pinecone/core-2020-05-10-deduplication huggingface dataset:
Error Estimates and Convergence Rates for Variational Hermite Interpolation

AbstractThis paper considers the variational problem of Hermite interpolation and its error bounds. The optimal Hermite interpolant, which minimises the semi-norm of the reproducing kernel Hilbert spaceChdetermined by givenr-CPDmfunctionh, is just theh-spline Hermite interpolant. The results on error estimation and convergence rate of theh-spline interpolant generalise those of W. R. Madych and S. A. Nelson (1988,Approx. Theory Appl.4, 77–89; 1990,Math. Comp.54, 211–230), Z. Wu and R. Schabach (1993,IMA J. Numer. Anal.13, 13–27), and W. Light and H. Wayne (1994,in“Approximation Theory, Wavelets and Applications” (S. P. Singh, Ed.), pp. 215–246, Kluwer Academic, Dordrecht/Norwell, MA) to the case of Hermite interpolation
Monotonicity Properties of Musielak–Orlicz Spaces and Dominated Best Approximation in Banach Lattices

AbstractCriteria for strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity and uniform monotonicity of a Musielak–Orlicz space endowed with the Amemiya norm and its subspace of order continuous elements are given in the cases of nonatomic and the counting measure space. To complete the results of Kurc (J. Approx. Theory69(1992), 173–187), criteria for upper local uniform monotonicity of these spaces equipped with the Luxemburg norm are also given. Some applications to dominated best approximation are presented
New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field

AbstractIn this paper we use techniques from the theory of ODEs and also from inverse scattering theory to obtain a variety of results on the regularity and support properties of the equilibrium measure for logarithmic potentials on the finite interval [−1, 1], in the presence of an external fieldV. In particular, we show that ifVisC2, then the equilibrium measure is absolutely continuous with respect to Lebesgue measure, with a density which is Hölder-12 on (−1, 1), and with at worst a square root singularity at ±1. Moreover, ifVis real analytic then the support of the equilibrium measure consists of a finite number of intervals. In the cases whereV=txm,m=1, 2, 3, or 4, the equilibrium measure is computed explicitly for allt∈R. For these cases the support of the equilibrium measure consists of 1, 2, or 3 intervals, depending ontandm. We also present detailed results for the general monomial caseV=txm, for allm∈N. The regularity results for the equilibrium measure are obtained by careful analysis of the Fekete points associated to the weightenV(x)dx. The results on the support of the equilibrium measure are obtained using two different approaches: (i) an explicit formula of the kind derived by physicists for mean-field theory calculations; (ii) detailed perturbation theoretic results of the kind that are needed to analyze the zero dispersion limit of the Korteweg–de Vries equation in Lax–Levermore theory. The implications of the above results for a variety of related problems in approximation theory and the theory of orthogonal polynomials are also discussed
Best Constants in Global Smoothness Preservation Inequalities for Some Multivariate Operators

AbstractThe Bernstein operator on the standardk-simplex and other analogousk-variate operators allow for a probabilistic representation in terms of the successive increments of a real valued superstationary stochastic process (a notion introduced in the paper) starting at the origin and having nondecreasing paths. For this class of operators, we obtain estimates of the best constants in preservation of the first modulus of continuity corresponding to thel1-norm, and in preservation of classes of functions defined by concave moduli of continuity. We also show that, in some special cases, such best constants do not depend upon the dimensionk. To show our results, we use probabilistic tools such as couplings and Wasserstein distances for multivariate probability distributions. The general results are applied to the computation of the aforementioned constants for several classical multivariate operators
On the Asymptotic Approximation with Bivariate Operators of Bleimann, Butzer, and Hahn

