The term "Fourier function" is not a standard mathematical term. When people refer to Fourier functions, they usually mean functions that can be represented by Fourier series or analyzed using Fourier transforms. These are functions that can be decomposed into combinations of sine and cosine waves, like the example shown here.
Fourier series allows us to represent periodic functions as infinite sums of sine and cosine terms. For example, a square wave can be approximated by adding more and more sine wave harmonics. As we increase the number of terms, the approximation becomes more accurate, demonstrating how complex periodic functions can be built from simple trigonometric components.
The Fourier Transform extends Fourier analysis to non-periodic functions. Instead of discrete frequency components like in Fourier series, the transform decomposes functions into continuous frequency spectra. For example, a Gaussian pulse in the time domain transforms to another Gaussian in the frequency domain, showing how localized signals spread across frequencies.
The basis functions in Fourier analysis are sine and cosine functions, or equivalently, complex exponentials. These functions are orthogonal, meaning they are mathematically independent of each other. By combining different frequencies of these basis functions, we can represent any function that satisfies certain mathematical conditions. This is the fundamental principle that makes Fourier analysis so powerful.
To conclude, the term "Fourier function" encompasses functions that can be represented by Fourier series for periodic cases, functions analyzable by Fourier transforms for non-periodic cases, and the fundamental sine and cosine basis functions used in Fourier analysis. These mathematical tools are essential in many fields including signal processing, quantum mechanics, and engineering, providing powerful methods for analyzing and understanding complex waveforms and signals.