pinecone/core-2020-05-10-deduplication huggingface dataset:
On the Behaviour of Zeros of Jacobi Polynomials

AbstractDenote by xn,k(α,β) and xn,k(λ)=xn,k(λ−1/2,λ−1/2) the zeros, in decreasing order, of the Jacobi polynomial P(α,β)n(x) and of the ultraspherical (Gegenbauer) polynomial Cλn(x), respectively. The monotonicity of xn,k(α,β) as functions of α and β, α,β>−1, is investigated. Necessary conditions such that the zeros of P(a,b)n(x) are smaller (greater) than the zeros of P(α,β)n(x) are provided. A. Markov proved that xn,k(a,b)<xn,k(α,β) (xn,k(a,b)>xn,k(α,β)) for every n∈N and each k, 1⩽k⩽n if a>α and b<β (a<α and b>β). We prove the converse statement of Markov's theorem. The question of how large the function fn(λ) could be such that the products fn(λ)xn,k(λ), k=1,…,[n/2] are increasing functions of λ, for λ>−1/2, is also discussed. Elbert and Siafarikas proved that fn(λ)=(λ+(2n2+1)/(4n+2))1/2 obeys this property. We establish the sharpness of their result
Design Approximation Problems for Linear-Phase Nonrecursive Digital Filters

AbstractThe topic of this paper is the study of four real, linear, possibly constrained minimum norm approximation problems, which arise in connection with the design of linear-phase nonrecursive digital filters and are distinguished by the type of used trigonometric approximation functions. In the case of unconstrained minimax designs these problems are normally solved by the Parks–McClellan algorithm, which is an application of the second algorithm of Remez to these problems and which is one of the most popular tools in filter design. In this paper the four types of approximation problems are investigated for all Lp and lp norms, respectively. It is especially proved that the assumptions for the Remez algorithm are satisfied in all four cases, which has been claimed, but is not obvious for three of them. Furthermore, results on the existence and uniqueness of solutions and on the convergence and the rate of convergence of the approximation errors are derived
Classification Theorems for General Orthogonal Polynomials on the Unit Circle

AbstractThe set P of all probability measures σ on the unit circle T splits into three disjoint subsets depending on properties of the derived set of {|ϕn|2dσ}n⩾0, denoted by Lim(σ). Here {ϕn}n⩾0 are orthogonal polynomials in L2(dσ). The first subset is the set of Rakhmanov measures, i.e., of σ∈P with {m}=Lim(σ), m being the normalized (m(T)=1) Lebesgue measure on T. The second subset Mar(T) consists of Markoff measures, i.e., of σ∈P with m∉Lim(σ), and is in fact the subject of study for the present paper. A measure σ, belongs to Mar(T) iff there are ε>0 and l>0 such that sup{|an+j|:0⩽j⩽l}>ε, n=0,1,2,…,{an} is the Geronimus parameters (=reflectioncoefficients) of σ. We use this equivalence to describe the asymptotic behavior of the zeros of the corresponding orthogonal polynomials (see Theorem G). The third subset consists of σ∈P with {m}⫋Lim(σ). We show that σ is ratio asymptotic iff either σ is a Rakhmanov measure or σ satisfies the López condition (which implies σ∈Mar(T)). Measures σ satisfying Lim(σ)={ν} (i.e., weakly asymptotic measures) are also classified. Either ν is the sum of equal point masses placed at the roots of zn=λ, λ∈T, n=1,2,…, or ν is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism z→zn, n=1,2,…, of a closed arc J (including J=T) with removed open concentric arc J0 (including J0=∅). Next, weakly asymptotic measures are completely described in terms of their Geronimus parameters. Finally, we obtain explicit formulae for the parameters of the equilibrium measures ν and show that these measures satisfy {ν}=Lim(ν)
On Lp-Generalization of a Theorem of Adamyan, Arov, and Kreın

AbstractThe paper deals with problems relating to the theory of Hankel operators. Let G be a bounded simple connected domain with the boundary Γ consisting of a closed analytic Jordan curve. Denote by Mn,p(G), 1⩽p<∞, the class of all meromorphic functions on G that can be represented in the form h=β/α, where β belongs to the Smirnov class Ep(G), α is a polynomial degree at most n, α≢0. We obtain estimates of s-numbers of the Hankel operator Af constructed from f∈Lp(Γ), 1⩽p<∞, in terms of the best approximation Δn,p of f in the space Lp(Γ) by functions belonging to the class Mn,p(G)
The Diagonalisation of the Multivariate Bernstein Operator

