Approximation to the function z α by rational fractions in a domain with zero external angle

Rational approximations to the function z ^α , α ∈ ? ?, were studied by Newman, Gonchar, Bulanov, Vyacheslavov, Andersson, Stahl, and others. The present paper deals with the order of best rational approximations to this function in a domain with zero external angle and vertex at the point z = 0. In particular, the obtained results show that the conditions imposed on the boundary of the domain in the Jackson-type inequality proved by the author in 2001 for the best rational approximations in Smirnov spaces cannot be weakened significantly.     

On uniform convergence of rational,Newton-Padé interpolants of type (n,n) with free poles asn→∞

Letf be meromorphic in . We show that there exists a sequence of distinct interpolation points {z _j } _j=1 , and forn1, rational functions,R _n (z) of type (n, n) solving the Newton-Padé (Hermite) interpolation problem,

A moment problem associated to rational Szegő functions

Leta ₁,...,a _p be distinct points in the finite complex plane ?, such that |a _j|>1,j=1,..., p and let \(b_j = 1/\bar \alpha _j ,\) j=1,..., p. Let μ₀, μ _π ^(j) , ν _π ^(j) j=1,..., p;n=1, 2,... be given complex numbers. We consider the following moment problem. Find a distribution ψ on [?π, π], with infinitely many points of increase, such that $$\begin{array}{l} \int_{ - \pi }^\pi {d\psi (\theta ) = \mu _0 ,} \\ \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - a_j )^n }} = \mu _n^{(j)} ,} \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - b_j )^n }} = v_n^{(j)} ,} j = 1,...,p;n = 1,2,.... \\ \end{array}$$ It will be shown that this problem has a unique solution if the moments generate a positive-definite Hermitian inner product on the linear space of rational functions with no poles in the extended complex plane ?^* outside {a ₁,...,a _p,b ₁,...,b _p}.     

On three-term recurrence and Christoffel–Darboux identity for orthogonal rational functions on the real line

First, we give an algebraic proof to the Christoffel–Darboux identity of formal orthogonal rational functions on the real line by exposing some underlying algebraic properties. This proof does not involve the three-term recurrence relationship. Besides, it is shown that if a family of rational functions satisfies the Christoffel–Darboux relation, then it also admits a three-term recurrence relationship. Thus, the equivalence between both relations is revealed.     

Approximation by rational functions with prescribed numerator degree in L p spaces for 1<p< ∞

The present paper establishes a complete result on approximation by rational functions with prescribed numerator degree in L ^pspaces for 1 < p < ∞ and proves that if f(x)∈L ^p _[-1,1] changes sign exactly l times in (-1,1), then there exists r(x)∈R _n ^l such that

Improved King’s methods with optimal order of convergence based on rational approximations

In this paper, we present three-point and four-point methods for solving nonlinear equations. The methodology is based on King’s family of fourth order methods [R.F. King, A family of fourth order methods for nonlinear equations, SIAM J. Numer. Anal. 10 (1973) 876–879] and further developed by using rational function approximations. The three-point method requires four function evaluations and has the order of convergence eight, whereas the four-point method requires five function evaluations and has the order of convergence sixteen. Therefore, the methods are optimal in the sense of Kung–Traub hypothesis. The proposed schemes are compared with closest competitors in a series of numerical examples. Moreover, theoretical order of convergence is verified in the examples.     

What generalizations do students achieve with respect to trigonometric functions in the transition from angles in degrees to real numbers?

The aim of this study is to examine prospective mathematics teachers’ generalizations of trigonometric functions from the unit circle to the Cartesian coordinate system. The researcher developed a test that aimed to reveal students’ generalizations, as well as the possible differences between them. The test was administered to 30 students who were near completion of their university degree program. The findings showed that the students were unable to establish the link between the unit circle and the Cartesian coordinate representation system; and therefore, they were not able to interpret the outputs of trigonometric functions with input of a real number that is not a multiple of π. The researcher also found that the students had developed certain misconceptions regarding the properties of trigonometric functions. To improve the teaching of trigonometric functions an instructional sequence and an alternative definition for trigonometric functions is proposed.     

Generalizations of Young’s Theorem to real functions of several variables

In 1907 W. H. Young classified the real-valued Baire one functions on the line which have the Darboux (intermediate-value) property as those which are bilaterally approachable. Here we investigate generalizations of this theorem to the setting of real-valued Baire one functions of several variables which possess various “Darboux-like” properties.      

Fekete-Szegö problem for close-to-convex functions with respect to a certain convex function dependent on a real parameter

Given α ∈ [0, 1], let h _α (z):= z/(1 - αz), z ∈ D:= {z ∈ D: |z| < 1}. An analytic standardly normalized function f in D is called close-to-convex with respect to h _α if there exists δ ∈ (-π/2, π/2) such that Re{e^iδ zf′(z)/h _α (z)} > 0, z ∈ D. For the class ? (h _α ) of all close-to-convex functions with respect to h _α , the Fekete-Szegö problem is studied.     

The pluripolar hull of {w=e −1/z }

In this paper we show that the pluripolar hull ofE={(z, ω)∈C²:ω=e ^−1/z,z≠0} is equal toE. This implies thatE is plurithin at 0, which answers a question of Sadullaev. The result remains valid ife ^−1/z is replaced by certain other holomorphic functions with an essential singularity at 0.     

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.     

Calculation of poles of meromorphic functions with q-d,r-s and ε-algorithms. Acceleration of these processes

We study (Sections 1 and 2) the speed of convergence of the q-d algorithm to poles of a meromorphic function in order to accelerate it.In Section 3 and 4, we show that the quotient of two successive vertical terms of two well known algorithms (r-s and ε) approaches also these poles. At last, a theorem of acceleration is given.     

Geometric Properties of Symmetric Spaces with Applications to Orlicz–Lorentz Spaces

We deal with the basic convexity properties –rotundity, and uniform, local uniform and full rotundity –- for symmetric spaces. A characterization of Orlicz–Lorentz spaces with the Kadec–Klee property for pointwise convergence is given. These results are applied to obtain criteria of convexity properties for Orlicz–Lorentz sequence spaces, and some new proofs of the sufficiency part of criteria for rotundity and uniform rotundity for Orlicz–Lorentz function spaces.     

A Geometric Uncertainty Principle with an Application to Pleijel’s Estimate

Let \({\Omega \subset \mathbb{R}^2}\) be an open, bounded domain and \({\Omega = \bigcup_{i = 1}^{N} \Omega_{i}}\) be a partition. Denote the Fraenkel asymmetry by \({0 \leq \mathcal{A}(\Omega_i) \leq 2}\) and write $$D(\Omega_i) := \frac{|\Omega_{i}| - {\rm min}_{1 \leq j \leq N}{|\Omega_{j}|}}{|\Omega_{i}|}$$ with \({0 \leq D(\Omega_{i}) \leq 1}\) . For N sufficiently large depending only on \({\Omega}\) , there is an uncertainty principle $$\left(\sum_{i=1}^{N}{\frac{|\Omega_{i}|}{|\Omega|}{\mathcal{A}}(\Omega_i)}\right) + \left(\sum_{i=1}^{N}{\frac{|\Omega_i|}{|\Omega|}D(\Omega_i)}\right) \geq \frac{1}{60000}.$$ The statement remains true in dimensions \({n \geq 3}\) for some constant \({c_{n} > 0}\) . As an application, we give an (unspecified) improvement of Pleijel’s estimate on the number of nodal domains of a Laplacian eigenfunction and an improved inequality for a spectral partition problem.