Bounded approximation by polynomials whose zeros lie on a circle

We are given a set of weighted unit disks and a set of points in Euclidean plane. The minimum weight unit disk cover (WUDC) problem asks for a subset of disks of minimum total weight that covers all given points. WUDC is one of the geometric set cover problems, which have been studied extensively for the past two decades (for many different geometric range spaces, such as (unit) disks, halfspaces, rectangles, triangles). It is known that the unweighted WUDC problem is NP-hard and admits a polynomial-time approximation scheme (PTAS). For the weighted WUDC problem, several constant approximations have been developed. However, whether the problem admits a PTAS has been an open question. In this paper, we answer this question affirmatively by presenting the first PTAS for WUDC. Our result implies the first PTAS for the minimum weight dominating set problem in unit disk graphs. Combining with existing ideas, our result can also be used to obtain the first PTAS for the maxmimum lifetime coverage problem and an improved constant approximation ratio for the connected dominating set problem in unit disk graphs.

State space methods for solving tangential interpolation problems for matrix functions in a given class have been developed extensively. These are interpolation problems in which the values in certain directions of the interpolant matrix functions, and several of their derivatives, are prescribed at given points in the complex plane. For the most part, the matrix functions under consideration are rational functions or polynomials (perhaps with symmetries); however, other important classes of matrix functions have been studied as well, notably, the Nevanlinna classes on the unit disk and the upper half-plane, and the Hardy class H/sub 2/ on the unit disk. This development is motivated by many important applications in systems and control. We present some recent results and open problems in this area.

Let α e [0,1]p be an analytic function in the unit disc E q and h be analytic and univalent functions in E, Φ be an analytic function in a certain domain D⊃ q (E). The authors study the differential subordinations and where z⊂E Examples of the application of these results are given.

Introduction.We extend the basic result on the subordination of analytic functions to analytic functions with values in a Banach space.The fundamental form of this theorem is as follows [2, p. 421]: Let/and g be two analytic (scalarvalued) functions defined on the unit disc D with g univalent and the range off contained in the range of g.Then fand g are analytically related, i.e. there is an analytic function to mapping D into itself with/(z)=g(cu(z)) for all z in D. This result extends to vector-valued analytic functions (for definitions see [1]) but there are subtleties: (1) the proof for scalar-valued functions consists of noting that co(z)=g~1(f(z)) is analytic; for vector-valued functions not only is the proper notion of analyticity for g'1 unavailable but, as we show below, g"1 need not even be continuous, and (2) the result for scalar-valued functions is true for arbitrary domains (usually stated for D only to avoid superfluous generality) while for vector-valued functions we cannot allow punctures in the domain of g.The basic theorem on subordination has as a simple consequence the interesting result that iff and g are nonconstant analytic (scalar-valued) functions on D with intersecting ranges then they are locally analytically related, i.e. there is a neighborhood Vin D and an analytic function oe mapping F into D with/(z)=g(co(z)) for z in V.This result also extends to vector-valued analytic functions but in this case the appropriate hypothesis is that/and g have ranges intersecting in an uncountable set.For f and g scalar-valued, if the ranges off and g intersect at all, then they intersect in an uncountable set by the open mapping theorem; so, as in the basic theorem, it is the failure of the open mapping theorem for vector-valued analytic functions which lends interest to the generalization and makes the proof more delicate.We will show that for vector-valued/and g their ranges may intersect in a set containing an infinite number of accumulation points and yet not be locally analytically related.I. Global theorem.Theorem.Let Xbe a complex Banach space and fand g be two analytic functions mapping the unit disc D into X.If g is one-to-one with the range of g containing the range off, then there is an analytic function u> mapping the disc into itself with f(z)=g(w(z))for all z in D.

Functions f in the class $ K_H $ are convex, univalent, harmonic, and sense preserving in the unit disk. Such functions can be expressed as $ f = h + \\overline {g} $ where h and g are analytic functions. If $ f \\in K_H $ has $ h(0) = 0, g(0) = 0, h'(0) = 1$ , and $ g'(0) = 0 $ , then $ f \\in K_H^0 $ . For $ f \\in K_H^0 $ and } analytic in the unit disk, an integral representation for $ f\ ilde {*}\\varphi = h*\\varphi + \\overline {g*\\varphi } $ is found. With } a strip mapping, $ f\ ilde {*}\\varphi $ is shown to be in $ K_H^0 $ . In a 1958 paper, Pólya and Schoenberg conjectured that if f and g are conformal mappings of the unit disk onto convex domains, then the Hadamard product f 2 g of f and g has the same property. It is known that the analogue of that result for harmonic mappings is false. In this paper, some examples are given in which the property of convexity is preserved for Hadamard products of certain convex harmonic mappings. In addition, an integral formula is used to determine the geometry of the Hadamard product from the geometry of the factors. This is true in particular for the convolution of strip mappings with certain functions $ f_n \\in K_H^0 $ which take the unit disk to regular n -gons.

