Hurwitz rational functions

A generalization of Hurwitz stable polynomials to real rational functions is considered. We establish an analog of the Hurwitz stability criterion for rational functions and introduce a new type of determinants that can be treated as a generalization of the Hurwitz determinants.     

Landen transformations and the integration of rational functions

We present a rational version of the classical Landen transformation for elliptic integrals. This is employed to obtain explicit closed-form expressions for a large class of integrals of even rational functions and to develop an algorithm for numerical integration of these functions.

     

On semiconjugate rational functions

We investigate semiconjugate rational functions, that is rational functions A, B related by the functional equation \({A \circ X = X \circ B}\), where X is a rational function. We show that if A and B is a pair of such functions, then either A can be obtained from B by a certain iterative process, or A and B can be described in terms of orbifolds of non-negative Euler characteristic on the Riemann sphere.     

An extension of the associated rational functions on the unit circle

A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the previous ones, the coefficients of this linear combination being self-reciprocal rational functions. We show that, under very general conditions on the self-reciprocal coefficients, this new sequence satisfies orthogonality conditions as well as a recurrence relation. Further, we identify the Carathéodory function of the corresponding orthogonality measure in terms of such self-reciprocal coefficients.The new class under study includes the associated rational functions as a particular case. As a consequence of the previous general analysis, we obtain explicit representations for the associated rational functions of arbitrary order, as well as for the related Carathéodory function. Such representations are used to find new properties of the associated rational functions.     

Global minimization of rational functions and the nearest GCDs

This paper discusses the global minimization of rational functions with or without constraints. We propose sum of squares relaxations to solve these problems, and study their properties. Some special features are discussed. First, we consider minimization of rational functions without constraints. Second, as an application, we show how to find the nearest common divisors of polynomials via unconstrained minimization of rational functions. Third, we discuss minimizing rational functions under some constraints which are described by polynomials.     

Hecke-symmetry and rational period functions

In this paper, we continue our work in the direction of a characterization of rational period functions on the Hecke groups. We examine the role that Hecke-symmetry of poles plays in this setting, and pay particular attention to non-symmetric irreducible systems of poles for a rational period function. This gives us a new expression for a class of rational period functions of any positive even integer weight on the Hecke groups. We illustrate these properties with examples of specific rational period functions. We also correct the wording of a theorem from an earlier paper.     

Approximation by rational functions in complex regions

Approximation by rational functions R_n (z) (in the C and L_p metrics) on plane compacta is investigated. The possibility is studied of the coincidence of rational and polynomial approximations for all n, and some functions are described for which this coincidence holds. Approximations on finite sets of points are investigated, and an explanation is given of why there are functions which cannot be approximated by rational functions of degree not higher than n (in the C metric).Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 121–130, February, 1971.Several questions concerning this work were discussed with the author by A. A. Gonchar. The author wishes to thank A. A. Gonchar for his help.     

Equiconvergence in rational approximation of meromorphic functions

Generalizing the Walsh theorem, E. B. Saff, A. Sharma, and R. S. Varga showed that there is a close relation between the rational interpolants in roots of unity and Padé approximants of certain meromorphic functions. The purpose of this paper is to extend this result, replacing the Padé approximant with other rational functions so as to obtain a larger region of equiconvergence.     

On the spiral-likeness of rational functions

The object of this paper is to extend some results concerning the univalence, starlikeness, and convexity of rational functions recently obtained by Reade, Silverman, and Todorov. The domain of variability of log{f(z)/z} for a fixedz and for such functionsf ranging over the class of λ-spirallike functions of order α are also determined.     

Symbolic-numeric integration of rational functions

Numerical Algorithms - We consider the problem of symbolic-numeric integration of symbolic functions, focusing on rational functions. Using a hybrid method allows the reliable yet efficient...     

A simple method of rational approximation of functions

It is shown that it is possible to apply a method proposed by V. K. Dzyadyk for the construction of rational functions that realize a near optimal approximation of entire elementary functions. The method may be termed linear in the sense that all the coefficients of the rational functions are determined from two systems of linear algebraic equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 7, pp. 998–1000, July, 1992.     

On the magnitudes of deviations of entire functions of infinite order from rational functions

The magnitudes of deviations b(a, f) of entire functions of infinite order from rational functions are studied.     

Spectral methods for orthogonal rational functions

We present an operator theoretic approach to orthogonal rational functions based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two alternatives are discussed, leading to representations which are linear fractional transformations with matrix coefficients acting on infinite Hessenberg or five-diagonal unitary matrices. This approach permits us to recover the orthogonality measure throughout the spectral analysis of an infinite matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg or five-diagonal matrices. As an application we use operator perturbation theory results to obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation for the orthogonal rational functions.     

Schur-Nevanlinna-Potapov sequences of rational matrix functions

We study particular sequences of rational matrix functions with poles outside the unit circle. These Schur-Nevanlinna-Potapov sequences are recursively constructed based on some complex numbers with norm less than one and some strictly contractive matrices. The main theme of this paper is a thorough analysis of the matrix functions belonging to the sequences in question. Essentially, such sequences are closely related to the theory of orthogonal rational matrix functions on the unit circle. As a further crosslink, we explain that the functions belonging to Schur-Nevanlinna-Potapov sequences can be used to describe the solution set of an interpolation problem of Nevanlinna-Pick type for matricial Schur functions.     

Reidemeister zeta functions of low-dimensional almost-crystallographic groups are rational

We prove that the Reidemeister zeta functions of automorphisms of crystallographic groups with diagonal holonomy ?₂ are rational. As a result, we obtain that Reidemeister zeta functions of automorphisms of almost-crystallographic groups up to dimension 3 are rational.     

Topological regluing of rational functions

Regluing is a surgery that helps to build topological models for rational functions. It also has a holomorphic interpretation, with the flavor of infinite dimensional Thurston–Teichmüller theory. We will discuss a topological theory of regluing, and just trace a direction, in which a holomorphic theory can develop.     

Residues and telescopers for bivariate rational functions

We give necessary and sufficient conditions for the existence of telescopers for rational functions of two variables in the continuous, discrete and q-discrete settings and characterize which operators can occur as telescopers. Using this latter characterization, we reprove results of Furstenberg and Zeilberger concerning diagonals of power series representing rational functions. The key concept behind these considerations is a generalization of the notion of residue in the continuous case to an analogous concept in the discrete and q-discrete cases.     

有理Fourier级数在变差条件下的收敛性研究

本文把Fourier级数的一些经典结论推广到有理Fourier级数的情况下. 首先给出了有理Fourier级数和共轭有理Fourier级数在有界变差条件下的收敛速度估计. 利用此结论, 得到了类似于Fourier级数的Dirichlet-Jordan定理和W. H. Young定理. 最后, 证明了这两个定理在调和有界变差条件下也成立.     

Differential inequalities for entire functions of finite degree and rational functions with prescribed poles

We establish new differential inequalities for the entire functions of finite degree with a majorant an entire function without zeros in the lower half-plane, for the entire functions with constraints on zeros and, as a consequence, for the rational functions with prescribed poles. All cases of equality in the main results are found. The estimates obtained generalize and strengthen some inequalities by Bernstein, Gardner, and Govil for entire functions of finite degree; by Smirnov, Aziz, and Shah for algebraic polynomials; and by Borwein and Erdelyi, Aziz and Shah, and the others for rational functions.