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The paper generalizes the instruction, suggested by B. Sz.-Nagy and C. Foias, for operatorfunction induced by the Cauchy problem $$T_t:\left\{\begin{array}{c}t{h}^{\text{'}\text{'}}\left(t\right)+(1-t)h^{\text{'}}\left(t\right)+Ah\left(t\right)=0\hfill \\ h\left(0\right)=h_0\left(t{h}^{\text{'}}\right)\left(0\right)=h_1\hfill \end{array}\right.$$ A unitary dilatation for $T_t$ is constructed in the present paper. then a translational model for the family $T_t$ is presented using a model construction scheme, suggested by Zolotarev, V., [3]. Finally, we derive a discrete functional model of family $T_t$ and operator $A$ applying the Laguerre transform $$f\left(x\right)\to {\int }_0^{\infty }f\left(x\right)P_n\left(x\right)e^{-x}dx$$ where $P_n\left(x\right)$ are Laguerre polynomials [6, 7]. We show that the Laguerre transform...

The present paper is devoted to further development of commutative nonstationary field themes; the first studies in this area were performed by K. Kirchev and V. Zolotarev [4, 5]. In this paper a more complicated variant of commutative field with nonstationary rank 2, carrying into more general situation for correlation function is studied. A condition of consistency (see (7) below) for commutative field is placed in the basis of the method proposed in [4, 5] and developed in this paper. The following...