Starlike Sets Research Articles

The core of the unit sphere of a Banach space

<abstract><p>A geometric invariant or preserver is essentially a geometric property of the unit sphere of a real Banach space that remains invariant under the action of a surjective isometry onto the unit sphere of another real Banach space. A new geometric invariant of the unit ball of a real Banach space was introduced and analyzed in this manuscript: the core of the unit sphere. This geometric invariant consists of all points in the unit sphere of a real Banach space, which are contained in a unique maximal face. It is, in a geometrical sense, the opposite of fractal-like sets such as starlike sets. Classical geometric properties, such as smoothness and strict convexity, were employed to characterize the core of the unit sphere. Also, the core was related to a recently introduced new index: the index of strong rotundity. A characterization of the core in terms of the index of strong rotundity was provided. Finally, applications to longstanding open problems, such as Tingley's problem, were provided by presenting a new notion: Mazur-Ulam classes of Banach spaces.</p></abstract>

Nearest Neighbor Recurrence Relations for Meixner–Angelesco Multiple Orthogonal Polynomials of the Second Kind

This paper studies a new family of Angelesco multiple orthogonal polynomials with shared orthogonality conditions with respect to a system of weight functions, which are complex analogs of Pascal distributions on a legged star-like set. The emphasis is placed on the algebraic properties, such as the raising operators, the Rodrigues-type formula, the explicit expression of the polynomials, and the nearest neighbor recurrence relations.

On Two Families of Discrete Multiple Orthogonal Polynomials on a Star-like Set

On Two Families of Discrete Multiple Orthogonal Polynomials on a Star-like Set

Null, recursively starlike-equivalent decompositions shrink

AbstractA subset E of a metric space X is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $\mathbb{R}^n$ for some n, sending E to a starlike set. A subset $E\subset X$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets $\{E_i\}_{i=0}^{N+1}$ such that $E_{i}/E_{i+1}\subset X/E_{i+1}$ is starlike-equivalent for each i and $E_{N+1}$ is a point. A decomposition $\mathcal{D}$ of a metric space X is said to be recursively starlike-equivalent, if there exists $N\geq 0$ such that each element of $\mathcal{D}$ is recursively starlike-equivalent of filtration length N. We prove that any null, recursively starlike-equivalent decomposition $\mathcal{D}$ of a compact metric space X shrinks, that is, the quotient map $X\to X/\mathcal{D}$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman–Starbird and Freedman and is applicable to the proof of Freedman’s celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological 4-manifolds, including the four-dimensional Poincaré conjecture.

Geometric Invariants of Surjective Isometries between Unit Spheres

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

Multiple orthogonal polynomials associated with confluent hypergeometric functions

We introduce and analyse a new family of multiple orthogonal polynomials of hypergeometric type with respect to two measures supported on the positive real line which can be described in terms of confluent hypergeometric functions of the second kind. These two measures form a Nikishin system. Our focus is on the multiple orthogonal polynomials for indices on the step line. The sequences of the derivatives of both type I and type II polynomials with respect to these indices are again multiple orthogonal and they correspond to the original sequences with shifted parameters. For the type I polynomials, we provide a Rodrigues-type formula. We characterise the type II polynomials on the step line, also known as d-orthogonal polynomials (where d is the number of measures involved so that here d=2), via their explicit expression as a terminating generalised hypergeometric series, as solutions to a third-order differential equation and via their recurrence relation. The latter involves recurrence coefficients which are unbounded and asymptotically periodic. Based on this information we deduce the asymptotic behaviour of the largest zeros of the type II polynomials. We also discuss limiting relations between these polynomials and the multiple orthogonal polynomials with respect to the modified Bessel weights. Particular choices on the parameters for the 2-orthogonal polynomials under discussion correspond to the cubic components of the already known threefold symmetric Hahn-classical multiple orthogonal polynomials on star-like sets.

Nikishin systems on star-like sets: Ratio asymptotics of the associated multiple orthogonal polynomials, II

In this paper we continue the investigations initiated in López-García and López Lagomasino (2018) on ratio asymptotics of multiple orthogonal polynomials and functions of the second kind associated with Nikishin systems on star-like sets. We describe in detail the limiting functions found in López-García and López Lagomasino (2018), expressing them in terms of certain conformal mappings defined on a compact Riemann surface of genus zero. We also express the limiting values of the recurrence coefficients, which are shown to be strictly positive, in terms of certain values of the conformal mappings. As a consequence, the limits depend exclusively on the location of the intervals determined by the supports of the measures that generate the Nikishin system.

Nikishin systems on star-like sets: algebraic properties and weak asymptotics of the associated multiple orthogonal polynomials

We investigate polynomials , , that are multi- orthogonal with respect to a Nikishin system of compactly supported measures over the star-like set of rays . We prove that the Nikishin system is normal, that the polynomials satisfy a three-term recurrence relation of order of the form with for all , and that the nonzero roots of are all simple and located in . Under the assumption that the measures generating the Nikishin system are regular (in the sense of Stahl and Totik), we describe the asymptotic zero distribution and weak behaviour of the polynomials in terms of a vector equilibrium problem for logarithmic potentials. Under the same regularity assumptions, we prove a theorem on the convergence of the Hermite-Padé approximants to the Nikishin system of Cauchy transforms. Bibliography: 16 titles.

