Properties Of Rational Functions Research Articles

On Rational Functions Related to Algorithms for a Computation of Roots. II

We discuss a nice composition properties related to algorithms for computation of N-roots. Our approach gives direct proof that a version of Newton's iterative algorithm is of order 2. A basic tool used in this note are properties of rational function Φ(w; z) = z-w/(z+w), which was used earlier in [1] in the case of algorithms for computations of square roots.

Ergodic Properties of Rational Functions that Preserve Lebesgue Measure on ℝ

We prove that all negative generalized Boole transformations are conservative, exact, pointwise dual ergodic, and quasi-finite with respect to Lebesgue measure on the real line. We then provide a formula for computing the Krengel, Parry, and Poisson entropy of all conservative rational functions that preserve Lebesgue measure on the real line.

Property of Rational Functions Related to Band-Pass Transformation With Application to Symmetric Filters Design

This paper considers the condition when a rational function $F(s)$ may be represented as the function of the argument $s+(1/s)$ . If this condition is satisfied then $F(s)$ is the ratio of recursive (symmetric) polynomials. This paper investigates the network properties of such rational functions and their realization. Then the symmetric polynomials are applied for synthesis of symmetric band-pass filters. Substituting $p=s+(1/s)$ into symmetric band-pass filter transfer function one obtains its low-pass generating filter. The slew rate and overshoot of generating filter step-response is closely connected with the step-response duration of symmetric band-pass filter. The choice of generating filter becomes an additional factor of symmetric band-pass filter design. As the generating filters the paper proposes using Lommel polynomial filters which have easy control of overshoot and slew rate. An example of six order symmetric band-pass filter is given.

A Simple Proof for a More Generalized Form of the AZ-identity

This paper gives a simple proof for a generalized form of the AZ-identity of one or two set systems. We also extend the results further and establish some new identities related to the AZ-identity. Our method uses some properties of rational functions.

Orthogonal on the circle rational functions with fixed poles II

The paper is devoted to the algebraic properties of rational functions which are orthogonal on the unit circle and have fixed poles.

Some Properties of Rational Functions with Prescribed Poles

AbstractLet P(z) be a polynomial of degree not exceeding n and let where |aj| > 1, j = 1, 2,…,n. If the rational function r(z) = P(z)/W(z) does not vanish in |z| < k, then for k = 1 it is known thatwhere B(Z) = W*(z)/W(z) and . In the paper we consider the case when k > 1 and obtain a sharp result. We also show thatwhere , and as a consquence of this result, we present a generalization of a theorem of O’Hara and Rodriguez for self-inversive polynomials. Finally, we establish a similar result when supremum is replaced by infimum for a rational function which has all its zeros in the unit circle.

A discrete approximation framework for hereditary systems

A discrete approximation framework for initial-value problems involving certain classes of linear functional differential equations (FDE) of the retarded type is constructed. An equivalence between the FDE and abstract evolution equations (AEE) in an appropriately chosen Hilbert space is established. This equivalence is then employed in the development of discrete approximation schemes in which the infinite-dimensional AEE is replaced by a finite-dimensional system of difference equations. Convergence and rates of convergence are demonstrated via the properties of rational functions with operator arguments and both classical and recent results from linear semigroup theory. Two examples of families of approximation schemes which are included in the general framework and which may be implemented directly on high-speed computing machines are developed. A numerical study of examples which illustrates the application and feasibility of the approximation techniques in a variety of problems together with a summary and analysis of the numerical results are also included.

Invariant subspace methods in linear multivariable distributed systems and lumped distributed network synthesis

Linear multivariable-distributed systems and synthesis problems for lumped-distributed networks are analyzed. The methods used center around the invariant subspace theory of Helson-Lax and the theory of vectorial Hardy functions. State-space and transfer function models are studied and their relations analyzed. We single out a class of systems and networks with nonrational transfer functions (scattering matrices), for which several of the well-known results for lumped systems and networks are generalized. In particular we develop the relations between singularities of transfer functions and natural modes of the systems, a degree theory for infinite-dimensional finear systems and a synthesis via lossless embedding of the scattering matrix. Finally coprime factorizations for this class of systems are developed. These factorizations play an essential role in the development and show that properties of Hardy functions are of fundamental importance for this class of distributed systems as properties of rational functions are for lumped systems.

Invariant subspace methods in linear multivariable-distributed systems and lumped-distributed network synthesis

Linear multivariable-distributed systems and synthesis problems for lumped-distributed networks are analyzed. The methods used center around the invariant subspace theory of Helson-Lax and the theory of vectorial Hardy functions. State-space and transfer function models are studied and their relations analyzed. We single out a class of systems and networks with nonrational transfer functions (scattering matrices), for which several of the well-known results for lumped systems and networks are generalized. In particular we develop the relations between singularities of transfer functions and natural modes of the systems, a degree theory for infinite-dimensional finear systems and a synthesis via lossless embedding of the scattering matrix. Finally coprime factorizations for this class of systems are developed. These factorizations play an essential role in the development and show that properties of Hardy functions are of fundamental importance for this class of distributed systems as properties of rational functions are for lumped systems.

On a Property of Rational Functions. II

On a Property of Rational Functions. II

On a property of rational functions. II

It is shown that if r n ( z ) {r_n}(z) is a rational function of degree n n such that r n ( 0 ) = 1 , lim | z | → ∞ | r n ( z ) | = 0 {r_n}(0) = 1,{\lim _{|z| \to \infty }}|{r_n}(z)| = 0 and all its poles lie in | ζ 1 | ≦ | z | ≦ 1 |{\zeta _1}| \leqq |z| \leqq 1 then max | z | = ρ > | ζ 1 | | r n ( z ) | ≧ 1 / ( 1 − ρ n ) {\max _{|z| = \rho > |{\zeta _1}|}}|{r_n}(z)| \geqq 1/(1 - {\rho ^n}) .