2.6.1.4 Fourier integral

Problem: Find the Fourier image \( F[f(x)](v) \) \( =\int_{-\infty}^{+\infty} f(t) e^{-i v t} d t \) of function \( f(t) \), using equalities \( F[ \) rect \( t](v)=\operatorname{sinc} \frac{v}{2} \) and \( F[\Lambda(t)](v)=\operatorname{sinc}^{2} \frac{v}{2} \). The graph of function \( f(t) \) consists a polygonal chain, connecting points \( A, B, C \) and \( D \), as well as from parts of the \( \mathrm{x} \)-axis (to the left of \( A \) and to the right of \( D \) ). \begin{tabular}{|c|c|c|c|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline\( (-1,1) \) & \( (1,-2) \) & \( (4,-2) \) & \( (6,1) \) \\ \hline \end{tabular}