D-Optimal designs for weighted polynomial regression—A functional approach

This paper is concerned with the problem of computing approximateD-optimal design for polynomial regression with analytic weight function on a interval [m ₀-a,m ₀+a]. It is shown that the structure of the optimal design depends ona and weight function. Moreover, the optimal support points and weights are analytic functions ofa ata=0. We make use of a Taylor expansion to provide a recursive procedure for calculating theD-optimal designs.     

Polynomial approximation on the boundary and strictly inside

We investigate the possibility of approximating a function on a compact setK of the complex plane in such a way that the rate of approximation is almost optimal onK, and the rate inside the interior ofK is faster than on the whole ofK. We show that ifK has an external angle smaller than π at some point z_o∈δK, then geometric convergence insideK is possible only for functions that are analytic at z_o. We also consider the possibility of approximation rates of the form exp(?cn ^β), for approximation insideK, where β is related to the largest external angle ofK. It is also shown that no matter how slowly the sequence {γ _n } tends to zero, there is aK and a Lip β, β<1, functionf such that approximation insideK cannot have order {γ _n }.     

A sufficient condition for a polynomial to be stable

A real polynomial is called Hurwitz (stable) if all its zeros have negative real parts. For a given n∈N we find the smallest possible constant dn>0 such that if the coefficients of F(z)=a₀+a₁z+?+anzn are positive and satisfy the inequalities akak+1>dnak−1ak+2 for k=1,2,…,n−2, then F(z) is Hurwitz.     

On the monotonicity of some functionals in the family of univalent functions

LetS denote the class of regular and univalent functions in |z|<1 with the normalizationf(0)=0,f′(0)=1. Denoted _f=inf _f∈s {|α||f(z)≠ α, |z|<1} and letS(d)={f¦f∈S,d _f=d, 1/4≦d≦1}. The analytic functionf(z) is univalent in |z|<1 if and only if $$log\frac{{f(z) - f(\zeta )}}{{z - \zeta }} = \sum\limits_{m,n = 0}^\infty {d_{mn} z^m \zeta ^n } $$ converges in the bicylinder |z|<1, |ξ|<1. LetC _mn =√mnd _mn andC _nn (d)= Max _fεS(d){Re(C _nn )}. The paper deals with the monotonicity ofc _nn(d) and related functionals.     

Order-constrained uniform approximations to the exponential based on restricted rationals

LetR _n/m(z∶γ)=P _n(z∶γ)/(1?γz) ^m be a rational approximation to exp (z),z ∈C, of ordern for all real positiveγ. In this paper we show there exists exactly one value ofγ in each of min(n+1,m) interpolation intervals such that the uniform error overR ^? is at a local minimum.     

Fekete configuration, quantitative equidistribution and wandering critical orbits in non-archimedean dynamics

Let f be a rational function of degree d > 1 on the projective line over a possibly non-archimedean algebraically closed field. A well-known process initiated by Brolin considers the pullbacks of points under iterates of f, and produces an important equilibrium measure. We define the asymptotic Fekete property of pullbacks of points, which means that they mirror the equilibrium measure appropriately. As application, we obtain an error estimate of equidistribution of pullbacks of points for C ¹-test functions in terms of the proximity of wandering critical orbits to the initial points, and show that the order is ${O(\sqrt{kd^{-k}})}$ upto a specific exceptional set of capacity 0 of initial points, which is contained in the set of superattracting periodic points and the omega-limit set of wandering critical points from the Julia set or the presingular domains of f. As an application in arithmetic dynamics, together with a dynamical Diophantine approximation, these estimates recover Favre and Rivera-Letelier’s quantitative equidistribution in a purely local manner.     

Approximation Properties of Julia Polynomials

Let G be a finite simply connected domain in the complex plane C, bounded by a rectifiable Jordan curve L, and let w = φ0 （z） be the Riemann conformal mapping of G onto D （0, r0） ：= {E-mail： ： ｜｜ 〈 r0}, normalized by the conditions φ0 （z0） = 0, φ＇0 （z0） = 1. In this work, the rate of approximation of φ0 by the polynomials, defined with the help of the solutions of some extremal problem, in a closed domain G is studied. This rate depends on the geometric properties of the boundary L.     

On the Convergence of Rational Functions of Best Lp-Approximation

In the present paper, for sequences of rational functions, analytic in some domain, a theorem of Montel’s type is proved. As an application, sequences of rational functions of best Lp — approximation of a function ?(z) ∈ L_p(Γ), with Γ being a closed rectifiable analytic curve, are considered. The case ?(z) ∈ Hp is discussed, too.     

On the complexity of a piecewise linear algorithm for approximating roots of complex polynomials

LetS be a triangulation of andf(z) = z ^d +a _d–1 z ^d–1++a ₀, a complex polynomial. LetF be the piecewise linear approximation off determined byS. For certainS, we establish an upper bound on the complexity of an algorithm which finds zeros ofF. This bound is a polynomial in terms ofn, max{a _i } _i , and measures of the sizes of simplices inS.     

