Generalized Fourier series

Remarks

If the $\varphi_k$ are orthonormal, $c_k = \ip{f}{\varphi_k}$. The partial sum $\sum_{k \le N} c_k \varphi_k$ is the orthogonal projection of $f$ onto the span of $\varphi_0, \dots, \varphi_N$, the closest point of that span to $f$. By Bessel's inequality, $\sum_k \abs{c_k}^2 \norm{\varphi_k}^2 \le \norm{f}^2$. In a Hilbert space the series converges to the projection of $f$ onto the closed span of the family. It converges to $f$ for every $f$ exactly when the family is complete, and then Parseval's identity holds.

Example: in $L^2(-1, 1)$, take the Legendre polynomials $\varphi_0 = 1$, $\varphi_1 = x$, $\varphi_2 = \frac12 (3x^2 - 1)$, and so on, and $f(x) = x^2$. Then $\ip{f}{\varphi_0} = \frac23$, $\ip{\varphi_0}{\varphi_0} = 2$, $\ip{f}{\varphi_1} = 0$, $\ip{f}{\varphi_2} = \frac{4}{15}$ and $\ip{\varphi_2}{\varphi_2} = \frac25$. So $c_0 = \frac13$, $c_1 = 0$, $c_2 = \frac23$, and indeed $x^2 = \frac13 + \frac23 \cdot \frac12(3x^2 - 1)$; all later coefficients vanish.

Non-example: the formula fails to reconstruct $f$ when the family is not orthogonal. In $L^2(0, 1)$ with $\varphi_0 = 1$ and $\varphi_1 = x$, which are not orthogonal, the function $f = x$ gives $c_0 = \frac12$ and $c_1 = 1$, so $c_0 \varphi_0 + c_1 \varphi_1 = x + \frac12 \ne f$.