pinecone/core-2020-05-10-deduplication huggingface dataset:
A Dual Approach to Constrained Interpolationfrom a Convex Subset of Hilbert Space

AbstractMany interesting and important problems of best approximationare included in (or can be reduced to) one of the followingtype: in a Hilbert spaceX, find the best approximationPK(x) to anyx∈Xfrom the setK≔C∩A−1(b),whereCis a closed convex subset ofX,Ais a bounded linearoperator fromXinto a finite-dimensional Hilbert spaceY, andb∈Y. The main point of this paper is to show thatPK(x)isidenticaltoPC(x+A*y)—the best approximationto a certain perturbationx+A*yofx—from the convexsetCor from a certain convex extremal subsetCbofC. Thelatter best approximation is generally much easier to computethan the former. Prior to this, the result had been known onlyin the case of a convex cone or forspecialdata sets associatedwith a closed convex set. In fact, we give anintrinsic characterizationof those pairs of setsCandA−1(b) for which this canalways be done. Finally, in many cases, the best approximationPC(x+A*y) can be obtained numerically from existingalgorithms or from modifications to existing algorithms. Wegive such an algorithm and prove its convergenc
Uniform Bounds for Sampling Expansions

AbstractLetf∈B2σ, i.e.,f∈L2(R) and its Fourier transformF(s)=∫Rf(t)e−2πistdtvanishes outside of [−σ, σ], then the Shannon sampling theorem says thatfcan be reconstructed by its infinitely many sampling points at {k/(2σ)},k∈Z, i.e.,f(t)=∑k=−∞∞fk2σsinπ(2σt−k)π(2σt−k),∀t∈R.But, in practice, only finitely many samples are available, so one would like to study the truncation errorTN(t)=f(t)−∑k=−NNfk2σsinc(2σt−k),∀f∈B2σ.The error bounds commonly seen in literature are not uniform. In this paper, the author gives uniform bounds for the truncation error forf∈B2σ, when its Fourier transform satisfies some smooth conditions
On Almost Interpolation

AbstractWe obtain some characterizations of almost interpolation configurations of points with respect to finite-dimensional functional spaces. Particularly, a Schoenberg–Whitney type characterization which is valid for any multivariate spline space relative to an arbitrary partition of a domainA⊂Rmis presented. As a closely related problem we investigate sectional structure of finite-dimensional spaces of real functions on a topological spaceA. It is shown that under some reasonable restrictions onAany space of this sort may be considered as piecewise almost Chebyshev
Zeros and Critical Points of Sobolev Orthogonal Polynomials

AbstractUsing potential theoretic methods we study the asymptotic distribution of zeros and critical points of Sobolev orthogonal polynomials, i.e., polynomials orthogonal with respect to an inner product involving derivatives. Under general assumptions it is shown that the critical points have a canonical asymptotic limit distribution supported on the real line. In certain cases the zeros themselves have the same asymptotic limit distribution, while in other cases we can only ascertain that the support of a limit distribution lies within a specified set in the complex plane. One of our tools, which is of independent interest, is a new result on zero distributions of asymptotically extremal polynomials. Our results are illustrated by numerical computations for the case of two disjoint intervals. We also describe the numerical methods that were used
Construction Techniques for Highly Accurate Quasi-Interpolation Operators

AbstractUnder mild additional assumptions this paper constructs quasi-interpolants in the form fh(x)=∑j=−∞+∞f(hj)ϕh(xh−j),x∈R,h>0, ((0.1))with approximation order ℓ−1, whereϕh(x) is a linear combination of translatesψ(x−jh) of a functionψinCℓ(R). Thus the order of convergence of such operators can be pushed up to a limit that only depends on the smoothness of the functionψ. This approach can be generalized to the multivariate setting by using discrete convolutions with tensor products of odd-degreeB-splines
Chain Sequences, Orthogonal Polynomials, and Jacobi Matrices

AbstractChain sequences are positive sequences {an} of the forman=gn(1−gn−1) for a nonnegative sequence {gn}. This concept was introduced by Wall in connection with continued fractions. In his monograph on orthogonal polynomials, Chihara conjectured that ifan⩾14 for eachnthen ∑(an−14)⩽14. We prove this conjecture and give other precise estimates foran. We also characterize the chain sequences {an} whose terms are greater than 14. We show connections to Jacobi matrices and orthogonal polynomials. In particular, we characterize the maximal chain sequences in terms of integrability properties of the spectral measure of the associated Jacobi matrix
On the Monotonicity of Positive Linear Operators