AbstractThe concern of this paper is a recent generalizationLn(f(t1,t2);x,y) for the operators of Bleimann, Butzer, and Hahn in two variables which is distinct from a tensor product. We present the complete asymptotic expansion for the operatorsLnasntends to infinity. The result is in a form convenient for applications. All coefficients ofn−k(k=1,2,…) are calculated explicitly in terms of Stirling numbers of the first and second kind. As a special case we obtain a Voronovskaja-type theorem for the operatorsLn. The result for the one-dimensional case was previously derived by the author
Best Constants in Preservation Inequalities Concerning the First Modulus and Lipschitz Classes for Bernstein-Type Operators

AbstractWe consider families (Lt, t∈T) of positive linear operators such that eachLtis representable in terms of a stochastic process starting at the origin and having nondecreasing paths and integrable stationary increments. For these families, we give probabilistic characterizations of the best possible constants both in preservation inequalities concerning the first modulus and in preservation of Lipschitz classes of first order. As an application, we compute such constants for the Bernstein, Szász, Gamma, Baskakov, and Beta operators
Asymptotic Behaviour of Solutions of Linear Recurrences and Sequences of Möbius-Transformations

AbstractThis paper is mainly concerned with the study of recurrences defined by Möbius-transformations, whose solutions are the orbits of points on the Riemann-sphere under a sequence of Möbius-transformations. We study the asymptotic behaviour of such solutions in relation to the asymptotic behaviour of the coefficients of the Möbius-transformations. Most of the theorems give sufficient conditions in order that there exist converging solutions, but a section of examples is added where examples are given of recurrences whose solutions do not converge because one or several of the conditions of the theorems are violated. One of the most important results of this paper is that if the fixpoints of the Möbius-transformations are of bounded variation and converge to distinct limits, then the behaviour of the solutions depends entirely on the products of the derivatives in the fixpoints. Several methods will be proposed to deal with the case that the fixpoints converge to one single limit. The paper starts with a few results onnth order recurrences and matrix recurrences and concludes with an investigation of the asymptotic behaviour of the solutions of linear second-order recurrences having coefficients that are asymptotic expressions in fractional powers of the indexn. A number of examples are added in order to show how some of the theorems can be applied
Successive Approximate Algorithm for Best Approximation from a Polyhedron

AbstractSupposeKis the intersection of a finite number of closed half-spaces {Ki} in a Hilbert spaceX, andx∈X\K. Dykstra's cyclic projections algorithm is a known method to determine an approximate solution of the best approximation ofxfromK, which is denoted byPK(x). Dykstra's algorithm reduces the problem to an iterative scheme which involves computing the best approximation from the individualKi. It is known that the sequence {xj} generated by Dykstra's method converges to the best approximationPK(x). But since it is difficult to find the definite value of an upper bound of the error ‖xj−PK(x)‖, the applicability of the algorithm is restrictive. This paper introduces a new method, called thesuccessive approximate algorithm, by which one can generate a finite sequencex0, x1, …, xkwithxk=PK(x). In addition, the error ‖xj−PK(x)‖ is monotone decreasing and has a definite upper bound easily to be determined. So the new algorithm is very applicable in practice
Average CaseL∞-Approximation in the Presence of Gaussian Noise

AbstractWe consider the average caseL∞-approximation of functions fromCr([0, 1]) with respect to ther-fold Wiener measure. An approximation is based onnfunction evaluations in the presence of Gaussian noise with varianceσ2>0. We show that the n th minimal average error is of ordern−(2r+1)/(4r+4)ln1/2n, and that it can be attained either by the piecewise polynomial approximation using repetitive observations, or by the smoothing spline approximation using non-repetitive observations. This completes the already known results forLq-approximation withq<∞ andσ⩾0, and forL∞-approximation withσ=0
On a Generalized Best Approximation Problem

