Fourier analysis – a term named after the French mathematician Joseph Fourier, is the process of breaking down a complex function and expressing it as a combination of simpler functions. The opposite process of combining simpler functions to reconstruct the complex function is termed as Fourier Synthesis.
Mostly, the simpler functions are chosen to be sine and cosine functions. Thus, the term “Fourier analysis” expresses a complex function in terms of sine and cosine terms and the term “Fourier Analysis” reconstructs the complex function from the sine and cosine terms.
Frequency is the measure of number of repetitive occurrences of a particular event. By definition, a sine wave is a smooth curve that repeats at a certain frequency. Thus, the term “frequency” and sine are almost synonymous. A cosine wave is also a sine wave but with 90* phase shift. Therefore, when you talk about sine and cosine functions, you are taking in terms of “frequencies”. That is why in signal processing, the Fourier analysis is applied in frequency (or spectrum) analysis.
Applied on functions that are periodic. A periodic function is broken down and expressed in terms of sine and cosine terms. In mathematics, the term “series” represents a sum of sequence of numbers. For example we can make a series with a sequence of numbers that follows Geometric Progression (common ratio between the numbers).