pinecone/core-2020-05-10-deduplication huggingface dataset:
Upward Extension of the Jacobi Matrix for Orthogonal Polynomials

AbstractOrthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrixrnew rows and columns, so that the original Jacobi matrix is shifted downward. Thernew rows and columns contain 2rnew parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials, and the 2rnew parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the 1?orthogonalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials
Multiquadric B-splines

AbstractOur purpose in this paper is to show that the analogy between polynomial splines and generalized multiquadrics is very strong. In particular, combinations of multiquadrics, calledψ-splines are defined that are analogues of polynomial B-splines. This paper includes global linear independence, polynomial reproduction, and quasi-interpolation results for the span of theψ-splines on non-uniform bi-infinite meshes which parallel those for polynomial B-splines. There are also results concerning the relationship between certain semi-infinite and bi-infinite combinations ofψ-splines. These results enable us to obtain error estimates for quasi-interpolation schemes involving multiquadrics based on a finite number of centres
On Shifted Cardinal Interpolation by Gaussians and Multiquadrics

AbstractA radial basis function approximation is a linear combination of translates of a fixed functionϕ: Rd→R. Such functions possess many useful and interesting properties when the translates are integers andϕis radially symmetric. We study the closely related problem for which the fixed function is the shifted Gaussianϕ=G(·−α), whereG(x)=exp(−λ‖x‖22) andα∈Rd. Specifically, we exploit the theory of elliptic functions to establish the invertibility of the Toeplitz operator[formula]whenαhas no half-integer components; it is singular otherwise. This implies the existence of ashifted Gaussian cardinal function, that is, a linear combinationχof integer translates of the shifted Gaussian satisfyingχ(j)=δ0j. We also study shifted cardinal functions when the parameterλtends to zero. In particular, we discover their uniform convergence to the sinc function when the shift vectorαpossesses no half-integer components. Our methods are based in part on similar results established by the first author when the basis function is the Hardy multiquadric. Several intriguing links with the theory of shifted B-spline cardinal interpolation are described in the finale
Orthogonal Polynomials on Arcs of the Unit Circle II. Orthogonal Polynomials with Periodic Reflection Coefficients

AbstractFirst we give necessary and sufficient conditions on a set of intervalsEl=∪lj=1[ϕ2j−1, 2j], 1<…<2l and 2l−ϕ1⩽2π, such that onElthere exists a real trigonometric polynomialτN(ϕ) with maximal number, i.e.,N+l, of extremal points onEl. The associated algebraic polynomialTN(z)=zN/2τN(z),z=eiϕ, is called the complex Chebyshev polynomial. Then it is shown that polynomials orthogonal onElhave periodic reflection coefficients if and only if they are orthogonal onElwith respect to a measure of the form[formula]certain point measures, whereAis a real trigonometric polynomial with no zeros onEland there exists a complex Chebyshev polynomial onEl. Let us point out in this connection that Geronimus has shown that orthogonal polynomials generated by periodic reflection coefficients of absolute value less than 1 are orthogonal with respect to a measure of the above type. Furthermore, we derive explicit representations of the corresponding orthogonal polynomials with the help of the complex Chebyshev polynomials. Finally, we provide a characterization of those definite functionals to which orthogonal polynomials with periodic reflection coefficients of modulus unequal to one are orthogonal
An Example in Tangential Meromorphic Approximation

AbstractIt is known that all sets of meromorphic uniform approximation inCwhich satisfy an additional condition involving the Gleason parts of the algebraR(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, although it is a set of tangential approximation, does not satisfy this extra condition on parts and, thus, showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem
Approximation Order of Bivariate Spline Interpolation

AbstractIn [G. Nürnberger and Th. Riessinger,Numer. Math.71(1995), 91–119], we developed an algorithm for constructing point sets at which unique Lagrange interpolation by spaces of bivariate splines of arbitrary degree and smoothness on uniform type triangulations is possible. Here, we show that similar Hermite interpolation sets yield (nearly) optimal approximation order. This is shown for differentiable splines of degree at least four defined on non-rectangular domains subdivided in uniform type triangles. Therefore, in practice we use Lagrange configurations which are “close” to these Hermite configurations. Applications to data fitting problems and numerical examples are given
Strictly Positive Definite Functions