AbstractWe provide sufficient conditions for a sequence of positive linear approximation operators,Ln(f,x), converging tof(x) from above to imply the convexity off. We show that, for the convolution operators of Feller type,Kn(f,x), generated by a sequence of iid random variables taking values in an intervalI, having a finite moment generating function, the inequalitiesKn(f,x)⩾f(x) (x∈I,n⩾1) are necessary and sufficient conditions forfto be convex. This provides a converse of a well-known result of R. A. Khan (Acta. Math. Acad. Sci. Hungar.39(1980), 193–203). It contains, as a special case, the corresponding result for the Bernstein polynomials and extends two results obtained for bounded continuous functions by Horova for Szász and Baskakov operators. As examples, similar results are also provided for the beta, Meyer-König Zeller, Picard, and Bleiman, Butzer, and Hahn operators
On the Polynomials Orthogonal on Regular Polygons

AbstractThe two-parameter Pastro–Al-Salam–Ismail (PASI) polynomials are known to be bi-orthogonal on the unit circle with continuous weight function when 0<q<1. We study the case ofqa root of unity. It is shown that corresponding PASI polynomials are orthogonal on the unit circle with discrete measure located on the vertices of the regularN-gon. Cases leading to a positive weight function are analyzed. In particular, we obtain trigonometric analogs of the Askey–Szego&#x030B; polynomials which are orthogonal on regularN-gons with positive weight function
Approximation by Extreme Functions

AbstractForTa topological space andXa real normed space,Y=C(T,X) denotes the space of continuous and bounded functions fromTintoXendowed with the sup norm. We calculate a formula for the distanceα(f) fromfinYto the setY−1of functions inYwhich have no zeros. Namely, we prove thatα(f) is the infimum of numbersδ>0 for which the continuous functiont↦f(t)/‖f(t)‖ defined for everytwith ‖f(t)‖⩾δhas a continuous extensionefromTinto the unit sphere ofX. This permits us to get the general expression of the Aron–Lohmanλ-function ofYwhenXis strictly convex. We show that any function inYhas a best approximation inY−1which can be chosen to have the least possible norm. IfXis strictly convex andE(Y) denotes the set of extreme points of the unit ball ofY, this fact enables us to prove that dist(f,E(Y))=max{1−m(f)+α(f), ‖f‖−1} ∀f∈Y, wherem(f)=inf {‖f(t)‖:t∈T}. Moreover, we show thatE(Y) is proximinal inY−1and give sufficient conditions under whichfinY\Y−1admits a best approximation inE(Y)
Multiply Universal Holomorphic Functions

AbstractLet O⊂C, O≠C, be an open set with simply connected components. In Theorem 1 we prove the existence of a holomorphic functionφon O, which has together with all its derivatives and all its antiderivatives six universal properties at the same time (based on the behaviour of sequences of derivatives or antiderivatives, overconvergence-phenomena, or properties of translates). In Theorem 2 we show that the family of all functions with these universal properties is a dense subset of the metric spaceH(O) of all holomorphic functions on O, ifH(O) is endowed with the usual compact-open topology
Locally Supported Kernels for Spherical Spline Interpolation

AbstractBy the use of locally supported basis functions for spherical spline interpolation the applicability of this approximation method is extended, since the resulting interpolation matrix is sparse and thus efficient solvers can be used. In this paper we study locally supported kernels in detail. Investigations on the Legendre coefficients allow a characterization of the underlying Hilbert space structure. We show how spherical spline interpolation with polynomial precision can be managed with locally supported kernels, thus giving the possibility to combine approximation techniques based on spherical harmonic expansions with those based on locally supported kernels
Some Positive Results and Counterexamples in Comonotone Approximation

AbstractLetfbe a continuous function on [−1, 1], which changes its monotonicity finitely many times in the interval, saystimes. We discuss the validity of Jackson-type estimates for the approximation offby algebraic polynomials that are comonotone with it. While we prove the validity of the Jackson-type estimate involving the Ditzian–Totik modulus of continuity and a constant which depends only ons, we show by counterexamples that in many cases this is not so, even for functions which possess locally absolutely continuous derivatives. These counterexamples are given when there are certain relations betweens, the number of changes of monotonicity, andr, the number of derivatives. For other cases we do have some Jackson-type estimates and another paper will be devoted to that
Best Approximation by Periodic Smooth Functions

