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 \(\mathrm{f}\).

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

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

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

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

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 $\mathrm{f}, \mathrm{a}_{\mathrm{k}}, \mathrm{b}_{\mathrm{k}}, \mathrm{a}_{0}$.