Functional Analysis

Fourier Integrals and Fourier Transforms Exercise 1 : Fourier Series For the following periodic function: $$ f(t) = \begin{cases} -1 & \text{for } -\frac{T}{2} \leq t < 0 \\ 1 & \text{for } 0 < t \leq \frac{T}{2} \end{cases} $$ Calculate the Fourier series for $$ f(t) = \begin{cases} 0 & \text{for } -\frac{T}{2} \leq t < -\frac{t_0}{2} \\ -1 & \text{for } -\frac{t_0}{2} \leq t < 0 \\ 1 & \text{for } 0 < t \leq \frac{t_0}{2} \\ 0 & \text{for } \frac{t_0}{2} < t \leq \frac{T}{2} \end{cases} $$Exercise 2 : Limiting Spectrum (T \to \infty) Now perform the limiting process for $T \to \infty$ and obtain the amplitude spectrum. ...

Fourier Series; Harmonic Analysis Exercise 1 : Symmetric rectangular pulse Obtain the Fourier series of the 2\pi-periodic function $$ f(x)=\begin{cases} 0, & -\pi\le x< -\tfrac{\pi}{2},\\ 1, & -\tfrac{\pi}{2}\le x< \tfrac{\pi}{2},\\ 0, & \tfrac{\pi}{2}\le x\le \pi, \end{cases} $$State whether the periodic extension is even or odd, compute the nonzero Fourier coefficients, and write the series in sine/cosine form. Exercise 2 : Odd square wave For the 2\pi-periodic function $$ f(x)=\begin{cases} 1, & -\pi\le x<0,\\ -1, & 0\le x\le\pi, \end{cases} $$Compute the Fourier series, indicate whether only sine terms appear, and derive the explicit formula for the coefficients. ...