AbstractWe give a complete characterization of the strictly positive definite functions on the real line. By Bochner's theorem, this is equivalent to proving that if the separated sequence of real numbers {an} describes the points of discontinuity of a distribution function, there exists an almost periodic polynomial with the zeros {an}. We prove a useful necessary condition that every strictly normalized, positive definite functionfsatisfies |f(x)|<1 for allx≠0. It is a sufficient condition for strictly positive definiteness that if the carrier of a nonzero finite Borel measure on R is not a discrete set, then the Fourier–Stieltjes transformμofμis strictly positive definite
Completeness of Integer Translates in Function Spaces on R

AbstractWe show that each of the Banach spacesC0(R) andLp(R), 2<p<∞, contains a function whose integer translates are complete. This function can also be chosen so that one of the following additional conditions hold: (1) Its non-negative integer translates are already complete. (2) Its integer translates form an orthonormal system inL2(R). (3) Its integer translates form a minimal system. A similar result holds for the corresponding Sobolev space, for certain weightedL2spaces, and in the multivariate setting. We also prove some results in the opposite direction
Riesz Bases of Splines and Regularized Splines with Multiple Knots

AbstractThis paper deals withL2(R)-norm and Sobolev-norm stability of polynomial splines with multiple knots, and with regularized versions thereof. An essential ingredient is a result on Hölder continuity of the shift operator operating on a B-spline series. The stability estimates can be reformulated in terms of a Riesz basis property for the underlying spline spaces. These can also be employed to derive a result on stable Hermite interpolation on the real line. We point to the connection with the problem of symmetric preconditioning of bi-infinite interpolation matrices
Strong and Weak Weighted Convergence of Jacobi Series

AbstractGivenα, β>−1, letpn(x)=p(α, β)n(x),n=0, 1, 2,… be the sequence of Jacobi polynomials orthonormal on (−1, 1) with respect to the weightu(x)=(1−x)α (1+x)β. Denote by (SNf)(x) theNth partial sum of the Fourier–Jacobi series of the functionfon (−1, 1), so that (SNf)(x)=∑Nn=0anpn(x), withan=∫1−1f(x)pn(x)u(x)dx. For fixedp∈(1, ∞), we characterize the weightswsuch that limN→∞∫1−1|[(SNf)(x)−f(x)]w(x)|pu(x)dx=0 whenever ∫1−1|f(x)w(x)|pdx<∞, the weightswsuch that limN→∞supλ>0λ[∫ENλw(x)pu(x)dx]1/p=0 whenever ∫1−1|f(x)w(x)|pdx<∞, and the weightswsuch that limN→∞supλ>0λ[∫ENλw(x)pu(x)dx]1/p=0 whenever ∫∞0[∫Fλw(x)dx]1/pdλ<∞; here,ENλ={x∈(−1, 1) ;|(SNf(x)−f(x)|>λ} andFλ={x∈(−1, 1):|f(x)|>λ}
On the Computation of Wavelet Coefficients

AbstractWe consider fast algorithms of wavelet decomposition of a functionfwhen discrete observations off(suppf⊆[0, 1]d) are available. The properties of the algorithms are studied for three types of observation design which ford=1 can be described as follows: the regular design, when the observationsf(xi) are taken on the regular gridxi=i/N,i=1, …, N; the case of a jittered regular grid, when it is only known that for all 1⩽i⩽N,i/N⩽xi<i+1)/N; and the random design case; in whichxi,i=1, …, N, are independent and identically distributed random variables on [0, 1]. We show that these algorithms are in a certain sense efficient when the accuracy of the approximation is concerned. The proposed algorithms are computationally straightforward: the whole effort to compute the decomposition is orderNfor the sample sizeN
Convergence of Derivatives of Optimal Nodal Splines

AbstractOptimal nodal spline interpolantsWfof ordermwhich have local support can be used to interpolate a continuous functionfat a set of mesh points. These splines belong to a spline space with simple knots at the mesh points as well as atm−2 arbitrary points between any two mesh points and they reproduce polynomials of orderm. It has been shown that, for a sequence of locally uniform meshes, these splines converge uniformly for anyf∈Cas the mesh norm tends to zero. In this paper, we derive a set of sufficient conditions on the sequence of meshes for the uniform convergence ofDjWftoDjfforf∈Csandj=1, …, s<m. In addition we give a bound forDrWfwiths<r<m. Finally, we use optimal nodal spline interpolants for the numerical evaluation of Cauchy principal value integrals
Erdős–Turán-Type Theorems on Piecewise Smooth Curves and Arcs