Nikishin systems on star-like sets: Ratio asymptotics of the associated multiple orthogonal polynomials

We investigate the ratio asymptotic behavior of the sequence (Qn)n=0∞ of multiple orthogonal polynomials associated with a Nikishin system of p≥1 measures that are compactly supported on the star-like set of p+1 rays S+={z∈C:zp+1≥0}. The main algebraic property of these polynomials is that they satisfy a three-term recurrence relation of the form zQn(z)=Qn+1(z)+anQn−p(z) with an>0 for all n≥p. Under a Rakhmanov-type condition on the measures generating the Nikishin system, we prove that the sequence of ratios Qn+1(z)∕Qn(z) and the sequence an of recurrence coefficients are limit periodic with period p(p+1). Our results complement some results obtained by the first author and Miña-Díaz in a recent paper in which algebraic properties and weak asymptotics of these polynomials were investigated. Our results also extend some results obtained by the first author in the case p=2.

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

We investigate the asymptotic behaviour of a family of multiple orthogonal polynomials that is naturally linked with the normal matrix model with a monomial potential of arbitrary degree d + 1. The polynomials that we investigate are multiple orthogonal with respect to a system of d analytic weights defined on a symmetric (d + 1)-star centred at the origin. In the first part we analyse in detail a vector equilibrium problem involving a system of d interacting measures (μ1, …, μd) supported on star-like sets in the plane. We show that in the subcritical regime, the first component of the solution to this problem is the asymptotic zero distribution of the multiple orthogonal polynomials. It also characterizes the domain where the eigenvalues in the normal matrix model accumulate, in the sense that the Schwarz function associated with the boundary of this domain can be expressed explicitly in terms of . The second part of the paper is devoted to the asymptotic analysis of the multiple orthogonal polynomials. The asymptotic results are obtained again in the subcritical regime, and they follow from the Deift/Zhou steepest descent analysis of a Riemann–Hilbert problem of size (d + 1) × (d + 1). The vector equilibrium problem and the Riemann–Hilbert problem that we investigate are generalizations of those studied recently by Bleher–Kuijlaars in the case d = 2.

Approximation by polynomials on quaternionic compact sets

In this paper, we obtain several extensions to the quaternionic setting of some results concerning the approximation by polynomials of functions continuous on a compact set and holomorphic in its interior. To this end, we prove an analog of the Riemann mapping theorem for a subclass of open sets, whose validity involves precisely the slice regular functions for which the composition remains slice regular. The results include approximation on compact starlike sets and compact axially symmetric sets. The cases of some concrete particular sets are described in details, including also quantitative estimates. Copyright © 2014 John Wiley & Sons, Ltd.

Equivalence of compositional expressions and independence relations in compositional models

We generalize Jirousek's (right) composition operator in such a way that it can be applied to distribution functions with values in a semifield, and introduce (parenthesized) compositional expressions, which in some sense generalize Jirousek's of compositional models. We say that two compositional expressions are equivalent if their evaluations always produce the same results whenever they are defined. Our first result is that a set system H is star-like with centre X if and only if every two compositional expressions with H and X are equivalent. This result is stronger than Jirousek's result which states that, if H is star-like with centre X, then every two generating sequences with base scheme H and key X are equivalent. Then, we focus on canonical expressions, by which we mean compositional expressions such that the sequence of the sets featured in and arranged in order of appearance enjoys the running intersection property. Since every compositional expression, whose base scheme is a star-like set system with centre X and whose key is X, is a canonical expression, we investigate the equivalence between two canonical expressions with the same base scheme and the same key. We state a graphical characterization of those set systems H such that every two canonical expressions with base scheme H and key X are equivalent, and also provide a graphical algorithm for their recognition. Finally, we discuss the problem of detecting conditional independences that hold in a compositional model.

High-Order Three-Term Recursions, Riemann–Hilbert Minors and Nikishin Systems on Star-Like Sets

We study monic polynomials Q n (x) generated by a high-order three-term recursion xQ n (x)=Q n+1(x)+a n−p Q n−p (x) with arbitrary p≥1 and a n >0 for all n. The recursion is encoded by a two-diagonal Hessenberg operator H. One of our main results is that, for periodic coefficients a n and under certain conditions, the Q n are multiple orthogonal polynomials with respect to a Nikishin system of orthogonality measures supported on star-like sets in the complex plane. This improves a recent result of Aptekarev–Kalyagin–Saff, where a formal connection with Nikishin systems was obtained in the case when $\sum_{n=0}^{\infty}|a_{n}-a|<\infty$ for some a>0. An important tool in this paper is the study of ‘Riemann–Hilbert minors’, or equivalently, the ‘generalized eigenvalues’ of the Hessenberg matrix H. We prove interlacing relations for the generalized eigenvalues by using totally positive matrices. In the case of asymptotically periodic coefficients a n , we find weak and ratio asymptotics for the Riemann–Hilbert minors and we obtain a connection with a vector equilibrium problem. We anticipate that in the future, the study of Riemann–Hilbert minors may prove useful for more general classes of multiple orthogonal polynomials.

