Extension of entire functions of completely regular growth and right inverse to the operator of representation of analytic functions by quasipolynomial series

Abstract

A new fully automated method for the accurate evaluation of the logarithmic decrement 6 in damped torsional oscillations is presented. The method is based on the measurement of time intervals between successive passages of an oscillating cup at a fixed position corresponding to an angular displacement 4 different from zero. The logarithmic decrement can be evaluated by means of a best-fit procedure from the dependence of the time interval upon 6 and T, T being the period of undamped motion.The method was tested by means of a torus-shaped crucible viscometer in a series of measurements of the viscosity of mercury at 19.2'C. The relative errors in the measurements were h TIT= 10-5 and A6jS = 1.25 x lo-*, representing an improvement, by a facor of about 10, on the best measurements reported in the literature for both the factors. The measured viscosity of the mercury was also in excellent agreement with previously published data.

On A. F. Leont'ev's interpolating function

“…2) Suppose a set is bounded in . For the given , we choose ∈ N by condition (2). We then choose a sequence ∈ , ∈ N. Since it is bounded on each compact set in C, by Montel theorem there exists a subsequence ( ) ∈N converging uniformly on each compact set in C to a function ∈ (C).…”

“…In what follows is the space of entire functions defined in Section 1 and the family of function ( , ) , ∈N defining this space obeys conditions (1) and (2). Assume that is a complex locally convex space (LCS) possessing the following properties: Let us prove some properties of functional Ω .…”

“…Let us show that Ω ( , , ) = ( , ), , ∈ C, ∈ . To distinguish it from our functional, we have denoted it slightly different than in [2]. In the present case we can also take 0 ≡ 1.…”

“…For solving the problem on existence of linear continuous right inverse (LCRI) for the operator of representation by the series of quasi-polynomials of functions analytic in , in [2,Sect. 3] there was introduced interpolating functional Ω ( , , ) being an analogue of interpolating function ( , ) defined by some entire function ( , ) of two complex variables , .…”

“…Suppose that conditions(1) and(2) hold. For each ∈ , ∈ C there exists ∈ N such thatlim → ) − ( )( ) − = ′ ( + ) = ( ′ )( ) for each ∈ C. The maximum modulus principle and condition (1) yield that set { ( )− ( ) − : 0 < | − | 1} is bounded in some space , and thus, it is relatively compact in some space .…”