Approximation on the entire real axis to two functions conjugate in the sense of M. Riesz by means of integral operators of a special type

1968 ◽  
pp. 145-178
Author(s):  
B. L. Golinskiĭ
Keyword(s):  
Real Axis ◽  
Integral Operators ◽  
Entire Real Axis

scholarly journals Solutions of Linear Impulsive Differential Systems Bounded on the Entire Real Axis

2010 ◽  
Vol 2010 (1) ◽  
pp. 494379 ◽  
Author(s):  
Alexandr Boichuk ◽  
Martina Langerová ◽  
Jaroslava Škoríková
Keyword(s):  
Real Axis ◽  
Differential Systems ◽  
Entire Real Axis ◽  
Impulsive Differential Systems

25.—Spectrum of a Third-order Differential Operator with Large Coefficients

1975 ◽  
Vol 72 (4) ◽  
pp. 299-305
Author(s):  
K. Unsworth
Keyword(s):  
Differential Operator ◽  
Differential Expression ◽  
Real Axis ◽  
Sufficient Conditions ◽  
Order Differential Operator ◽  
Third Order ◽  
Minimal Operator ◽  
The Third ◽  
Entire Real Axis

SynopsisThis paper sets out to study the spectrum of self-adjoint extensions of the minimal operator associated with the third-order formally symmetric differential expression. The technique employed is the method of singular sequences. Sufficient conditions are established on the coefficients of the differential expression in order that the spectrum should cover the entire real axis. Particular cases in which the coefficients behave roughly as powers of x as the magnitude of x becomes large are then considered, and certain conclusions are drawn regarding the spectra under different restrictions on these powers of x.

scholarly journals Existence Conditions for Bounded Solutions of Weakly Perturbed Linear Impulsive Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Alexander Boichuk ◽  
Martina Langerová ◽  
Jaroslava Škoríková
Keyword(s):  
Small Parameter ◽  
Real Axis ◽  
Laurent Series ◽  
Existence Of Solutions ◽  
Bounded Solutions ◽  
Impulsive Systems ◽  
Entire Real Axis ◽  
Existence Conditions ◽  
Generating System

The weakly perturbed linear nonhomogeneous impulsive systems in the formẋ=A(t)x  +  εA1(t)x  +  f(t),  t∈R,  t∉T:={τi}Z,Δx|t=τi=γi+εA1ix(τi-),  τi∈T⊂R,  γi∈Rn, andi∈Zare considered. Under the assumption that the generating system (forε=0) does not have solutions bounded on the entire real axis for some nonhomogeneities and using the Vishik-Lyusternik method, we establish conditions for the existence of solutions of these systems bounded on the entire real axis in the form of a Laurent series in powers of small parameterεwith finitely many terms with negative powers ofε, and we suggest an algorithm of construction of these solutions.

scholarly journals Solutions of Linear Impulsive Differential Systems Bounded on the Entire Real Axis

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Alexandr Boichuk ◽  
Martina Langerová ◽  
Jaroslava Škoríková
Keyword(s):  
Real Axis ◽  
Differential Systems ◽  
Entire Real Axis ◽  
Impulsive Differential Systems

On Trigonometric Formulations of Polynomial Equations

2006 ◽  
Author(s):  
Harvey Lipkin
Keyword(s):  
Real Axis ◽  
Projective Transformation ◽  
Polynomial Equations ◽  
Displacement Analysis ◽  
Kinematic Chains ◽  
Real Roots ◽  
Entire Real Axis ◽  
Half Angle ◽  
Sine Polynomials ◽  
Circular Points

The displacement analysis of open and closed kinematic chains is based on polynomial equations whose variables are functions of relative joint displacements. The objective of this paper is to investigate new and interesting properties of the transformations between the canonical cosine-sine polynomials and the even degree tan-half angle polynomials associated with displacement kinematics. Using a homogeneous coordinate formulation, it is shown that the coefficients of the polynomials are linearly related by a projective transformation whose elements can be defined recursively. The canonical cosine-sine polynomial is then transformed to a cosine or a sine polynomial which can be rooted by usual techniques. However, all real roots are bracketed between −1 and +1 which can have numerical advantages over a corresponding tan-half angle polynomial for which the entire real axis must be searched. It is also demonstrated how polynomial solutions corresponding to circular points at infinity in the tan-half angle, which are typically introduced as extraneous roots via algebraic elimination, may be easily factored out by the transformation to the cosine-sine formulation.

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