a day ago
- Fourier series can approximate non-periodic functions on a finite interval by extending the period, but for truly non-repeating functions the Fourier transform is used.
- The Fourier transform is derived as the limit of Fourier series coefficients as the period approaches infinity, turning discrete frequencies into a continuous spectrum.
- The Fourier transform converts a time-domain function to a frequency-domain representation, and the inverse transform converts it back.
- Key properties of the Fourier transform include linearity, scaling, time shifting, derivative transformation, and the convolution theorem (convolution in time equals multiplication in frequency).
- A sufficient condition for the Fourier transform to exist is absolute integrability of the function.