Generalized Fourier series

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In mathematics, a Generalized Fourier series expands a square-integrable function defined on an interval over the real line. The constituent functions in the series expansion form an orthonormal basis of an inner product space. While a Fourier series expansion consists of only trigonometric functions, a generalized Fourier series is a decomposition involving any set of functions that satisfy the Sturm-Liouville eigenvalue problem. These expansions find common use in interpolation theory.^[1] It is expressed by a series of sinusoids that can be stated in various forms. In essence, a pair of functions is considered, where t is a variable (usually time), and m and n are real multipliers of t, reflecting the length of the interval.