Multivalued contractions

LetE be a Banach space,C a closed convex subset ofE, F a multivalued contraction fromC to itself with closed values. Ifx ₀ is a fixed point and ifF(x ₀) is not a singleton, then there exists a fixed pointx ₁ ofF which is different fromx ₀. We prove also that there is in the Euclidean space ?² a multivalued contraction with compact connected values having a nonconnected set of fixed points.     

Meromorphic representations of products of complex arithmetic progressions

Given a complex arithmetic sequence, a + nd, where a,d ? \mathbbC{a,d \in \mathbb{C}}, d ≠ 0, and n ? \mathbbZ⁺{n \in \mathbb{Z}^+}, define P^a_d(a+nd): = a(a+d)?(a+nd){\Pi^a_d(a+nd):= a(a+d)\cdots (a+nd)}. At first P_d^a{\Pi_d^a} is only defined on the terms of the arithmetic sequence. In this article, P_d^a{\Pi_d^a} is extended to a meromorphic function on \mathbbC \{-d,-2d,...}{\mathbb{C} \setminus\{-d,-2d,\dots\}} which satisfies the functional equation, P_d^a(z+d)=(z+d)P_d^a(z){\Pi_d^a(z+d)=(z+d)\Pi_d^a(z)}. This extension is represented in three ways: in terms of the classical P{\Pi} function; as a limit involving a Pochhammer-type symbol; as an infinite product involving a generalized Euler constant. The infinite product representation leads to a natural Multiplication Formula for the functions P_d^a{\Pi_d^a}, which, in turn, provides an easy way to prove Gauss’s Multiplication Formula for the Γ function.     

The holomorphic solutions of the generalized Dhombres functional equation

We study holomorphic solutions f of the generalized Dhombres equation f(zf(z))=φ(f(z)), z∈C, where φ is in the class E of entire functions. We show, that there is a nowhere dense set E₀⊂E such that for every φ∈E?E₀, any solution f vanishes at 0 and hence, satisfies the conditions for local analytic solutions with fixed point 0 from our recent paper. Consequently, we are able to provide a characterization of solutions in the typical case where φ∈E?E₀. We also show that for polynomial φ any holomorphic solution on C?{0} can be extended to the whole of C. Using this, in special cases like φ(z)=zk+1, k∈N, we can provide a characterization of the analytic solutions in C.     

Non-linear extremal problems for analytic two-dimensional Riemann-Stieltjes integrals

We examine certain non-linear extremal problems for two-dimensional Riemann-Stieltjes integrals \(\varphi (z) \equiv \int {_D \int {g(z,\zeta )d\mu (\zeta ),} \zeta \in D \equiv [\zeta |\left| \zeta \right|} \leqslant 1]\) ,z∈Δ≡[z‖|z|<1] whereg(z, ζ) is a continuous function in (z, ζ)∈[Δ×D] and an analytic function forz∈Δ and μ(ζ) is a unit mass measure onD. In particular, if the mass is distributed on the segment [a, b], we obtain the well-known Ruscheweyh results for the one-dimensional Riemann-Stieltjes integrals \(\varphi (z) \equiv \int_a^b {g(z,t)d\mu (t),z \in \Delta } \) . In particular, ifg(z,σ)≡z/(1—zσ), we determine the maximal domain of univalence and the radii of starlikeness and convexity of order α, ?∞<α<1, of the corresponding functions ?(z). A particular study is made of the functions of classesS ₁(D) andS ₂(D) which is similar to the study of the functions of the corresponding classesS ₁(C) andS ₂(C) of Schwarz analytic functions. In addition to obtaining maximal domains of univalence, we also determine the unique extremal functions for each of the functional studied.     

Sequentially equidistant points in metric spaces

A sequence (z ₀,z ₁,z ₂,, ...,z _n, z _n+1) of points fromp=z ₀ toq=z _n+1 in a metric spaceX is said to besequentially equidistant ifd(z _i−1,z _i)=d(z _i,z _i+1) for 1≦i≦n. If there is path inX fromp toq (or if a certain weaker condition holds), then such a sequence exists, with all points distinct, for every choice ofn, while ifX is compact and connected, then such a sequence exists at least forn=2. An example is given of a dense connected subspaceS ofR ^m ,m≧2, and an uncountable dense subsetE disjoint fromS for which there is no sequentially equidistant sequence of distinct points (n ≧ 2) inS ∪E between any two points ofE. Techniques of dimension theory are utilized in the construction of these examples, as well as in the proofs of some of the positive results. Supported in part by NSF Grant DMS-8701666.     

