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 \(f(x)\)

Find the Fourier cosine series over the interval \(0<\mathrm{x}<\mathrm{c}\) for the function \(\mathrm{f}(\mathrm{x})=\mathrm{x}\)

Find the Fourier sine series for the function defined by $$ \begin{array}{ll} f(x)=0 & 0 \leq x<\pi / 2 \\ f(x)=1 & \pi / 2

Define the following properties of a real valued function \(\mathrm{f}\) of a real variable: (a) The "limit from the right" of \(\mathrm{f}\) at \(\mathrm{x}_{\mathrm{o}}\) (b) The "limit from the left" of \(\mathrm{f}\) at \(\mathrm{x}_{\circ}\) (c) \(\mathrm{f}\) is piecewise continuous on \((\mathrm{a}, \mathrm{b})\). (d) The right and left hand derivatives of \(\mathrm{f}\) at \(\mathrm{x}_{0}\)

The Fourier series of the following functions have been found in previous problems but no convergence questions were discussed. Determine now which of these functions has Fourier series which are (i) pointwise convergent (ii) uniformly convergent and (iii) convergent in the mean: (a) $\mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}} \quad(-\pi<\mathrm{x}<\pi)$ (b) \(\mathrm{f}(\mathrm{x})=\mathrm{x} \quad, 0<\mathrm{x}<\pi\) and \(=0 \quad, \pi

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 \(\mathrm{f}\).