AbstractLetWnbe the set of 2π-periodic functions with absolutely continuous (n−1)th derivatives and nth derivatives with essential suprema bounded by one. Letn>1. Best uniform approximations to a periodic continuous function fromWnare characterized. The result depends upon an analysis of the relation between the zeros, knots, and signs of periodic splines with simple knots. An appendix by O. V. Davydov states an alternative characterisation and demonstrates that the two characterisations are equivalent
Some Applications of the Robin Function to Multivariable Approximation Theory

AbstractLetE⊂Cnbe compact, regular, and polynomially convex with pluricomplex Green functionVE. Given a sequence of polynomials {pj}j=1,2,…, the first result is a condition for (limj→∞(log|pj(z)|/deg(pj)))* to equalVEon Cn−E. The condition involves the Robin function ofEand the highest order homogeneous terms of thepjand generalizes one-variable results of Blatt–Saff. A second result gives a necessary and sufficient condition forf, the uniform limit of polynomials onE, to extend holomorphically toER={z|VE(z)<logR} forR>1. The condition involves highest order homogeneous terms of best approximating polynomials tofand the Robin function ofEand extends results of Szczepański
Extremal Solutions of the Two-DimensionalL-Problem of Moments, II

AbstractThe extremal solutions of the truncatedL-problem of moments in two real variables, with support included in a given compact set, are described as characteristic functions of semi-algebraic sets given by a single polynomial inequality. An exponential kernel, arising as the determinantal function of a naturally associated hyponormal operator with rank-one self-commutator, on the other hand provides a defining function for these semi-algebraic sets and, on the other hand, encodes in a closed form their moments. In order to understand the finite determination structure of the extremal sequences of moments we study analytic continuation properties of the corresponding exponential kernel and, separately, some cyclicity properties of the associated hyponormal operator. An intrinsic characterization of the exponential kernel is also discussed
On Power Functions and Error Estimates for Radial Basis Function Interpolation

AbstractThis paper discusses approximation errors for interpolation in a variational setting which may be obtained from the analysis given by Golomb and Weinberger. We show how this analysis may be used to derive the power function estimate of the error as introduced by Schaback and Powell. A simple error tool for the power function is presented, which is similar to one appearing in the work of Madych and Nelson. It is then shown that this tool is adequate to reproducing the original error analysis presented by Duchon. An interesting consequence of our work is that no explicit use is made of the polynomial reproduction properties of the interpolation operator
Rational Approximation with Locally Geometric Rates

AbstractWe investigate the rate of pointwise rational approximation of functions from two classes. The distinguishing feature of these classes is the essentially faster convergence of the best uniform rational approximants versus best uniform polynomial approximants. It is known that for piecewise analytic functions “near best” polynomials converging geometrically fast at every point of analyticity of the function exist. Here we construct rational approximants enjoying similar properties. We also show that our construction yields rates of convergence that are, in a certain sense, best possible
Determination of the Jumps of a Bounded Function by Its Fourier Series

AbstractThe well-known identity which determines the jumps of a function of bounded variation by its Fourier series is extended to larger classes of functions, such asVΦ,ΛBV, andV[v], under some conditions on the generalized variations. It is shown as well that the conditions on the generalized variations are definitive in some sense. Based on the above-mentioned results, an identity which determines the jumps of a bounded function by its Fourier series with respect to the system of generalized Jacobi polynomials is obtained for these function classes
Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle

29 pages, no figures.-- MSC2000 codes: 42C05, 30E10, 30E15.MR#: MR1604927 (99c:42041)Zbl#: Zbl 0897.42016Ratio and relative asymptotics are given for sequences of polynomials orthogonal with respect to measures supported on an arc of the unit circle, where their absolutely continuous component is positive almost everywhere. The results obtained extend to this setting known ones given by Rakhmanov and Máté, Nevai, and Totik for the case when the arc is the whole unit circle. Technically speaking, the main feature is the use of orthogonality with respect to varying measures.This research was partially supported by the Third World Academy of Science under Grant 93-277 RG/Maths/La.Publicad
Rational Approximation with Varying Weights, II

