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A basis function is a function from a selected set of functions used in approximating a function of interest. Modeling, e.g., using numerical approximations of partial differential equations (PDEs) is organized through the use of basis functions which are parameterized and combined (e.g., summed) to approximate a function of interest.
One example is the sine function, as used to form a Fourier series. The concept is also used in the finite element method and similar methods of solving PDEs.