Topics In Applied Math

Fourier Transform Of Single Frequency Sines And Cosines The Fourier transform (FT) of function $f(x)$ is defined as follows: $$ F(k) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(x) e^{-ikx} dx $$ The FT is dervived from a Fourier series in complex notation. The Fourier series expands a function into a weighted set of orthogonal basis functions. In case of the FT, the basis functions are sines and cosines. Sines and cosines are periodic on $2\pi$ meaning that the functions repeat themselves at increments of $2 \pi$. The principle idea is that we can represent most functions as a linear combination - a weighted sum - of sine and cosine basis functions. When a function of time $t$ is Fourier transformed, the components of the FT correspond to frequencies of the basis sines and cosines. Frequency is proportional to inverse time. The FT space is often called reciprocal space. If the function transformed were a function of position $x$, the FT is a function of inverse position. I