Characterization of a condition related to a class of preinvex functions

In the theory of invex functions, a very important role is played by the so-called Condition C. We present a convenient characterization of this condition in terms of starlike sets and projections.

Averages over starlike sets, starlike maximal functions, and homogeneous singular integrals

We improve some of the results in our 1999 paper concerning weighted norm estimates for homogeneous singular integrals with rough kernels. Using a representation of such integrals in terms of averages over starlike sets, we prove a two-weight L p L^{p} inequality for 1 &gt; p &gt; 2 1 &gt; p &gt; 2 which we were previously able to obtain only for p ≥ 2 p \geq 2 . We also construct examples of weights that satisfy conditions which were shown in our earlier paper to be sufficient for one-weight inequalities when 1 &gt; p &gt; ∞ 1&gt;p&gt;\infty .

Asymptotics of multiple orthogonal polynomials for a system of two measures supported on a starlike set

For a system of two measures supported on a starlike set in the complex plane, we study the asymptotic properties of the associated multiple orthogonal polynomials Q n and their recurrence coefficients. These measures are assumed to form a Nikishin-type system, and the polynomials Q n satisfy a three-term recurrence relation of order three with positive coefficients. Under certain assumptions on the orthogonality measures, we prove that the sequence of ratios { Q n + 1 / Q n } has four different periodic limits, and we describe these limits in terms of a conformal representation of a compact Riemann surface. Several relations are found involving these limiting functions and the limiting values of the recurrence coefficients. We also study the n th root asymptotic behavior and zero asymptotic distribution of Q n .

Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials

The asymptotic theory is developed for polynomial sequences that are generated by the three-term higher-order recurrence $$Q_{n+1}=zQ_{n}-a_{n-p+1}Q_{n-p},\quad p\in\mathbb{N},n\geq p,$$ where z is a complex variable and the coefficients ak are positive and satisfy the perturbation condition ∑n=1∞|an−a|<∞. Our results generalize known results for p=1, that is, for orthogonal polynomial sequences on the real line that belong to the Blumenthal–Nevai class. As is known, for p≥2, the role of the interval is replaced by a starlike set S of p+1 rays emanating from the origin on which the Qn satisfy a multiple orthogonality condition involving p measures. Here we obtain strong asymptotics for the Qn in the complex plane outside the common support of these measures as well as on the (finite) open rays of their support. In so doing, we obtain an extension of Weyl’s famous theorem dealing with compact perturbations of bounded self-adjoint operators. Furthermore, we derive generalizations of the classical Szegő functions, and we show that there is an underlying Nikishin system hierarchy for the orthogonality measures that is related to the Weyl functions. Our results also have application to Hermite–Pade approximants as well as to vector continued fractions.

Coefficient inequalities for concave and meromorphically starlike univalent functions

Let D denote the open unit disk and f : D ! C be mero- morphic and univalent in D with the simple pole at p 2 (0,1) and satis- fying the standard normalization f(0) = f 0 (0) − 1 = 0. Also, let f have the expansion f(z) = 1 X n=−1 an(z − p) n , jz − pj< 1 − p, such that f maps D onto a domain whose complement with respect to C is a convex set (starlike set with respect to a point w0 2 C,w0 6 0 resp.). We call these functions as concave (meromorphically starlike resp.) univalent functions and denote this class by Co(p) (� s (p,w0) resp.). We prove some coefficient estimates for functions in the classes where the sharpness of these estimates is also achieved.

On the Increase of Capacity with Asymmetry

Let E be a compact, starlike set in ℝN, N ≥2, that is very close to a ball B of the same area or volume. This paper presents inequalities, for logarithmic capacity if N = 2 or for capacity if N ≥ 3, of the form \(lcap\ E \geq exp \lbrace K_2\alpha(E)^2\rbrace\ lcap\ \overline B\) or \(cap\ E \geq \lbrace 1+K_N\alpha(E)^2\rbrace\ cap\ \overline B\) where α(E) is a modulus of asymmetry that measures the departure of the shape of E from that of B. The results are far less general than those of Hansen and Nadirashvili for N = 2 and those of Hall, Hayman and Weitsman for N ≥ 3, but (for the particular sets considered) the present inequalities are somewhat sharper.

ON THE PRIMITIVES OF IFFERENTIAL ONE-MAPS

If a i .erential one-map between normed spaces is continuous on a star-like open set an is i .erentiable possibly except of one point then it has a primitive if an only if its erivative as a bilinear form is symmetric at every point where it exists.A generalization of the theorem on the existence of the potential operator is also obtained.The proof is based on the idea of the Goursat Lemma.