Evaluation criteria of representation error of signals as orthogonal series

It is proposed to decompose a signal into a series of functions that are orthogonal with weight, which can be both positive and negative values. It is shown that in this case there are two criteria for evaluating the mean square error of this approximation. It is expedient to use one of the criteria to assess the effect of noise for the correct reception. This criterion allows determining the form of the basis functions, which minimize the error. The second criterion should be used to determine the approximation signal error as finite sum of the series. If the basis functions are equidistant, the error is minimized by determining the displacement intervals of basis functions.