mathrm{n}=1}\left[\\{\cos (2
\mathrm{n}-1) \mathrm{x}\\} /\left\\{(2 \mathrm{n}-1)^{2}\right\\}\right]
\end{aligned}
$$
Without computing any Fourier coefficients, find the Fourier series for
$$
\begin{array}{ll}
g(x)=-1 & -\pi

Chapter 14

State the most general Pointwise Convergence Theorem for Fourier series (i.e. the one with the weakest premises). Discuss its meaning.

Chapter 14

Consider the infinite trigonometric series $\left(a_{0} / 2\right)+{ }^{\infty} \Sigma_{n=1}\left(a_{n} \cos n x+b_{n} \sin n x\right)$ and assume that it converges uniformly for all \(\mathrm{x} \in(-\pi, \pi)\). It can then be considered as a function \(f\) of \(x\) with period \(2 \pi\), i.e. $\mathrm{f}(\mathrm{x})=\left(\mathrm{a}_{0} / 2\right)+{ }^{\infty} \sum_{\mathrm{n}=1} \mathrm{a}_{\mathrm{n}} \cos \mathrm{n} \mathrm{x}+\mathrm{b}_{\mathrm{n}} \sin \mathrm{n} \mathrm{x}$ Determine the values of \(a_{n}, b_{n}\) in terms of \(f(x)\)

Chapter 14

Determine the Fourier series of the function given by $$ \begin{aligned} &f(x)=x^{2}, x \in(-\pi, \pi) \\ &f(x+2 \pi)=f(x), a 11 x \end{aligned} $$

Chapter 14

A piecewise continuous function \(\mathrm{f}(\mathrm{x})\) is to be approximated in the interval \((-\pi, \pi)\) by a trigonometric polynomial of the form $$ g_{n}(x)=\left(A_{0} / 2\right)+{ }^{n} \sum_{k=1} A_{k} \cos k x+B_{k} \sin k x $$ where \(\mathrm{A}_{\mathrm{k}}, \mathrm{B}_{\mathrm{k}}, \mathrm{A}_{\circ}\) are undetermined. Prove that the total square deviation $$ \mathrm{D}_{\mathrm{n}}=\pi \int_{-\pi}\left[\mathrm{f}(\mathrm{x})-\mathrm{g}_{\mathrm{n}}(\mathrm{x})\right]^{2} \mathrm{~d} \mathrm{x} $$ is minimized by choosing \(A_{0}, A_{k}, B_{k}\) to be the Fourier coefficients of $\mathrm{f}, \mathrm{a}_{\mathrm{k}}, \mathrm{b}_{\mathrm{k}}, \mathrm{a}_{0}$.

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