AbstractIfLis a Jordan curve or a Jordan arc andpnis a monic polynomial of degreenwe obtain estimates for the discrepancy between the equilibrium measureμLofLand the distributionνpnof the zeros ofpnbased on one-sided bounds for the differenceU(μL−νpn, z) of their logarithmic potentials. These new estimates generalize known results to the case thatLis not smooth, i.e., corners ofLare allowed, but cusps are not. Moreover, the results are independent of the angles at the corners. The method of proof shows that both situations—upperorlower bounds ofU(μL−νpn, z)—can be treated simultaneously. As an application, the distribution of Fekete points and extremal points of best uniform approximants can be investigated generalizing results of Kleiner [14] and Blatt and Grothmann [6] to Jordan curves and arcs with corners
Continuities of Metric Projection and GeometricConsequences

AbstractWe discuss the geometric characterization of a subsetKof a normed linear space via continuity conditions on the metricprojection ontoK. The geometric properties considered includeconvexity, tubularity, and polyhedral structure. The continuityconditions utilized include semicontinuity, generalized stronguniqueness and the non-triviality of the derived mapping. Infinite-dimensional space with the uniform norm we show thatconvexity is equivalent to rotation-invariant almost convexityand we characterize those sets every rotation of which has continuousmetric projection. We show that polyhedral structure underliesgeneralized strong uniqueness of the metric projection
A Class of Extremal Functions and Trigonometric Polynomials

AbstractIn this paper we present two classes of extremal approximating functions. These functions have the property that they are entire, have finite exponential type, and provide excellent approximations along the real line for a specific set of functions. One class of functions provides majorants and minorants, while the other class minimizes theL1-norm on the real line. As applications we construct extremal trigonometric polynomials and obtain an inequality involving almost periodic trigonometric polynomials
Cesàro Summability of Two-Parameter Walsh–Fourier Series

AbstractA new atomic decomposition of the two-parameter dyadic martingale Hardy spacesHpdefined by the quadratic variation is given. We introduceHp-quasi-local operators and prove that if a sublinear operatorVisHp-quasi-local and bounded fromL2toL2then it is also bounded fromHptoLp(0<p⩽1). By an interpolation theorem we get thatVis of weak type (H#1, L1) where the Hardy spaceH#1is defined by the hybrid maximal function. As an application it is shown that the maximal operator of the Cesàro means of a two-parameter martingale is bounded fromHptoLp(4/5<p⩽∞) and is of weak type (H#1, L1). So we obtain that the Cesàro means of a functionf∈H#1converge a.e. to the function in question. Finally, it is verified that if the supremum is taken over all two-powers, only, then the maximal operator of the Cesàro means is bounded fromHptoLpfor every 2/3<p⩽∞
On the Measure of Approximation for Some Linear Means of Trigonometric Fourier Series

AbstractMany approximation methods inC2πmay be generated via a certain functionϕ∈C[0, 1]withϕ(0)=1,ϕ(1)=0. The functionϕj(t)=cos(j−1/2)πt(j∈N) generates the Rogosinski approximation method [N. K. Bari, “A Treatise on Trigonometric Series,” Vols. I, II, Pergamon Press, New York, 1964]. Our idea consists in representingϕby the orthogonal systemϕjto extend results previously known for the Rogosinski method to arbitrary approximation methods. We illustrate this by proving two asymptotic estimates for the measure of approximation
Orthant–Monotonic Norms and Overdetermined Linear Systems

AbstractIn the paper the properties of ‖·‖-approximate solutions of real overdetermined linear systemsAx=bare investigated. We characterize the norms for which the approximate solutions are in the convex hull of the basic solutions ofAx=b. For this purpose we prove some properties of the orthant–monotonic norms. We derive an explicit formula which expresses the approximate solutions ofAx=b, with respect to some norms, as convex combinations of the basic solutions for ann+1 bynreal matrixAof rank n. Moreover, we consider the relation between the sets of the weighted approximate solutions for the orthant–monotonic norms satisfying some additional conditions
Asymptotic Behaviour of Orthogonal Polynomials on the Unit Circle with Asymptotically Periodic Reflection Coefficients