AbstractLetCbe a closed bounded convex subset of a Banach spaceEwhich has the origin ofEas an interior point and letpCdenote the Minkowski functional with respect toC. Given a closed setX⊂Eand a pointu∈Ewe consider a minimization problem minC(u, X) which consists in proving the existence of a pointx∈Xsuch thatpC(x−u)=λC(u, X), whereλC(u, X)=inf{pC(x−u)∣x∈X}. If such a point is unique and every sequence {xn}⊂Xsatisfying the condition limn→+∞pC(xn−u)=λC(u, X) converges to this point, the minimization problem min(u, X) is called well posed. Under the assumption that the modulus of convexity with respect topCis strictly positive, we prove that for every closed subsetXofE, the setEo(X) of allu∈Efor which the minimization problem minC(u, X) is well posed is a residual subset ofE. In fact we show more, namely that the setE\Eo(X) isσ-porous inE. Moreover, we prove that for most closed bounded subsetsXofE, the setE\Eo(X) is dense inE
On Some Classes of Polynomials Orthogonal on Arcs of the Unit Circle Connected with Symmetric Orthogonal Polynomials on an Interval

AbstractStarting from the Delsarte–Genin (DG) mapping of the symmetric orthogonal polynomials on an interval (OPI) we construct a one-parameter family of polynomials orthogonal on the unit circle (OPC). The value of the parameter defines the arc on the circle where the weight function vanishes. Some explicit examples of OPC connected with generic Askey–Wilson polynomials are constructed. These polynomials can be considered, as a four-parameter extension of the Askey–Szegő polynomials on the unit circle. We also present examples of OPC having finite and infinite purely discrete spectrum and addition masses inside and outside the unit circle. Connections of DG transformation with sieved OPI and OPC, chain sequences, and the Bauer's numericalg-algorithm (in approximation theory) are analyzed. In particular, we construct some classes of “semiclassical” OPI and OPC starting from periodic solutions of the Bauer'sg-algorithm
Three Conjectures on Shepard Interpolatory Operators

AbstractAfter establishing direct and converse approximation theorems for the Shepard interpolatory operators, J. Szabados (Approx. Theory Appl.7, No. 3, 1991, 63–76) left some open saturation problems (“the most intriguing questions” as he said), which he raised as three conjectures. The present paper proves the second parts of some conjectures, but constructs counterexamples to show that the first parts of three conjectures are not true. The constructive procedure uses some novel ideas and techniques
Gram Polynomials and the Kummer Function

AbstractLet {φk}nk=0,n<m, be a family of polynomials orthogonal with respect to the positive semi-definite bilinear form(g, h)d:=1m∑j=1mg(xj)h(xj),xj:=−1+(2j−1)/m.These polynomials are known as Gram polynomials. The present paper investigates the growth of |φk(x)| as a function ofkandmfor fixedx∈[−1, 1]. We show that whenn⩽2.5m1/2, the polynomials in the family {φk}nk=0are of modest size on [−1, 1], and they are therefore well suited for the approximation of functions on this interval. We also demonstrate that if the degreekis close tom, andm⩾10, thenφk(x) oscillates with large amplitude for values ofxnear the endpoints of [−1, 1], and this behavior makesφkpoorly suited for the approximation of functions on [−1, 1]. We study the growth properties of |φk(x)| by deriving a second order differential equation, one solution of which exposes the growth. The connection between Gram polynomials and this solution to the differential equation suggested what became a long-standing conjectured inequality for the confluent hypergeometric function1F1, also known as Kummer's function, i.e., that1F1((1−a)/2, 1, t2)⩽1F1(1/2, 1, t2) for alla⩾0. In this paper we completely resolve this conjecture by verifying a generalization of the conjectured inequality with sharp constants
The Lebesgue Function and Lebesgue Constant of Lagrange Interpolation for Erdoo&#x030B;s Weights

