# Fourier coefficients

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The coefficients $$c_i = \frac{\int_X f \phi_i dx}{\int_X \phi_i^2 dx} \ \ \text{or}\ \ c_i = \frac{\int_X f \bar{\phi_i}dx}{\int_X \phi_i \bar{\phi_i} dx} \tag{*}$$ in the expansion of a function $f$ defined on a space $X$ with respect to an orthogonal system of real-valued (complex-valued) functions on $X$. If $\{ \phi_i \}$ is an orthogonal system in a Hilbert (pre-Hilbert) space, then, given an element $f$ of this space, the numbers $c_i = (f,\phi_i)/(\phi_i,\phi_i)$ are also called the Fourier coefficients of $f$ with respect to the system $\{ \phi_i \}$. J. Fourier first investigated trigonometric series with coefficients defined by (*).