We investigate approximations of analytic functions determined by Cauchy-type integrals in Jordan domains of the complex plane. We develop, modify, and complete (in a certain sense) our earlier results. Special attention is given to the investigation of approximation of functions analytic in a disk by Taylor sums. In particular, we obtain asymptotic equalities for upper bounds of the deviations of Taylor sums on the classes of ψ-integrals of functions analytic in the unit disk and continuous in its closure. These equalities are a generalization of the known Stechkin's results on the approximation of functions analytic in the unit disk and having bounded rth derivatives (here, r is a natural number).

Since the transfer function (or matrix) of a stable linear system is analytic in |z| ≥ 1, the transformation z → z −1 guarantees its analyticity in the unit disk |z| ≤ 1. In this chapter we will study approximation by polynomials and “stable” rational functions in |z| < 1, using the Hardy spaces H 2 and H ∞ norms. Here, due to the reciprocal transformation, “stability” requires location of poles in |z| > 1. Approximation in the Hankel norm will be considered in the next chapter. In order to achieve a firm understanding of the rudiments of these important mathematical tools for optimal system reduction, filter design, model matching, feedback control, and disturbance rejection, etc., we will first consider SISO systems. Standard methods are sometimes applicable to generalize our discussions to MIMO systems which will be studied in Chap. 6. Here and throughout, P n will always denote the collection of all polynomials of degree no greater than n and R n the collection of all rational functions of degree n.KeywordsHardy SpaceFilter DesignBlaschke ProductExtremal FunctionInterpolation ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This note finds an approximation of every matrix-valued contractive holomorphic function on the open unit disc by rational inner functions. This theorem is then applied to obtain a uniform approximation result for holomorphic functions which extend continuously to the unit circle. Also, as applications, approximation results are obtained for holomorphic functions on the unit disc with values in certain other domains of interest.

Let f be an analytic function in the unit disc |z|<1 on the complex plane mathbb {C}. This paper is devoted to obtaining the correspondence between f(z) and zf'(z) at the point w, 0<|w|=R< 1, such that |f(w)|=min {|f(z)|: f(z)inpartial f(|z|leq R) }. We present several applications of the main result. A part of them improve the previous results of this type.

We investigate the generalization of the starlikeness of complex order and the generalization of convexity of complex order for the analytic functions in the unit disc D = {z: \z\ &lt; 1}.

Growth analysis of analytic functions is very important part of research inthe field of complex analysis and many researchers are involved in this areaduring past decades. Collecting ideas from Heittokangas et al. (Meromorphicfunctions of finite $\varphi $-order and linear q-difference equations, J.Difference Equ. Appl., 27 (9) (2021), 1280-1309) and Bela\"{\i}di etal. (Study of complex oscillation of solutions of a second order lineardifferential equation with entire coefficients of $(\alpha ,\beta ,\gamma )$%-order, WSEAS Trans. Math., 21 (2022), 361-370), here in this paper,we have defined the $(\alpha ,\beta ,\gamma )$-Nevanlinna order and\ $%(\alpha ,\beta ,\gamma )$-Nevanlinna type of an analytic function $f$ in theunit disc $U$. We have also established some growth properties of thecomposition of two analytic functions in the unit disc on the basis of their$(\alpha ,\beta ,\gamma )$-Nevanlinna order, $(\alpha ,\beta ,\gamma )$%-Nevanlinna lower order, $(\alpha ,\beta ,\gamma )$-Nevanlinna type and $%(\alpha ,\beta ,\gamma )$-Nevanlinna weak type as compared to the growth oftheir corresponding left and right factors, where $\alpha ,\beta ,\gamma $are continuous non-negative functions defined on $(-\infty ,+\infty )$.

