Regions of meromorphy and value distribution of geometrically converging rational functions

Let D be a region, {rn}_n∈N a sequence of rational functions of degree at most n and let each rn have at most m poles in D, for m∈N fixed. We prove that if {rn}_n∈N converges geometrically to a function f on some continuum S⊂D and if the number of zeros of rn in any compact subset of D is of growth o(n) as n→∞, then the sequence {rn}_n∈N converges m₁-almost uniformly to a meromorphic function in D. This result about meromorphic continuation is used to obtain Picard-type theorems for the value distribution of m₁-maximally convergent rational functions, especially in Padé approximation and Chebyshev rational approximation.     

A survey of truncation error analysis for Padé and continued fraction approximants

To compute the value of a functionf(z) in the complex domain by means of a converging sequence of rational approximants {f _n(z)} of a continued fraction and/or Padé table, it is essential to have sharp estimates of the truncation error ¦f(z)–f _n(z)¦. This paper is an expository survey of constructive methods for obtaining such truncation error bounds. For most cases dealt with, {f _n(z)} is the sequence of approximants of a continued fractoin, and eachf _n(z) is a (1-point or 2-point) Padé approximant. To provide a common framework that applies to rational approximantf _n(z) that may or may not be successive approximants of a continued fraction, we introduce linear fractional approximant sequences (LFASs). Truncation error bounds are included for a large number of classes of LFASs, most of which contain representations of important functions and constants used in mathematics, statistics, engineering and the physical sciences. An extensive bibliography is given at the end of the paper.Research supported in part by the U.S. National Science Foundation under Grants INT-9113400 and DMS-9302584.     

Kernels in edge-coloured orientations of nearly complete graphs

We call the digraph D an orientation of a graph G if D is obtained from G by the orientation of each edge of G in exactly one of the two possible directions. The digraph D is an m-coloured digraph if the arcs of D are coloured with m-colours.Let D be an m-coloured digraph. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike.A set N⊆V(D) is said to be a kernel by monochromatic paths if it satisfies the two following conditions: (i) for every pair of different vertices u,v∈N there is no monochromatic directed path between them and (ii) for every vertex x∈V(D)-N there is a vertex y∈N such that there is an xy-monochromatic directed path.In this paper we obtain sufficient conditions for an m-coloured orientation of a graph obtained from K_n by deletion of the arcs of K_1,r(0?r?n-1) to have a kernel by monochromatic.     

Generalized Padé approximations to the exponential function

The stability properties of the Padé rational approximations to the exponential function are of importance in determining the linear stability properties of several classes of Runge-Kutta methods. It is well known that the Padé approximationR _n,m (z) =N _n,m (z)/M _n,m (z), whereN _n,m (z) is of degreen andM _n,m (z) is of degreem, is A-stable if and only if 0 m – n 2, a result first conjectured by Ehle. In the study of the linear stability properties of the broader class of general linear methods one must generalize these rational approximations. In this paper we introduce a generalization of the Padé approximations to the exponential function and present a method of constructing these approximations for arbitrary order and degree. A generalization of the Ehle inequality is considered and, in the case of the quadratic Padé approximations, evidence is presented that suggests the inequality is both necessary and sufficient for A-stability. However, in the case of the cubic Padé approximations, the inequality is shown to be insufficient for A-stability. A generalization of the restricted Padé approximation, in which the denominator has a singlem-fold zero, is also introduced. A procedure for the construction of these restricted approximations is described, and results are presented on the A-stability of the restricted quadratic Padé approximations. Finally, to demonstrate the connection between a generalized Padé approximation and a general linear method, a specific general linear method is constructed with a stability region corresponding to a given quadratic Padé approximation.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part I. Maehly Type Approximants

Laurent Padé–Chebyshev rational approximants, A _m (z,z ^–1)/B _n (z,z ^–1), whose Laurent series expansions match that of a given function f(z,z ^–1) up to as high a degree in z,z ^–1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

Loewner chains and parametric representation in several complex variables

Let B be the unit ball of with respect to an arbitrary norm. We study certain properties of Loewner chains and their transition mappings on the unit ball B. We show that any Loewner chain f(z,t) and the transition mapping v(z,s,t) associated to f(z,t) satisfy locally Lipschitz conditions in t locally uniformly with respect to z∈B. Moreover, we prove that a mapping f∈H(B) has parametric representation if and only if there exists a Loewner chain f(z,t) such that the family {e^−tf(z,t)}_t?0 is a normal family on B and f(z)=f(z,0) for z∈B. Also we show that univalent solutions f(z,t) of the generalized Loewner differential equation in higher dimensions are unique when {e^−tf(z,t)}_t?0 is a normal family on B. Finally we show that the set S⁰(B) of mappings which have parametric representation on B is compact.     

