Biorthonormal expansion in signal spaces composed of spline functions

The authors derive the biorthonormal basis of the spline sampling basis to complete the biorthonormal expansion formula in the signal space composed of spline functions. This makes it possible to obtain an operator that evaluates the expansion coefficients as inner products of a given signal with the biorthonormal basis. It is also shown that to sample a signal by using the biorthonormal basis is to obtain sample values of a spline function that approximates the signal in the least mean squares sense. It is noted that one of the applications for the biorthonormal basis is to use it as an impulse response for approximation of waveforms or figures