AbstractWe establish pointwise as well as uniform estimates for Lebesgue functions associated with a large class of Erdo&#x030B;s weights on the real line. An Erdo&#x030B;s weight is of the formW≔exp(−Q), whereQ:R→Ris even and is of faster than polynomial growth at infinity. The archetypal examples areWk,α(x) ≔exp(−Qk,α(x)), ((i))whereQk,α(x) ≔expk(|x|α),α>1,k⩾1. Here expk≔exp(exp(exp(…))) denotes thekth iterated exponential.WA,B(x) ≔exp(−QA,B(x)), ((ii))whereQA,B(x) ≔exp(log(A+x2))B,B>1 andA>A0. For a carefully chosen system of nodesχn≔{ξ1,ξ2,…,ξn},n⩾1, our result imply in particular, that the Lebesgue constant ‖Λn(Wk,α,χn)‖L∞(R)≔supx∈R|Λn(Wk,α,χn)| (x) satisfies uniformly forn⩾N0, ‖Λn(Wk,α,χn)‖L∞(R)∼logn. Moreover, we show that this choice of nodes is optimal with respect to the zeros of the orthonormal polynomials generated byW2. Indeed, letUn≔{xj,n:1⩽j⩽n},n⩾1, where thexk,nare the zeros of the orthogonal polynomialspn(W2,·) generated byW2. Then in particular, we have uniformly forn⩾N, ‖Λn(Wk,α,Un)‖L∞(R)∼n1/6(∏kj=1logjn)1/6. Here,logj≔log(log(log(…))) denotes the j th iterated logarithm. We deduce sharp theorems of uniform convergence of weighted Lagrange interpolation together with rates of convergence. In particular, these results apply toWk,αandWA,B
Strong Approximation via Sidon Type Inequalities

AbstractThe aim of this paper is twofold. First we want to show how a duality relation provides a vehicle to deduce strong summability and approximation properties of Fourier series from some basic inequalities, called Sidon type inequalities. This way the technicalities concerning several strong summability and approximation problems can be reduced to proving such inequalities. On the other hand, we will isolate two properties that induce the sharpest version of these inequalities for a number of orthonormal systems, find their counterparts in terms of strong approximation, and show some of their consequences. We note that these results are known to be the best possible for the trigonometric syste
Non-Optimal Rates of Ergodic Limits and Approximate Solutions

AbstractThis paper is concerned with non-optimal rates of convergence for two processes {Aα} and {Bα}, which satisfy ‖Aα‖=O(1),BαA⊂ABα=I−Aα, ‖AAα‖=O(e(α)), whereAis a closed operator ande(α)→0. Under suitable conditions, we describe, in terms ofK-functionals, thosex(resp.y) for which the non-optimal convergence rate of {Aαx} (resp. {Bαy}) is of the orderO(f(α)), wherefis a function satisfyinge(α)⩽f(α)→0. In case thatf(α)/e(α)→∞, the sharpness of the non-optimal rate of {Aαx} is equivalent to thatAhas non-closed range. The result provides a unified approach to dealing with non-optimal rates for many particular mean ergodic theorems and for various methods of solving the equationAx=y. We discuss in particular applications toα-times integrated semigroups,n-times integrated cosine functions, and tensor product semigroups
Generalization of the Left Bernstein Quasi-Interpolants

AbstractP. Sablonnière introduced the so-called left Bernstein quasi-interpolant, and proved that the sequence of the approximating polynomials converges pointwise in high-order rate to each sufficiently smooth approximated function. On the other hand, Z.-C. Wu proved that the sequence of the norms of the operators is bounded. In this paper, we extract the essence why Sablonnière's operator exhibits good convergence and stability properties, and we clarify a sufficient condition for general operators to have similar properties. Moreover, regarding the family of the general operators, we derive detailed results about the derivatives of the approximating polynomials that estimate their uniform convergence degree, using a convenient differentiability condition on approximated functions. Our results readily imply all the preceding ones
Approximation ofk-Monotone Functions

