ABSTRACT
The usefulness of orthonormality and completeness of expansion functions can be readily appreciated in e.g. spectral analysis of time signals. In signal processing, one of the key concerns is obtaining an adequate approximation for the energy of the signal in every small time interval. For this purpose, it is very useful to use the concept of an ‘approximation in the mean’ over, e.g. the time interval ranging from zero to infinity with respect to a positive weight function. Solving the problem of ‘approximation in the mean’ reveals that the minimum of the integral is possible to attain if Fourier coefficients are defined through the usual unsymmetric scalar product. A connection between expansion basis sets with the so-called completeness and separability axioms of the Hilbert eigenspace in quantum mechanics is discussed.