Inversion of analytically perturbed linear operators that are singular at the origin

Let H and K be Hilbert spaces and for each z∈C let A(z)∈L(H,K) be a bounded but not necessarily compact linear map with A(z) analytic on a region |z|<a. If A(0) is singular we find conditions under which A⁻¹(z) is well defined on some region 0<|z|<b by a convergent Laurent series with a finite order pole at the origin. We show that by changing to a standard Sobolev topology the method extends to closed unbounded linear operators and also that it can be used in Banach spaces where complementation of certain closed subspaces is possible. Our method is illustrated with several key examples.²     

A Laurent-type expansion for solutions of Stokes' equation in sectorial regions

We study solutions of Stokes' equation in a regionD (in the complex plane) being the intersection of a sector with vertex at the origin and a ring about the origin, subject to no-slip boundary conditions on the radial boundaries ofD. Using a result of Kratz and Peyerimhoff, we represent solutionsv(z) by means of two analytic functionsv ₁(z) andv ₂(z), and for these we obtain expansions into infinite series, quite analogous to Laurent series, but in complex powers ofz, the exponents depending upon the angular opening ofD. Forv(z), this leads to an expansion quite analogous to the one stated without proof by Moffat in 1964 in a more special situation.     

具有缺项系数的几类解析函数族的性质

S*表示所有在单位圆盘 D 内解析且满足条件 f(0)=f′ (0)-1=0的星形函数族, K 表示所有在 D内解析且满足条件 f(0)=f′ (0)-1=0 的凸函数族, P 表示所有在 D 内解析且满足条件p(0)=1, Rep(z)>0 的函数族. 设P_n={p(z): p(z)=1+a_nzⁿ+a_n+1zⁿ⁺¹+…∈ P}, S*_n={f (z): f(z)=z+a_nzⁿ+a_n+1zⁿ⁺¹+…∈ S*}, K_n={f (z): f (z)=z+a_nzⁿ+a_n+1zⁿ⁺¹+…∈ K}. L_Sn*={g(z)=ln f(z)/z, f ∈ S_n*}, 其中对数函数取使得ln1=0的那个单值解析分支. 该文研究了函数族S_n^*, K_n和L_Sn*的性质, 找出了解析函数族L_Sn*的极值点与支撑点,并对S*_n与K_n的极值点和支撑点作了一些探讨.     

U(1)-invariant special Lagrangian 3-folds. II. Existence of singular solutions

This is the second of three papers studying special Lagrangian 3-submanifolds (SLV 3-folds) N in invariant under the U(1)-action e^iθ:(z₁,z₂,z₃)?(e^iθz₁,e^−iθz₂,z₃), using analytic methods. If N is such a 3-fold then |z₁|²−|z₂|²=2a on N for some . Locally, N can be written as a kind of graph of functions satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy.The first paper studied the case when a is nonzero. Then u,v are smooth and N is nonsingular. It proved existence and uniqueness for solutions of two Dirichlet problems derived from the equations on u,v. This implied existence and uniqueness for a large class of nonsingular U(1)-invariant SL 3-folds in , with boundary conditions.In this paper and its sequel we focus on the case a=0. Then the nonlinear Cauchy-Riemann equation is not always elliptic. Because of this there may be points (x,0) where u,v are not differentiable, corresponding to singular points of N. This paper is concerned largely with technical analytic issues, and the sequel with the geometry of the singularities of N. We prove a priori estimates for derivatives of solutions of the nonlinear Cauchy-Riemann equation, and use them to show existence and uniqueness of weak solutions u,v to the two Dirichlet problems when a=0, which are continuous and weakly differentiable. This gives existence and uniqueness for a large class of singular U(1)-invariant SL 3-folds in , with boundary conditions.     

On the Rate of Approximation of Closed Jordan Curves by Lemniscates

As proved by Hilbert, it is, in principle, possible to construct an arbitrarily close approximation in the Hausdorff metric to an arbitrary closed Jordan curve Γ in the complex plane {z} by lemniscates generated by polynomials P(z). In the present paper, we obtain quantitative upper bounds for the least deviations H _n (Γ) (in this metric) from the curve Γ of the lemniscates generated by polynomials of a given degree n in terms of the moduli of continuity of the conformal mapping of the exterior of Γ onto the exterior of the unit circle, of the mapping inverse to it, and of the Green function with a pole at infinity for the exterior of Γ. For the case in which the curve Γ is analytic, we prove that H _n (Γ) = O(q _n ), 0 ≤ q = q(Γ) < 1, n → ∞.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 861–876.Original Russian Text Copyright ©2005 by O. N. Kosukhin.     

Identity theorem for bounded p-adic meromorphic functions

Let K be a complete ultrametric algebraically closed field. We investigate several properties of sequences (an)_n∈N in a disk d(0,R⁻) with regards to bounded analytic functions in that disk: sequences of uniqueness (when f(an)=0∀n∈N implies f=0), identity sequences (when limn→+∞f(an)=0 implies f=0) and analytic boundaries (when lim supn→∞|f(an)|=‖f‖). Particularly, we show that identity sequences and analytic boundary sequences are two equivalent properties. For certain sequences, sequences of uniqueness and identity sequences are two equivalent properties. A connection with Blaschke sequences is made. Most of the properties shown on analytic functions have continuation to meromorphic functions.