AbstractLet Bn be the multivariate Bernstein operator of degree n for a simplex in Rs. In this paper, we show that Bn is diagonalisable with the same eigenvalues as the univariate Bernstein operator, i.e.,λ(n)k≔n!(n−k)!1nk, k=1,…,n, 1=λ(n)1>λ(n)2>···>λ(n)n>0 , and we describe the corresponding eigenfuctions and their properties. Since Bn reproduces only the linear polynomials, these are the eigenspace for λ(n)1=1. For k>1, the λ(n)k-eigenspace consists of polynomials of exact degree k, which are uniquely determined by their leading term. These are described in terms of the substitution of the barycentric coordinates (for the underlying simplex) into elementary eigenfunctions. It turns out that there are eigenfunctions of every degree k which are common to each Bn, n⩾k, for sufficiently large s. The limiting eigenfunctions and their connection with orthogonal polynomials of several variables is also considered
On Carleman and Knopp's Inequalities

AbstractA sharpened version of Carleman's inequality is proved. This result unifies and generalizes some recent results of this type. Also the “ordinary” sum that serves as the upper bound is replaced by the corresponding Cesaro sum. Moreover, a Carleman-type inequality with a more general measure is proved and this result may also be seen as a generalization of a continuous variant of Carleman's inequality, which is usually referred to as Knopp's inequality. A new elementary proof of (Carleman–)Knopp's inequality and a new inequality of Hardy–Knopp type is pointed out
Direct and Inverse Approximation Theorems for Local Trigonometric Bases

AbstractWe investigate the approximation of smooth functions by local trigonometric bases. In particular, we are interested in the local behavior of the approximation error in the Lp-norm. We derive direct and inverse approximation theorems that describe the best approximation on an interval by a finite linear combination of basis functions with support in this interval. As a result, we characterize the Besov spaces on an interval as approximation spaces with respect to a local trigonometric basis. These local results are generalized to the approximation on the real line by linear combinations which are locally finite. The proofs are based on the classical inequalities of Jackson and Timan which are applied to local trigonometric bases by the means of folding and unfolding operators
A Construction of Compactly Supported Biorthogonal Scaling Vectors and Multiwavelets on R2

AbstractIn Kessler (Appl. Comput. Harmonic Anal.9 (2000), 146–165), a construction was given for a class of orthogonal compactly supported scaling vectors on R2, called short scaling vectors, and their associated multiwavelets. The span of the translates of the scaling functions along a triangular lattice includes continuous piecewise linear functions on the lattice, although the scaling functions are fractal interpolation functions and possibly nondifferentiable. In this paper, a similar construction will be used to create biorthogonal scaling vectors and their associated multiwavelets. The additional freedom will allow for one of the dual spaces to consist entirely of the continuous piecewise linear functions on a uniform subdivision of the original triangular lattice
Coconvex Approximation

AbstractLet f∈C[−1,1] change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials which are coconvex with it, namely, polynomials that change their convexity exactly at the points where f does. We discuss some Jackson-type estimates where the constants involved depend on the location of the points of change of convexity. We also show that in some cases the constants may be taken independent of the points of change of convexity, but that in other cases this dependence is essential. But mostly we obtain such estimates for functions f that themselves are continuous piecewise polynomials on the Chebyshev partition, which form a single polynomial in a small neighborhood of each point of change of convexity. These estimates involve the k modulus of smoothness of the piecewise polynomials when they themselves are of degree k−1
Oblique Projections and Abstract Splines

AbstractGiven a closed subspace S of a Hilbert space H and a bounded linear operator A∈L(H) which is positive, consider the set of all A-self-adjoint projections onto S: P(A,S)={Q∈L(H): Q2=Q, Q(H)=S, AQ=Q*A}. In addition, if H1 is another Hilbert space, T:H→H1 is a bounded linear operator such that T*T=A and ξ∈H, consider the set of (T,S) spline interpolants to ξ: s p(T,S,ξ)=η∈ξ+S:∣∣Tη∣∣=minσ∈S∣∣T(ξ+σ)∣∣. A strong relationship exists between P(A,S) and sp(T,S,ξ). In fact, P(A,S) is not empty if and only if s p(T,S,ξ) is not empty for every ξ∈H. In this case, for any ξ∈H\S it holds s p(T,S,ξ)={(1−Q)ξ:Q∈P(A,S)} and for any ξ∈H, the unique vector of s p(T,S,ξ) with minimal norm is (1−PA,S)ξ, where PA,S is a distinguished element of P(A,S). These results offer a generalization to arbitrary operators of several theorems by de Boor, Atteia, Sard and others, which hold for closed range operators
On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

