STATIONARY CONNECTED CURVES IN HILBERT SPACES

Hatamleh Hatamleh

doi:10.3844/jmssp.2014.262.266

Abstract

In this article the structure of non-stationary curves which are stationary connected in Hilbert space is studied using triangular models of non-self-adjoint operator. The concept of evolutionary representability plays here an important role. It is proved that if one of two curves in Hilbert space is evolutionary representable and the curves are stationary connected, then another curve is evolutionary representable too. These curves are studied firstly. The structure of a cross-correlation function in the case when operator, defining the evolutionary representation, has one-dimensional non-Hermitian subspace (the spectrum is discreet and situated in the upper complex half-plane or has infinite multiplicity at zero (Volterra operator)) is studied.

Highlights

INTRODUCTION(linearly) representable (Pugachev and Sinitsyn, 2001; Livshits and Yantsevich, 1979), i.e., may be expressed
It is well known (Rozanov, 1967; Hannan, 2009; Pugachev and Sinitsyn, 2001) that if two stationary random processes of the second order ξ1 (t) and ξ2 (t)
In this article the structure of non-stationary curves which are stationary connected in Hilbert space is studied using triangular models of non-self-adjoint operator

Summary

INTRODUCTION

(linearly) representable (Pugachev and Sinitsyn, 2001; Livshits and Yantsevich, 1979), i.e., may be expressed. It is well known (Rozanov, 1967; Hannan, 2009; Pugachev and Sinitsyn, 2001) that if two stationary random processes of the second order ξ1 (t) and ξ2 (t). (in what follows we consider that Mξα (t) = 0 ) are stationary connected, in the corresponding space. Is the value space of random processes) they correspond to the stationary curves of the form ξα (t) = Ut ξ0α , where. The solution of problem may be found in the framework of the Hilbert approach to the construction of the correlation theory of random processes.

EVOLUTIONARY REPRESENTABLE STATIONARY

CONCLUSION