AbstractWe consider two problems concerning uniform approximation by weighted rational functions {wnrn}∞n=1, wherern=pn/qnhas real coefficients, degpn⩽[αn] and degqn⩽[βn], for givenα>0 andβ⩾0. Forw(x):=exwe show that on any interval [0, a] witha∈(0, â(α, β)), every real-valued functionf∈C([0, a]) is the uniform limit of some sequence {wnrn}. An implicit formula forâ(α, β) was given in the first part of this series of papers; in particular,â(1, 1)=2π. Forw(x):=xθwithθ>1 we show that uniform approximation of real-valuedf∈C([b, 1]) on [b, 1] by weighted rationalswnrnis possible for anyb∈(b(θ; α, β),1), whereb(θ; α, β) was also found in Part I; in particular,b(θ; 1, 1)=tan4((π/4)((θ−1)/θ)). Both of the mentioned results are sharp in the sense that approximation is no longer possible ifâis replaced byâ+εorbis replaced byb−εwithε>0. We use potential theoretic methods to prove our theorems
Best Approximation of Functions like |x|λexp(−A|x|−α)

AbstractWe determine the exact order of best approximation by polynomials and entire functions of exponential type of functions likeϕλ, α(x)=|x|λexp(−A|x|−α). In particular, it is shown thatE(ϕλ, α, Pn, Lp(−1, 1))∼n−(2λp+αp+2)/2p(1+α)×exp(−(1+α−1)(Aα)1/(1+α)cosαπ/2(1+α)nα/(1+α)), whereE(ϕλ, α, Pn, Lp(−1, 1)) denotes best polynomial approximation ofϕλ, αinLp(−1, 1),λ∈R,α∈(0, 2],A>0, 1⩽p⩽∞. The problem, concerning the exact order of decrease ofE(ϕ0, 2, Pn, L∞(−1, 1)), has been posed by S. N. Bernstein
Shape Preserving Properties of Some Positive Linear Operators on Unbounded Intervals

AbstractIn this paper we study some shape preserving properties of particular positive linear operators acting on spaces of continuous functions defined on the interval [0, +∞[, which are strongly related to the semigroups generated by a large class of degenerate elliptic second order differential operators. We study the conditions under which these operators leave invariant the class of increasing functions, as well as the class of convex functions and Hölder continuous functions. As a consequence, we derive some regularity results concerning the related semigroups
On the Approximation of Positive Functions by Power Series, II

AbstractWe consider positive functionsh=h(x) defined forx∈R+0. Conditions for the existence of a power seriesN(x)=∑cnxn,cn⩾0, with the propertyd1⩽h(x)/N(x)⩽d2,x⩾0,for some constantsd1, d2∈R+, are investigated in [J. Clunie and T. Kövari,Canad. J. Math.20(1968), 7–20; P. Erdős and T. Kövari,Acta Math. Acad. Sci. Hung.7(1956), 305–316; U. Schmid,Complex Variables18(1992), 187–192; U. Schmid, J.Approx. Theory83(1995), 342–346]. In this paper, methods are discussed which allow for a given functionhthe construction of the coefficientscn,n∈N0, for the above defined power seriesNand to find suitable constantsd1andd2. We also study the power seriesH(x)=∑xn/un, where we setun=sup{xn/h(x), x⩾0}, forn∈N0, and the relation betweenhandHconcerning the above stated inequalities
Error Estimates for Interpolation by Compactly Supported Radial Basis Functions of Minimal Degree

AbstractWe consider error estimates for interpolation by a special class of compactly supported radial basis functions. These functions consist of a univariate polynomial within their support and are of minimal degree depending on space dimension and smoothness. Their associated “native” Hilbert spaces are shown to be norm-equivalent to Sobolev spaces. Thus we can derive approximation orders for functions from Sobolev spaces which are comparable to those of thin-plate-spline interpolation. Finally, we investigate the numerical stability of the interpolation process
Coefficients of Polynomials of Restricted Growth on the Real Line

AbstractLetφ:(−∞, ∞)→(0, ∞) be a given continuous even function and letmbe a positive integer. We show that, with some additional restrictions onφ, there exist decreasing sequencesx1, …, xmandy1, …, ym−1of symmetrically located points on (−∞, ∞) and corresponding polynomialsPandQof degreesm−1 andm, respectively, satisfying|P(x)|⩽φ(x)m,|Q(x)|⩽φ(x)m,−∞<x<∞,where equality holds with alternating signs at the corresponding sequence of points (and also at ±∞ forQ). Moreover, for any polynomialpof degree at mostm, (a)if |p(xj)|⩽φ(xj)mforj=1, …, m, then |p(k)(0)|⩽|P(k)(0)| wheneverkandmhave opposite parity and 0⩽k<m; (b)if |p(yj)|⩽φ(yj)mforj=1, …, m−1 and if limsupy→∞|p(y)|/φ(y)m⩽1, then |p(k)(0)|⩽|Q(k)(0)| wheneverkandmhave the same parity and 0⩽k⩽m. We give two computational methods for determining these sequences of points and thusPandQ
Connections between the Support and Linear Independence of Refinable Distributions