On normal families and differential polynomials for meromorphic functions

We consider the normality criterion for a families F meromorphic in the unit disc Δ, and show that if there exist functions a(z) holomorphic in Δ, a(z)≠1, for each z∈Δ, such that there not only exists a positive number ε₀ such that |an(a(z)−1)−1|?ε₀ for arbitrary sequence of integers an(n∈N) and for any z∈Δ, but also exists a positive number B>0 such that for every f(z)∈F, B|f^′(z)|?|f(z)| whenever f(z)f^″(z)−a(z)(f^′2(z))=0 in Δ. Then is normal in Δ.     

Compositions of analytic functions of the form Fn(z) = Fn−1(fn(z)), fn(z) → f(z)

Frequently, in applications, a function is iterated in order to determine its fixed point, which represents the solution of some problem. In the variation of iteration presented in this paper fixed points serve a different purpose. The sequence {F_n(z)} is studied, where F₁(z) = f₁(z) and F_n(z) = F_n−1(f_n(z)), with f_n → f. Many infinite arithmetic expansions exhibit this form, and the fixed point, α, of f may be used as a modifying factor (z = α) to influence the convergence behaviour of these expansions. Thus one employs, rather than seeks the fixed point of the function f.     

Arithmetical properties of linear recurrent sequences

Let F(z)∈R[z] be a polynomial with positive leading coefficient, and let α>1 be an algebraic number. For r=degF>0, assuming that at least one coefficient of F lies outside the field Q(α) if α is a Pisot number, we prove that the difference between the largest and the smallest limit points of the sequence of fractional parts {F(n)αn}_n=1,2,3,… is at least 1/?(Pr+1), where ? stands for the so-called reduced length of a polynomial.     

On the divergence of polynomial interpolation

Consider a triangular interpolation scheme on a continuous piecewise C¹ curve of the complex plane, and let Γ be the closure of this triangular scheme. Given a meromorphic function f with no singularities on Γ, we are interested in the region of convergence of the sequence of interpolating polynomials to the function f. In particular, we focus on the case in which Γ is not fully contained in the interior of the region of convergence defined by the standard logarithmic potential. Let us call Γ_out the subset of Γ outside of the convergence region.In the paper we show that the sequence of interpolating polynomials, {P_n}_n, is divergent on all the points of Γ_out, except on a set of zero Lebesgue measure. Moreover, the structure of the set of divergence is also discussed: the subset of values z for which there exists a partial sequence of {P_n(z)}_n that converges to f(z) has zero Hausdorff dimension (so it also has zero Lebesgue measure), while the subset of values for which all the partials are divergent has full Lebesgue measure.The classical Runge example is also considered. In this case we show that, for all z in the part of the interval (−5,5) outside the region of convergence, the sequence {P_n(z)}_n is divergent.     

Nevanlinna-Pick interpolation by rational functions with a single pole inside the unit disk

We devise an efficient algorithm that, given points z₁,…,z_k in the open unit disk D and a set of complex numbers {f_i,0,f_i,1,…,f_i,ni−1} assigned to each z_i, produces a rational function f with a single (multiple) pole in D, such that f is bounded on the unit circle by a predetermined positive number, and its Taylor expansion at z_i has f_i,0,f_i,1,…,f_i,ni−1 as its first n_i coefficients.     

Hole probability for entire functions represented by Gaussian Taylor series

Consider the Gaussian entire functionf(z) = ?? _n=0 ^?? ?? _n a _n z ⁿ , where {?? _n } is a sequence of independent and identically distributed standard complex Gaussians and {a _n } is some sequence of non-negative coefficients, with a ₀ > 0. We study the asymptotics (for large values of r) of the hole probability for f (z), that is, the probability P _H (r) that f(z) has no zeros in the disk {|z| < r}. We prove that log P _H (r) = ?S(r) + o(S(r)), where S(r) = 2·?? _n??0log⁺(a _n r ⁿ ) as r tends to ?? outside a deterministic exceptional set of finite logarithmic measure.     

