Determinant representation of Pade joint approximations

Determinant formulas for numerators and denominators of joint approximations of Pade $\pi_j(z), j=1,2 \dots,r$ of perfect system of functions $\{f_j(z)\}^r_{j=1}$ are established. The analogue of the Pade theorem is proved and the obvious kind of the remainders is found at the approach $f_j(z)$ by the rational function $\pi_j(z)$ . The received theorems supplement and generalise the known results of Ermite, Pade, K. Mahler, E.M. Nikishin, A.I. Aptekarev and other authors.