We present an exact expression for the $ L _{ 2 } $ error that occurs when one approximates a periodic signal in a basis of shifted and scaled versions of a generating function. This formulation is applicable to a wide variety of linear approximation schemes including wavelets, splines, and bandlimited signal expansions. The formula takes the simple form of a Parseval's-like relation, where the Fourier coefficients of the signal are weighted against a frequency kernel that characterizes the approximation operator. We use this expression to analyze the behavior of the error as the sampling step approaches zero. We also experimentally verify the expression of the error in the context of the interpolation of closed curves.