AbstractIt is shown that an algebraic polynomial of degree ⩽k−1 which interpolates ak-monotone functionfatkpoints, sufficiently approximates it, even if the points of interpolation are close to each other. It is well known that this result is not true in general for non-k-monotone functions. As an application, we prove a (positive) result on simultaneous approximation of ak-monotone function and its derivatives inLp,0<p<1, metric, and also show that the rate of the best algebraic approximation ofk-monotone functions (with bounded (k−2)nd derivatives inLp, 1<p<∞, iso(n−k/p)
Nearly Comonotone Approximation

AbstractWe discuss the degree of approximation by polynomials of a functionfthat is piecewise monotone in [−1, 1]. We would like to approximatefby polynomials which are comonotone with it. We show that by relaxing the requirement for comonotonicity in small neighborhoods of the points where changes in monotonicity occur and near the endpoints, we can achieve a higher degree of approximation. We show here that in that case the polynomials can achieve the rate ofω3. On the other hand, we show in another paper, that no relaxing of the monotonicity requirements on sets of measures approaching 0 allowsω4estimates
On Polynomials Related with Hermite–Padé Approximations to the Exponential Function

AbstractWe investigate the polynomialsPn,Qm, andRs, having degreesn,m, ands, respectively, withPnmonic, that solve the approximation problem [formula]We give a connection between the coefficients of each of the polynomialsPn,Qm, andRsand certain hypergeometric functions, which leads to a simple expression forQmin the casen=s. The approximate location of the zeros ofQm, whenn⪢mandnequals;s, are deduced from the zeros of the classical Hermite polynomial. Contour integral representations ofPn,Qm,Rs, andEnmsare given and, using saddle point methods, we derive the exact asymptotics ofPn,Qm, andRsasn,m, andstend to infinity through certain ray sequences. We also discuss aspects of the more complicated uniform asymptotic methods for obtaining insight into the zero distribution of the polynomials, and we give an example showing the zeros of the polynomialsPn,Qm, andRsfor the casen=s=40,m=45
Jackson Theorems for Erdo&#x030B;s Weights inLp(0<p⩽∞)

AbstractAn Erdo&#x030B;s weight is of the formW≔e−QwhereQis even and of faster than polynomial growth at ∞. For example, we can takeQ(x)≔expk(|x|α),k⩾1,α>0,x∈R,where expkdenotes thekth iterated exponential. We prove Jackson theorems in weightedLpspaces with norm ‖fW‖Lp(R)for all 0<p⩽∞. These are the first proper Jackson theorems for Erdo&#x030B;s weights even inL∈. An interesting feature is a Timan– Nikolskii–Brudnyi effect: The degree of approximation improves towards the endpoints of a certain interval. By contrast, there is no such feature for Freud weights
Some Best Constants in the Landau Inequality on a Finite Interval

AbstractAn additive form of the Landau inequality forf∈Wn∞[−1,1],‖f(m)‖⩽1cm1−mnT(m)n(1)‖f‖+cn−m2n−1n!mnT(m)n(1)‖f(n)‖is proved for 0<c⩽(cos(π/2n))−2, 1⩽m⩽n−1, with equality forf(x)=Tn(1+(x−1)/c), 1⩽c⩽(cos(π/2n))−2, whereTnis the Chebyshev polynomial. From this follows a sharp multiplicative inequality,‖f(m)‖⩽(2n−1n!)−m/nT(m)n(1)‖f|1−m/n‖f(n)‖m/nfor ‖f(n)‖⩾σ‖f‖, 2n−1n!cos2n(π/2n)⩽σ⩽2n−1n!, 1⩽m⩽n−1. For these values ofσ, the result confirms Karlin's conjecture on the Landau inequality for intermediate derivatives on a finite interval. For the proof of the additive inequality a Duffin and Schaeffer-type inequality for polynomials is shown
On Rational Interpolation to |x| at the Adjusted Chebyshev Nodes

