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...

Following the ideas of R. DeMarr, we establish a Galois connection between distance functions on a set $S$ and inequality relations on $X_S=S\times ℝ$. Moreover, we also investigate a relationship between the functions of $S$ and $X_S$.

We show that Whitney?s approximation theorem holds in a general setting including spaces of (ultra)differentiable functions and ultradistributions. This is used to obtain real analytic modifications for differentiable functions including optimal estimates. Finally, a surjectivity criterion for continuous linear operators between Fréchet sheaves is deduced, which can be applied to the boundary value problem for holomorphic functions and to convolution operators in spaces of ultradifferentiable functions...

Let $(L,|\left|·\right|{|}_L)$ be a Banach lattice of equivalence classes of real-valued measurable functions on a σ-finite measure space and $T=T\left(u\right):u=(u₁,...,u_d),u_i>0,1\le i\le d$ be a strongly continuous locally bounded d-dimensional semigroup of positive linear operators on L. Under suitable conditions on the Banach lattice L we prove a general differentiation theorem for locally bounded d-dimensional processes in L which are additive with respect to the semigroup T.

Let L be a Banach lattice of real-valued measurable functions on a σ-finite measure space and T=$T_t$: t < 0 be a strongly continuous semigroup of positive linear operators on the Banach lattice L. Under some suitable norm conditions on L we prove a general differentiation theorem for superadditive processes in L with respect to the semigroup T.

Although there is an extensive literature on various means of two positive operators and their applications, these means do not typically readily extend to means of three and more operators. It has been an open problem to define and prove the existence of these higher order means in a general setting. In this paper we lay the foundations for such a theory by showing how higher order means can be inductively defined and established in general metric spaces, in particular, in convex metric spaces....

The goal of this paper is to establish a general homogenization result for linearized elasticity of an eigenvalue problem defined over perforated domains, beyond the periodic setting, within the framework of the $H^0$-convergence theory. Our main homogenization result states that the knowledge of the fourth-order tensor $A^0$, the $H^0$-limit of $A^{\epsilon }$, is sufficient to determine the homogenized eigenvalue problem and preserve the structure of the spectrum. This theorem is proved essentially by using Tartar’s method...