AbstractFor n⩾1, let {xjn}j=1n be n distinct points and let Ln[·] denote the corresponding Lagrange Interpolation operator. Let W : R→[0,∞). What conditions on the array {xjn}1⩽j⩽n, n⩾1 ensure the existence of p>0 such limn→∞∥(f−Ln[f])Wφb∥Lp(R)=0 for every continuous f :R→R with suitably restricted growth, and some “weighting factor” φb? We obtain a necessary and sufficient condition for such a p to exist. The result is the weighted analogue of our earlier work for interpolation arrays contained in a compact set
Sharp Estimates for Jacobi Matrices and Chain Sequences

AbstractChain sequences are positive sequences {an} of the form an=gn(1−gn−1) for a nonnegative sequence {gn}. This concept has been introduced by Wall in connection with continued fractions. These sequences are very useful in determining the support of orthogonality measure for orthogonal polynomials. Equivalently, they can be used for localizing spectra of Jacobi matrices associated with orthogonal polynomials through the recurrence relation.We derive sharp estimates for chain sequences which in turn give sharp estimates for the norms of Jacobi matrices. We also give applications to unbounded essentially self-adjoint Jacobi matrices. In particular, we show how to determine whether their spectrum admits gaps around 0, and derive some integrability properties of the spectral measure
Blumenthal's Theorem for Laurent Orthogonal Polynomials

AbstractWe investigate polynomials satisfying a three-term recurrence relation of the form Bn(x)=(x−βn)Bn−1(x)−αnxBn−2(x), with positive recurrence coefficients αn+1,βn (n=1,2,…). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. We analyse in more detail the case where αn→α and βn→β and show that the zeros of Bn are dense on an interval and that the support of the Laurent orthogonality measure is equal to this interval and a set which is at most denumerable with accumulation points (if any) at the endpoints of the interval. This result is the Laurent version of Blumenthal's theorem for orthogonal polynomials
Isotonic Approximation in L1

AbstractLet (Ω,A,P) be a measurable space and L⊂A a sub-σ-lattice of the σ-algebra A. For X∈L1(Ω,A,P) we denote by PLX the set of conditional 1-mean (or best approximants) of X given L1(L) (the set of all L-measurable and integrable functions). In this paper, we obtain characterizations of the elements in PLX, similar to those obtained by Landers and Rogge for conditional s-means with 1<s<∞. Moreover, using these characterizations we can extend the operator PL to a bigger space L0(Ω,A,P). When, in certain sense, Ln goes to L∞, we will be able to prove theorems about convergence and we will obtain bounds for the maximal function ∣supnPLnX∣. A sharper characterization of conditional 1-means for certain particular σ-lattice was proved in previous papers. In the last section of this paper we generalize those results to all totally ordered σ-lattices
On Strong Approximations of USC Nonconvex-Valued Mappings

AbstractFor any upper semicontinuous and compact-valued (usco) mapping F : X→Y from a metric space X without isolated points into a normed space Y we prove the existence of a single-valued continuous mapping f : X→Y such that the Hausdorff distance between graphs ΓF and Γf is arbitrarily small, whenever “measure of nonconvexity” of values of F admits an appropriate common upper estimate. Hence, we prove a version of the Beer–Cellina theorem, under controlled withdrawal of convexity of values of multifunctions. We also give conditions for such strong approximability of star-shaped-valued upper'semicontinuous (usc) multifunctions in comparison with Beer's result for Hausdorff continuous star-shaped-valued multifunctions
Weighted Sobolev spaces on curves

