Quantitative L2 approximation error of a probability density estimate given by its samples

We present a new result characterized by an exact integral expression for the approximation error between a probability density and an integer shift invariant estimate obtained from its samples. Unlike the Parzen window estimate, this estimate avoids recomputing the complete probability density for each new sample - only a few coefficients are required, making it practical for real-time applications. We also show how to obtain the exact asymptotic behavior of the approximation error when the number of samples increases and provide the trade-off between the number of samples and the sampling step size.