This paper is devoted to an investigation of a topological ring of analytic functions. Specifically, this ring, denoted by R, is the set of functions analytic on the unit disc with the usual addition and scalar multiplication, the Hadamard product for its ring multiplication, and the compact-open topology. The ring R is identified algebrai cally with a subring RA of the ring of continuous functions on the non-negative integers X. The operations in fT are the usual pointwise operations, ana the structure of R is determined by considering its isomorph iT. In Chapter I we are concerned with the problems of identifying the maximal ideal space of R and describing the maximal ideals intrinsically. We first show, using theorems on general rings of continuous functions, that the maximal ideals are in one-to-one correspondence with the points of the Stone-Cech compactification px of X. We next give an intrinsic description of the maximal ideals, using the properties of the power series expansions for analytic functions. Using this description we strengthen the prev ious theorem appreciably and show that the maximal ideal space with the hul1-kernel topology is homeomorphic to pX. Finally, the Hadamard product is used to give a simple iv characterization of the dual space of the topological linear space of analytic functions on the unit disc. This dual space is isomorphic to the set of functions in R whose radius of convergence exceeds one, which is exactly the intersection of the maximal ideals corresponding to points of pX -X (the dense maximal ideals of R) . In Chapter II we continue the investigation of the maximal ideals by studying the structure of their associated residue class rings. The complex number field 0 is isomorphically embedded in R/M, where M is a maximal ideal of R. If M corresponds to a point of X, then R/M and the isomorph 0* of 0 are identical; whereas, if M corresponds to a point of ^X -X, then R/M is a transcendental extension of 0* having transcendence degree c, the cardinality of the continuum. Moreover, we show, in the second case, that R/M is algebraically closed. Using theorems on transcendental extensions and algebraically closed fields, we show that, in either case, R/M and 0 are isomorphic fields. The two classes of maximal ideals are distinguished by the fact that their residue class rings admit radically different types of complex-valued isomorphisms. In Chapter III we are concerned primarily with the structure of the closed ideals of R. The basic tool used is the rotational completeness theorem for analytic v functions, which we proved using the methods and results of harmonic analysis. We show that the closure of every principal ideal is principal, give a necessary and suffi cient condition that a principal ideal be closed, and show that every closed ideal is a principal ideal generated by an idempotent element of R. Using these theorems we indicate connections with the general theory of dual rings, of which R is an example, and raise several questions for further i investigation in the direction of releasing some of the restrictions with which most of the results so far have been obtained.

For an analytic function ऒ on the unit disk , we show that the L2 integral mean of ऒ on c &lt; |z| &lt; r with respect to the weighted area measure (1 − |z|2)αd A(z) is a logarithmically convex function of r on (c, 1), where −3 ≤ ∞ ≤ 0 and c ∈ [0, 1). Moreover, the range [−3, 0] for ∞ is best possible. When c = 0, our arguments here also simplify the proof for several results we obtained in earlier papers.

Suppose $\\varphi$ is an analytic self-map of open unit disk $\\mathbf{D}$ and $\\psi$ is an analytic function on $\\mathbf{D}$. Then a weighted composition operator induced by $\\varphi$ with weight $\\psi$ is given by $(W_{\\psi, \\varphi}f)(z)= \\psi(z)f(\\varphi(z))$, for $z \\in \\mathbf{D}$ and $f$ analytic on $\\mathbf{D}$. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators $W_{\\psi, \\varphi}$. In terms of fixed points in the closed unit disk $\\overline{\\mathbf{D}}$, conditions under which $W_{\\psi, \\varphi}$ is compact are given. Necessary conditions for the compactness of $C_{\\varphi}$ are given in terms of the angular derivative $\\varphi^{\\prime}(\\zeta)$ where $\\zeta$ is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten $p$-class $S_{p}(H^s(\\beta_1), H^q(\\beta_2))$, where the inducing map has supremum norm strictly smaller than~$1$.

The problem of describing the analytic functions g on the unit disc such that the integral operator T_g(f)(z)=int _0^zf(zeta )g'(zeta ),dzeta is bounded (or compact) from a Banach space (or complete metric space) X of analytic functions to the Hardy space H^infty is a tough problem, and remains unsettled in many cases. For analytic functions g with non-negative Maclaurin coefficients, we describe the boundedness and compactness of T_g acting from a weighted Dirichlet space D^p_omega , induced by an upper doubling weight omega , to H^infty . We also characterize, in terms of neat conditions on omega , the upper doubling weights for which T_g: D^p_omega rightarrow H^infty is bounded (or compact) only if g is constant.