Talk:Video/The imaginary number i and the Fourier Transform

Fourier Series There are functions that are hard to work with for example discontinuous, fractal, and not smooth. On the other hand, there were functions has wonderful properties for example sin, cos, and Linear function. In the Fourier series, we can use functions with wonderful properties to proximate functions that are hard to work with. for example sin and cos to proximate fractal functions. In the Fourier series, we approximate the functions by adding up function together and the Fourier series will tell us what coefficient to use in our combination. However, In Fourier Series, you can only approximate a function on an interval.

Fourier transforms integral equation : ${\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi \xi x}\,dx,\quad \forall \ \xi \in \mathbb {R} .$

Example for Fourier transform: We have a signal called x(t) in time we will represent it in terms of the time domain. We also can represent it in another way which is called x(f) we will represent it in terms of the frequency domain and This is why we called transformation. Fourier Transform is an equivalent representation of the signal.

Convolution equation : $(f*g)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .$