Twisted rational functions and series

We study rings of twisted rational functions and twisted Laurent series over simple artinian rings. We determine the centers of such rings and investigate the structure of subalgebras of these rings. We extend to infinite dimensional division rings and to simple artinian rings results proven in a previous paper for finite dimensional division algebras. We investigate “Galois subrings” of rings of fractions.     

Overconvergent series of rational functions and universal Laurent series

In this paper, series of rational functions with fixed poles, which have restricted growth near the poles are considered. If they converge with a geometric rate on a continuum, a phenomenon of overconvergence takes place, in the sense that the convergence extends to a certain maximal domain. From this result, some properties of universal Laurent series are derived.     

Conditions for the representability of analytic functions by series of rational functions

With a functionf(z), analytic in the unit circle, we associate by a specific rule the series \(\sum\nolimits_{n = 1}^\infty {\frac{{A_n }}{{1 - \lambda _n z}},\left| {\lambda _n } \right|< 1} \) . we derive a (necessary and sufficient) condition for the convergence of the series in the unit circle. We derive further conditions under which the series converges to the functionf(z) itself.     

Representation of functions analytic in a closed domain by series of rational functions

A method is given for representing functions analytic in a closed Jordan domain by series of the form (where the poles _n lie outside the domain) and a bound is obtained for the coefficients A_n.Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 191–200, August, 1968.     

Approximation orders of formal Laurent series by Oppenheim rational functions

We study formal Laurent series which are better approximated by their Oppenheim convergents. We calculate the Hausdorff dimensions of sets of Laurent series which have given polynomial or exponential approximation orders. Such approximations are faster than the approximation of typical Laurent series (with respect to the Haar measure).     

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.     

A factorization for formal laurent series and lattice path enumeration

If f = Σ_n=?∞^∞a_ntⁿ is a formal Laurent series with certain restrictions on the a_n, then f = f_?f₀f₊, where f_? contains only negative powers of t, f₊ contains only positive powers of t, and f₀ is independent of t. Applications include Lagrange's formula for series reversion, the problem of counting lattice paths below a diagonal, and a theorem of Furstenberg that the diagonal of a rational power series in two variables is algebraic.     

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.     

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.     

On the rate of convergence of the partial sums of dirichlet series and rational approximation of analytic functions

We study the asymptotic behavior of the rate of convergence of Dirichlet series that are absolutely convergent in a half-plane; the results obtained are applicable to rational approximation of functions analytic in the unit disk with nonnegative Taylor coefficients. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 116–122.     

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.     

Fast decreasing rational functions

Necessary and sufficient conditions are obtained for the existence of sequences of rational functions of the formr _n(x) =p _n(x)/p_n(−x), withp _n a polynomial of degreen, that decrease geometrically on (0, 1] in accordance with a specified rate function. The technique of proof involves minimum energy problems for Green potentials in the presence of an external field. Applications are given for the construction of rational approximations of |x| and sgn(x) on [−1, 1] having geometric rates of convergence forx ≠ 0. The research of this author was supported, in part, by National Science Foundation grant DMS-9501130.     

Derivations and linear functions along rational functions

The main purpose of this paper is to give characterization theorems on derivations as well as on linear functions. Among others the following problem will be investigated: Let ${n \in \mathbb{Z}, f, g\colon\mathbb{R} \to\mathbb{R}}$ be additive functions, ${\left(\begin{array}{cc} a&b\\ c&d \end{array} \right) \in \mathbf{GL}_{2}(\mathbb{Q})}$ be arbitrarily fixed, and let us assume that the mapping $$ \phi(x)=g\left(\frac{ax^{n}+b}{cx^{n}+d}\right)-\frac{x^{n-1}f(x)}{(cx^{n}+d)^{2}} \quad \left(x\in\mathbb{R}, cx^{n}+d\neq 0\right)$$ satisfies some regularity on its domain (e.g. (locally) boundedness, continuity, measurability). Is it true that in this case the above functions can be represented as a sum of a derivation and a linear function? Analogous statements ensuring linearity will also be presented.