AbstractThe purpose of this paper is to study the relationships between the support of a refinable distributionφand the global and local linear independence of the integer translates ofφ. It has been shown elsewhere that a compactly supported distributionφhas globally independent integer translates if and only ifφhas minimal convex support. However, such a distribution may have “holes” in its support. By insisting thatφ∈L2(R) and generates a multiresolution analysis, Lemarié and Malgouyres have ensured that no such holes can occur. In this article we generalize this result to refinable distributions. We also give a result on the local linear independence of the integer translates ofφ. We work with integer dilation factorN⩾2 throughout this paper
Lattices and Algorithms for Bivariate Bernstein, Lagrange, Newton, and Other Related Polynomial Bases Based on Duality betweenL-Bases andB-Bases

AbstractL-Bases andB-bases are two important classes of polynomial bases used for representing surfaces in approximation theory and computer aided geometric design. It is well known that the Bernstein and multinomial (or Taylor) bases are special cases of bothL-bases andB-bases. We establish that certain proper subclasses of bivariate Lagrange and Newton bases areL-bases. Furthermore, we present a rich collection of lattices (or point-line configurations) that admit unique Lagrange or Hermite interpolation problems which can be solved quite naturally in terms of Lagrange and NewtonL-bases. A new geometric point-line duality betweenL-bases andB-bases is described: lines inL-bases correspond to points or vectors inB-bases and concurrent lines map to collinear points and vice versa. This duality betweenL-bases andB-bases is then used to establish that certain proper subclasses of power bases areB-bases and are dual to LagrangeL-bases. This geometric duality is further used to describe the lattices that admit powerB-bases.B-bases dual to NewtonL-bases are also investigated. Duality can also be used to develop change of basis algorithms with computational complexityO(n3) between any twoL-bases and/orB-bases. We describe, in particular, a new change of basis algorithm from a bivariate LagrangeL-basis to a bivariate Bernstein basis with computational complexityO(n3)
Proximinality inLp(μ, X)

AbstractLetXbe a Banach space and letYbe a closed subspace ofX. Let 1⩽p⩽∞ and let us denote byLp(μ, X) the Banach space of allX-valued Bochnerp-integrable (essentially bounded forp=∞) functions on a certain positive completeσ-finite measure space (Ω, Σ, μ), endowed with the usualp-norm. In this paper we give a negative answer to the following question: “IfYis proximinal inX, isLp(μ, Y) proximinal inLp(μ, X)?” We also show that the answer is affirmative for separable spacesY. Some consequences of this are obtained
Accuracy of Lattice Translates of Several Multidimensional Refinable Functions

AbstractComplex-valued functionsf1,…,fronRdarerefinableif they are linear combinations of finitely many of the rescaled and translated functionsfi(Ax−k), where the translateskare taken along a latticeΓ⊂RdandAis adilation matrixthat expansively mapsΓinto itself. Refinable functions satisfy arefinement equationf(x)=∑k∈Λckf(Ax−k), whereΛis a finite subset ofΓ, theckarer×rmatrices, andf(x)=(f1(x),…,fr(x))T. Theaccuracyoffis the highest degreepsuch that all multivariate polynomialsqwith degree(q)<pare exactly reproduced from linear combinations of translates off1,…,fralong the latticeΓ. In this paper, we determine the accuracypfrom the matricesck. Moreover, we determine explicitly the coefficientsyα,i(k) such thatxα=∑ri=1∑k∈Γyα,i(k)fi(x+k). These coefficients are multivariate polynomialsyα,i(x) of degree |α| evaluated at lattice pointsk∈1
Generalized Multivariate Padé Approximants

AbstractA new definition of multivariate Padé approximation is introduced, which is a natural generalization of the univariate Padé approximation and consists in replacing the exact interpolation problem by a least squares interpolation. This new definition allows a straightforward extension of the Montessus de Ballore theorem to the multivariate case. Except for the particular case of the so-called homogeneous Padé approximants, this extension has up to now been impossible to obtain in the classical formulation of the multivariate Padé approximation. Besides, the least squares formulation can also be applied to the univariate case, and provides an alternative to the classical Padé interpolation