45 pages, no figures.-- MSC1987 codes: 41A10, 46E35, 46G10.MR#: MR1934626 (2003j:46038)Zbl#: Zbl 1019.46026In this paper we present a definition of weighted Sobolev spaces on curves and find general conditions under which the spaces are complete for non-closed compact curves. We also prove the density of the polynomials in these spaces and, finally, we find conditions under which the multiplication operator is bounded in the space of polynomials.Research of second (D.P.), third (J.M.R.) and fourth (E.R.) authors was partially supported by a grant from DGI (BFM 2000-0206-C04-01), Spain.Publicad
Stable Summation of Orthogonal Series with Noisy Coefficients

AbstractWe study the recovery of continuous functions from Fourier coefficients with respect to certain given orthonormal systems, blurred by noise. For deterministic noise this is a classical ill-posed problem. Emphasis is laid on a priori smoothness assumptions on the solution, which allows to apply regularization to reach the best possible accuracy. Results are obtained for systems obeying norm growth conditions. In the white noise setting mild additional assumptions have to be made to have accurate bounds. We finish our study with the recovery of functions from noisy coefficients with respect to the Haar system
Dense Linear Manifolds of Monsters

AbstractIn this paper the new concept of totally omnipresent operators is introduced. These operators act on the space of holomorphic functions of a domain in the complex plane. The concept is more restrictive than that of strongly omnipresent operators, also introduced by the authors in an earlier work, and both of them are related to the existence of functions whose images under such operators exhibit an extremely wild behaviour near the boundary. Sufficient conditions for an operator to be totally omnipresent as well as several outstanding examples are provided. After extending a statement of the first author about the existence of large linear manifolds of hypercyclic vectors for a sequence of suitable continuous linear mappings, it is shown that there is a dense linear manifold of holomorphic monsters in the sense of Luh, so completing earlier nice results due to Luh and Grosse-Erdmann
On Some Extremal Properties of Lagrange Interpolatory Polynomials

AbstractIn this paper, we will show that Lagrange interpolatory polynomials are optimal for solving some approximation theory problems concerning the finding of linear widths.In particular, we will show thatinfLn∈ℓnsupp∈Pn+1 ∥p−Lnp∥C[−1,1]=12n , where Ln is a set of the linear operators with finite rank n+1 defined on C−1,1], and where Pn+1 denotes the set of polynomials p=∑i=0n+1aixi of degree⩽n+1 such that ∣an+1∣⩽1. The infimum is achieved for Lagrange interpolatory polynomial for nodes 2k+12(n+1), k=0,…,n
Universal Overconvergence of Polynomial Expansions of Harmonic Functions

AbstractFor each compact subset K of RN let H(K) denote the space of functions that are harmonic on some neighbourhood of K. The space H(K) is equipped with the topology of uniform convergence on K. Let Ω be an open subset of RN such that 0∈Ω and RN\Ω is connected. It is shown that there exists a series ∑Hn, where Hn is a homogeneous harmonic polynomial of degree n on RN, such that (i) ∑Hn converges on some ball of centre 0 to a function that is continuous on Ω and harmonic on Ω, (ii) the partial sums of ∑Hn are dense in H(K) for every compact subset K of RN\Ω with connected complement. Some refinements are given and our results are compared with an analogous theorem concerning overconvergence of power series
Asymptotic Analysis for the Dunkl Kernel

AbstractThis paper studies the asymptotic behavior of the integral kernel of the Dunkl transform, the so-called Dunkl kernel, when one of its arguments is fixed and the other tends to infinity either within a Weyl chamber of the associated reflection group, or within a suitable complex domain. The obtained results are based on the asymptotic analysis of an associated system of ordinary differential equations. They generalize the well-known asymptotics of the confluent hypergeometric function 1F1 to the higher-dimensional setting and include a complete short-time asymptotics for the Dunkl-type heat kernel. As an application, it is shown that the representing measures of Dunkl's intertwining operator are generically continuous
Mass Points of Measures and Orthogonal Polynomials on the Unit Circle

AbstractOrthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szego&#x030B; recurrences. We assume that the reflection coefficients tend to some complex number a with 0<∣a∣<1. The orthogonality measure μ then lives essentially on the arc {eit :α⩽t⩽2π−α} where sinα2=def∣a∣ with α∈(0,π). Under the certain rate of convergence it was proved in (Golinskii et al. (J. Approx. Theory96 (1999), 1–32)) that μ has no mass points inside this arc. We show that this result is sharp in a sense. We also examine the case of the whole unit circle and some examples of singular continuous measures given by their reflection coefficients
Approximation Numbers of Identity Operators between Symmetric Sequence Spaces