A note on approximation properties of q-Durrmeyer operators

In this paper, the approximation properties of q-Durrmeyer operators D_n,q(f;x) for f∈C[0,1] are discussed. The exact class of continuous functions satisfying approximation process lim_n→∞D_n,q(f;x)=f(x) is determined. The results of the paper provide an elaboration of the previously-known ones on operators D_n,q.     

Log-concavity and LC-positivity

A triangle {a(n,k)}_0?k?n of nonnegative numbers is LC-positive if for each r, the sequence of polynomials is q-log-concave. It is double LC-positive if both triangles {a(n,k)} and {a(n,n−k)} are LC-positive. We show that if {a(n,k)} is LC-positive then the log-concavity of the sequence {xk} implies that of the sequence {zn} defined by , and if {a(n,k)} is double LC-positive then the log-concavity of sequences {xk} and {yk} implies that of the sequence {zn} defined by . Examples of double LC-positive triangles include the constant triangle and the Pascal triangle. We also give a generalization of a result of Liggett that is used to prove a conjecture of Pemantle on characteristics of negative dependence.     

Estimates for derivatives of holomorphic functions in a hyperbolic domain

Let f(z) be a holomorphic function in a hyperbolic domain Ω. For 2?n?8, the sharp estimate of |f(n)(z)/f^′(z)| associated with the Poincaré density λΩ(z) and the radius of convexity ρΩc(z) at z∈Ω is established for f(z) univalent or convex in each Δc(z) and z∈Ω. The detailed equality condition of the estimate is given. Further application of the results to the Avkhadiev-Wirths conjecture is also discussed.     

Quasi-convex sequences in the circle and the 3-adic integers

In this paper, we present families of quasi-convex sequences converging to zero in the circle group T, and the group J₃ of 3-adic integers. These sequences are determined by increasing sequences of integers. For an increasing sequence , put gn=an+1−an. We prove that

(a)

the set {0}∪{±³−(an+1)|n∈N} is quasi-convex in T if and only if a₀>0 and gn>1 for every n∈N;

(b)

the set {0}∪{±an³|n∈N} is quasi-convex in the group J₃ of 3-adic integers if and only if gn>1 for every n∈N.

Moreover, we solve an open problem from [D. Dikranjan, L. de Leo, Countably infinite quasi-convex sets in some locally compact abelian groups, Topology Appl. 157 (8) (2010) 1347-1356] providing a complete characterization of the sequences such that {0}∪{±²−(an+1)|n∈N} is quasi-convex in T. Using this result, we also obtain a characterization of the sequences such that the set {0}∪{±²−(an+1)|n∈N} is quasi-convex in R.     

Second order asymptotic behaviour of subordinated sequences with longtailed subordinator

Suppose that {a(n)} is a discrete probability distribution on the set N₀={0,1,2,…} and {p(n)} is some non-negative sequence defined on the same set. The equation defines a new sequence {b(n)}. Here {a*k(n)} denotes the k-fold convolution of the distribution {a(n)}. In the paper the asymptotic behaviour of the sequence {b(n)} is investigated. It is known that for the large classes of the sequences {a(n)} and {p(n)}, b(n)∼σp([σn]), where 1/σ is the mean of the distribution {a(n)}. The main object of the present work is to estimate the difference b(n)−σp([σn]) for some classes of the sequences {a(n)} and {p(n)}.     

On generalized shift bases for the wiener disc algebra

LetW(D) denote the set of functionsf(z)=Σ _n=0 ^∞ A _n Z ⁿ a _nzⁿ for which Σ_n=0 ^∞|a _n |<+∞. Given any finite set lcub;f _i (z)rcub; _i=1 ⁿ inW(D) the following are equivalent: (i) The generalized shift sequence lcub;f ₁(z)z ^kn ,f ₂(z)z ^kn+1, …,f _n (z)z ^(k+1)n−1rcub; _k=0 ^∞ is a basis forW(D) which is equivalent to the basis lcub;z ^m rcub; _m=0 ^∞ . (ii) The generalized shift sequence is complete inW(D), (iii) The function has no zero in |z|≦1, wherew=e ^2πiti /n.