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Problem 400

Let \(\mathrm{f}(\mathrm{x})\) be a real valued function of one variable, \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\), with a period \(2 c\) (i.e. \(\mathrm{f}(\mathrm{x}+2 \mathrm{c})=\mathrm{f}(\mathrm{x})\) for all \(\mathrm{x} \in \mathrm{R}\) ). Define the Fourier series of \(\mathrm{f}\).

Expert verified

The Fourier series representation of a real-valued periodic function \(f(x)\) with period \(2c\) is given by:
\[ f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos{(\frac{n\pi x}{c})} + b_n \sin{(\frac{n\pi x}{c})}) \]
The coefficients can be found using the following formulas:
1. \( a_0 = \frac{1}{c} \int_{-c}^{c} f(x)dx \)
2. \( a_n = \frac{1}{c} \int_{-c}^{c} f(x)\cos{(\frac{n\pi x}{c})}dx \)
3. \( b_n = \frac{1}{c} \int_{-c}^{c} f(x)\sin{(\frac{n\pi x}{c})}dx \)
For n = 1, 2, 3, ...
To compute the Fourier coefficients, substitute the given function \(f(x)\) (if provided) into the formulas and integrate over the interval \([-c, c]\).

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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

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

Find the Fourier series of the function \(\mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}},-\pi<\mathrm{X}<\pi\)

Chapter 14

Find the Fourier sine series of \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}\) over the interval \((0,1)\).

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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