AbstractRecently Brutman and Passow considered Newman-type rational interpolation to |x| induced by arbitrary sets of symmetric nodes in [−1,1] and showed that under mild restrictions on the location of the interpolation nodes, the corresponding sequence of rational interpolants converges to |x|. They also studied the special case where the interpolation nodes are the roots of the Chebyshev polynomials and proved that for this case the exact order of approximation isO(1/nlogn), which, in view of Werner's result, is the same as for rational interpolation at equidistant nodes. In the present note we consider the set of interpolation nodes obtained by adjusting the Chebyshev roots to the interval [0,1] and then extending this set to [−1,1] in a symmetric way. We show that this procedure improves the quality of approximation, namely we prove that in this case the exact order of approximation isO(1/n2)
Inequalities for the Associated Legendre Functions

AbstractIn this paper bounds for the associated Legendre functions of the first kindPmn(x) for realx∈[−1,1] and integersm,nare proved. A relation is derived that allows us to generalize known bounds of the Legendre polynomialsPn(x)≡P0n(x) for the Legendre functionsPmn(x) of non-zero orderm. Furthermore, upper and lower bounds of the typeA(α,n,m)⩽maxx∈[−1,1]|(1−x2)α/2Pmn(x)|⩽B(α,,n,m) are proved for all 0⩽α⩽1/2 and 1⩽|m|⩽n. Forα=0 andα=1/2 these upper bounds are improvements and simplifications of known results
Perturbation of Orthogonal Polynomials on an Arc of the Unit Circle, II

AbstractOrthogonal polynomials on the unit circle are fully determined by their reflection coefficients through the Szegő recurrences. Assuming that the reflection coefficients converge to a complex numberawith 0<|a|<1, or, in addition, they form a sequence of bounded variation, we analyze the orthogonal polynomials by comparing them with orthogonal polynomials with constant reflection coefficients which were studied earlier by Ya. L. Geronimus and N. I. Akhiezer. In particular, we present asymptotic relations under certain assumptions on the rate of convergence of the reflection coefficients. Under weaker conditions we still obtain useful information about the orthogonal polynomials and also about the measure of orthogonality
Stability of the Shifts of a Finite Number of Functions

AbstractLetφ1 ,…,φnbe compactly supported distributions inLp(Rs) (0<p⩽∞). We say that the shifts ofφ1,…,φnareLp-stable if there exist two positive constantsC1andC2such that, for arbitrary sequences a1,…,an∈lp(Zs),C1∑k=1n‖aklp(Zs)⩽∑k=1n∑α∈Zsφk(·−α)ak(α)Lp(Rs)⩽C2∑k=1n‖aklp(Zs).In this paper we prove that the shifts ofφ1,…,φnareLp-stable if and only if, for anyξ∈Rs, the sequences (φk(ξ+2βπ))β∈Zs(k=1,…,n) are linearly independent, whereφdenotes the Fourier transform ofφ. This extends the previous results of Jia and Micchelli on a characterization ofLp-stability (1⩽p⩽∞) of the shifts of a finite number of compactly supported functions to the case 0<p⩽∞
Matrix Continued Fractions

AbstractA matrix continued fraction is defined and used for the approximation of a function F known as a power series in 1/zwith matrix coefficientsp×q, or equivalently by a matrix of functions holomorphic at infinity. It is a generalization of P-fractions, and the sequence of convergents converges to the given function. These convergents have as denominators a matrix, the columns of which are orthogonal with respect to the linear matrix functional associated to F. The case where the algorithm breaks off is characterized in terms of F
Weak Copositive and Intertwining Approximation

AbstractIt is known that shape preserving approximation has lower rates than unconstrained approximation. This is especially true for copositive and intertwining approximations. Forf∈Lp, 1⩽p<∞, the former only has rateωϕ(f,n−1)p, and the latter cannot even be bounded byC‖f‖p. In this paper, we discuss various ways to relax the restrictions in these approximations and conclude that the most sensible way is the so-calledalmostcopositive/intertwining approximation in which one relaxes the restriction on the approximants in a neighborhood of radiusΔn(yj) of each sign changeyj