Two relations for generalized discrete Fourier transform coefficients

A new generalized discrete Fourier transform DFT that allows for sample shift δ 2 [0, T/N] in time-domain is defined. Two relations are proved for the sum of errors between generalized DFT coefficients and theirs theoretical values. The first is the equation for samples received for continuous and piecewise–smooth functions. The second relation is the inequality for samples generated by discontinuous functions. Moreover, the influence of samples shift on generalized DFT coefficients values, which leads to aliasing phenomenon, is presented.