In this paper, we will present the optimum interpolation functions minimizing various measures of approximation error simultaneously. For an ordinary interpolatory approximation using sample values of a band-limited signal and a FIR filterbank system having analysis filters Hm((omega) ) (m equals 0,1,...,M - 1), we outline necessary formulation for the time-limited interpolation functions (psi) m(t) realizing the optimum approximation in each limited block separately. Further, under some assumptions, we will present analytic or piece-wise analytic interpolation functions (phi) m(t) minimizing various measures of approximation error defined at discrete time samples tn equals n (n equals 0,+/- 1,+/- 2,...). In this discussion, (phi) m(n) are equal to (psi) m(n) (n equals 0,+/- 1,+/- 2,...). Since (phi) m(t) are time-limited, (phi) m(n) vanish outside of the finite set of n. Hence, one can use FIR filters if one wants to realize discrete synthesis filters which impulse responses are (phi) m(n).