Fourier transform as a limiting case of fourier series.it
Fourier series,a kind of periodic phenomena
Fourier series techniques is identified with the analysis of non periodic phenomena.
Analysis:[Function(signal)]:break up a signal into simple constituent parts.
Synthesis(合成):reassembling a signal or reassembling a function from its constituent parts.
Linear operations:integrals and series
Periodic phenomena often are either periodicity in time,a pattern repeats in time over and over again,as harmonic motion,or periodicity in space some object in space,one dimension,two dimensions.periodicity arises from the symmetry,
E.g,Heat on a circular ring:temperature on the ring,it’s periodic in the space.
Symmetry
The theory of groups
Periodicity in time :frequency,the number of repetitions
Periodicity in space:period
In th case of wave motion,there is a relationship between the frequency and the wavelength determined by the velocity,that can be two different phenomenon.periodicity in time and periodicity in space may not have anything to do with each other,but if you have a wave traveling,if you have a regularly repeating pattern over time,then they di have something to do with each other.