AbstractLet {an}n∈N0withan∈C,an+N=anand |an|<1 for alln∈N0, be a periodic sequence of reflection coefficients and let {Pn}n∈N0be the associated sequence of orthogonal polynomials generated byPn+1=zPn−ānP*n. Furthermore let {bn}n∈N0be an asymptotically periodic sequence of reflection coefficients which arises by a perturbation of the sequence {an}n∈N0and thus satisfies the conditions limν→∞bj+νN=ajforj=0, …, N−1, and |bn|<1 for alln∈N0. Let {Pn}n∈N0generated byPn+1=zPn−b&nP*nbe the disturbed orthogonal polynomials. Using the “periodic” polynomials {Pn}n∈N0as a comparison system we derive so-called comparative asymptotics for the disturbed polynomials on and off the support of the disturbed orthogonality measure, which consists essentially of several arcs of the unit circle. As a by-product of these results we obtain asymptotically a description of the location of the zeros of {Pn}n∈N0. Finally, a representation for the absolutely continuous part of the disturbed orthogonality measure is derived, and it is shown that there are at most finitely many point measures if thebn's converge geometrically fast to thean's
Asymptotic Zero Distribution of Laurent-Type Rational Functions

AbstractWe study convergence and asymptotic zero distribution of sequences of rational functions with fixed location of poles that approximate an analytic function in a multiply connected domain. Although the study of zero distributions of polynomials has a long history, analogous results for truncations of Laurent series have been obtained only recently by Edrei (Michigan Math. J.29(1982), 43–57). We obtain extensions of Edrei's results for more general sequences of Laurent-type rational functions. It turns out that the limiting measure describing zero distributions is a linear convex combination of the harmonic measures at the poles of rational functions, which arises as the solution to a minimum weighted energy problem for a special weight. Applications of these results include the asymptotic zero distribution of the best approximants to analytic functions in multiply connected domains, Faber–Laurent polynomials, Laurent–Padé approximants, trigonometric polynomials, etc
A Criterion for the Nonuniqueness of the Measure of Orthogonality

AbstractLet the polynomialsPn(x),n⩾1, be defned byP0(x)=0,P1(x)=1,anPn+1(x)+an−1Pn−1(x)+bnPn(x)=xPn(x),n⩾1. Ifan>0 andbnare real then there exists at least one measure of orthogonality for the polynomialsPn(x),n=1, 2, ... . The problem of finding conditions on the sequencesanandbnunder which this measure is unique or nonunique still remains open for large classes of sequencesanandbn. Here a new criterion for the nonuniqueness of the measure of orthogonality is proved. This was achieived by proving that the infinite-dimensional Jacobi matrix associated with the sequencesanandbnis not self-adjoint
Behaviour of Exponential Splines as Tensions Increase without Bound

AbstractSchweikert (J. Math. Phys.45(1966), 312–317) showed that for sufficiently high tensions an exponential spline would have no more changes in sign of its second derivative than there were changes in the sign of successive second differences of its knot sequence. Späth (Computing4(1969), 225–233) proved the analogous result for first derivatives, assuming uniform tension throughout the spline. Later, Pruess (J. Approx. Theory17(1976), 86–96) extended Späth's result to the case where the inter-knot tensionspimay not all be the same but tend to infinity at the same asymptotic growth rate, in the sense thatpi∈Θ(p1) for alli. This paper extends Pruess's result by showing his hypothesis of uniform boundedness of the tensions to be unnecessary. A corollary is the fact that for high enough minimum interknot tension, the exponential spline through monotone knots will be a C2monotone curve. In addition, qualitative bounds on the difference in slopes between the interpolating polygon and the exponential spline are developed, which show that Gibbs-like behaviour of the spline's derivative cannot occur in the neighbourhood of the knots
Orthogonal Rational Functions and Nested Disks