AbstractWe prove asymptotic formulas for the behavior of approximation quantities of identity operators between symmetric sequence spaces. These formulas extend recent results of Defant, Mastyło, and Michels for identities lpn↪Fn with an n-dimensional symmetric normed space Fn with p-concavity conditions on Fn and 1⩽p⩽2. We consider the general case of identities En↪Fn with weak assumptions on the asymptotic behavior of the fundamental sequences of the n-dimensional symmetric spaces En and Fn. We give applications to Lorentz and Orlicz sequence spaces, again considerably generalizing results of Pietsch, Defant, Mastyło, and Michels
A Chebyshev Set and its Distance Function

AbstractWe prove that in a Banach space X with rotund dual X* a Chebyshev set C is convex iff the distance function dC is regular on X\C iff dC admits the strict and Gâteaux derivatives on X\C which are determined by the subdifferential ∂∣∣x−x̄∣∣ for each x∈X\C and x̄∈PC(x)≔\{c∈C:∣∣x−c∣∣=dC(x)}. If X is a reflexive Banach space with smooth and Kadec norm then C is convex iff it is weakly closed iff PC is continuous. If the norms of X and X* are Fréchet differentiable then C is convex iff dC is Fréchet differentiable on X\C. If also X has a uniformly Gâteaux differentiable norm then C is convex iff the Gâteaux (Fréchet) subdifferential ∂−dC(x) (∂FdC(x)) is nonempty on X\C
On the Asymptotics of Fekete-Type Points for Univariate Radial Basis Interpolation

AbstractSuppose that K⊂Rd is compact and that we are given a function f∈C(K) together with distinct points xi∈K, 1⩽i⩽n. Radial basis interpolation consists of choosing a fixed (basis) function g : R+→R and looking for a linear combination of the translates g(|x−xj|) which interpolates f at the given points. Specifically, we look for coefficients cj∈R such that F(x)=∑j=1ncjg(|x−xj|) has the property that F(xi)=f(xi), 1⩽i⩽n. The Fekete-type points of this process are those for which the associated interpolation matrix [g(|xi−xj|)]1⩽i,j⩽n has determinant as large as possible (in absolute value). In this work, we show that, in the univariate case, for a broad class of functions g, among all point sequences which are (strongly) asymptotically distributed according to a weight function, the equally spaced points give the asymptotically largest determinant. This gives strong evidence that the Fekete points themselves are indeed asymptotically equally spaced
Special Uniform Approximations of Continuous Vector-Valued Functions. Part I: Special Approximations in CX(T)

AbstractIn this paper, we give special uniform approximations of functions u from the spaces CX(T) and C∞(T,X), with elements ū of the tensor products CΓ(T)⊗X, respectively C0(T,Γ)⊗X, for a topological space T and a Γ-locally convex space X. We call an approximation special, if ū satisfies additional constraints, namely supp v⊂u−1(X\{0}) and ū(T)⊂ co(u(T)) (resp. ⊂ co(u(T)∪{0})). In Section 3, we give three distinct applications, which are due exactly to these constraints: a density result with respect to the inductive limit topology, a Tietze–Dugundji's type extension new theorem and a proof of Schauder–Tihonov's fixed point theorem
Macrophage apoptosis in the central nervous system in experimental autoimmune encephalomyelitis

Using light and electron microscopy, we have demonstrated that macrophage apoptosis (programmed cell death) occurs in the central nervous system (CNS) in Lewis rats with acute experimental autoimmune encephalomyelitis (EAE) and chronic relapsing EAE. Apoptotic macrophages were identified by the presence of an apoptotic nucleus in a cell with cytoplasm containing myelin debris but no intermediate filaments. They were found in the meninges, perivascular spaces and in the parenchyma of the white and grey matter of the spinal cord. In acute EAE the apoptotic macrophages were most frequently seen at the time of maximal neurological signs and during the early stages of clinical recovery. Several possible mechanisms may be responsible for the macrophage apoptosis: the release or withdrawal of cytokines; T-cell cytotoxicity; the effect of activated macrophage products, such as nitric oxide; and a direct effect of endogenous glucocorticoids. Macrophage apoptosis, together with the T-cell apoptosis we have previously described in the CNS in EAE, may contribute to the down-regulation of this autoimmune disease