AbstractIn Akhiezer's book [“The Classical Moment Problem and Some Related Questions in Analysis,” Oliver & Boyd, Edinburghasol;London, 1965] the uniqueness of the solution of the Hamburger moment problem, if a solution exists, is related to a theory of nested disks in the complex plane. The purpose of the present paper is to develop a similar nested disk theory for a moment problem that arises in the study of certain orthogonal rational functions. Let {αn}∞n=0be a sequence in the open unit disk in the complex plane, letB0=1andBn(z)=∏k=0nαk|αk|αk−z1−αkz,n=1,2,…,(αk/|αk|=−1 whenαk=0), and letL=span{B:n=0,1,2,…}.We consider the following “moment” problem: Given a positive-definite Hermitian inner product ⦠·,·⦔ on L×L, find a non-decreasing functionμon [−π,π] (or a positive Borel measureμon [−π,π)) such that⦠f,g⦔=∫π−πf(eiθ)g(eiθ)dμ(θ) forf,g∈L.In particular we give necessary and sufficient conditions for the uniqueness of the solution in the case that∑n=1∞(1−|αn|)<∞.If this series diverges the solution is always unique
Best Approximation and Cyclic Variation Diminishing Kernels

AbstractWe study best uniform approximation of periodic functions from∫2π0K(x,y)h(y)dy:|h(y)|⩽1,where the kernelK(x,y) is strictly cyclic variation diminishing, and related problems including periodic generalized perfect splines. For various approximation problems of this type, we show the uniqueness of the best approximation and characterize the best approximation by extremal properties of the error function. The results are proved by using a characterization of best approximants from quasi-Chebyshev spaces and certain perturbation results
On the Convergence of Polynomial Approximation of Rational Functions

AbstractThis paper investigates the convergence condition for the polynomial approximation of rational functions and rational curves. The main result, based on a hybrid expression of rational functions (or curves), is that two-point Hermite interpolation converges if all eigenvalue moduli of a certainr×rmatrix are less than 2, whereris the degree of the rational function (or curve), and where the elements of the matrix are expressions involving only the denominator polynomial coefficients (weights) of the rational function (or curve). As a corollary for the special case ofr=1, a necessary and sufficient condition for convergence is also obtained which only involves the roots of the denominator of the rational function and which is shown to be superior to the condition obtained by the traditional remainder theory for polynomial interpolation. For the low degree cases (r=1, 2, and 3), concrete conditions are derived. Application to rational Bernstein–Bézier curves is discussed
A Refineable Space of Smooth Spline Surfaces of Arbitrary Topological Genus

AbstractIt is shown that parametrical smoothness conditions are sufficient for modeling smooth spline surfaces of arbitrary topology if degenerate surface segments are accepted. In general, degeneracy, i.e., vanishing partial derivatives at extraordinary points, is leading to surfaces with geometrical singularities. However, if the partial derivatives of higher order satisfy certain conditions, the existence of a regular smooth reparametrization can be guaranteed. So, degeneracy is no fundamental obstacle to generating surfaces which are smooth in the sense of differential geometry. Besides its striking simplicity the approach presented here admits the construction of smooth spline spaces which have a natural refinement property. Thus, various algorithms based on subdivision of tensor product B-spline surfaces become available for surfaces of general type
Quadrature Formulae and Polynomial Inequalities

AbstractIn this paper we prove several inequalities for polynomials and trigonometric polynomials. They are all obtained as applications of certain quadrature formulae, some of which are proved here for the first time. Such an application of a Gaussian quadrature formula was pointed out by Bojanov in 1986 (seeEast. J. Approx.1(1995), 37–46;J. Approx. Theory83(1995), 175–181). Coincidentally, in the same year, it was shown how an inequality for entire functions of exponential type belonging toL2(R) could be deduced from a Gaussian quadrature formula for the doubly infinite integral ∫∞−∞f(x)dx
Asymptotics of Diagonal Hermite–Padé Approximants toez

AbstractLetmbe a fixed positive integer. We consider Hermite–Padé approximants to the exponential functionR(z)=∑p=0mAp(z)epz=O(z^mn+n−1),where the degree of the polynomialsAp, 0⩽p⩽m, is less thann. Asn→∞, exact asymptotics for theAp's and the remainder termR, along with an upper bound on the zeros of the polynomialsAp, are given. These asymptotics show that shifted Hermite–Padé approximants asymptotically minimize exponential polynomials of the above form on a disk {|z|⩽ρ}, providedρdoes not exceedπ/m. These results generalize some of those obtained by Borwein (Const. Approx.2(1986), 291–302) on quadratic Hermite–Padé approximants