pinecone/core-2020-05-10-deduplication huggingface dataset:
Liouville–Green–Olver Approximations for Complex Difference Equations

AbstractLiouville–Green (LG) orWKBapproximations for second-order lineardifferenceequations withcomplexcoefficients are obtained. Preciseboundsfor the error term in the asymptotic representation of theLGrecessive solution are given, and thedouble asymptoticnature, with respect to both,nand additional parameters, is shown; all this is in the spirit of F. W. J. Olver's rigorous work on theLGasymptotics for differential equations. Theholomorphiccharacter of such error terms, and hence of theLGbasis, is also established, when the coefficients of the difference equation are holomorphic. Qualitative properties, such as oscillation and growth of theLGbasis solutions, are displayed.Second-orderasymptotics with bounds is also obtained, and an application to three-term recurrences satisfied by certain orthogonal polynomials (a subclass of the Blumenthal-Nevai class), is made for illustration. The special case ofultrasphericalfunctions of the second kind is worked out in detail
Hypercyclic Sequences of Differential and Antidifferential Operators

AbstractIn this paper, we provide some extensions of earlier results about hypercyclicity of some operators on the Fréchet space of entire functions of several complex variables. Specifically, we generalize in several directions a theorem about hyper- cyclicity of certain infinite order linear differential operators with constant coefficients and study the corresponding property for certain kinds of “antidifferential” operators which are introduced in the paper. In addition, the existence of hypercyclic functions for certain sequences of differential operators with additional properties, for instance, boundedness or with some nonvanishing derivatives, is established
Some Basic Lommel Polynomials

AbstractThe basic Lommel polynomials associated to the1ϕ1q-Bessel function and the Jacksonq-Bessel functions are considered as orthogonal polynomials inqν, whereνis the order of the corresponding basic Bessel functions. The corresponding moment problems are both indeterminate and determinate depending on a parameter. Using techniques of Chihara and Maki we derive an explicit orthogonality measure, which is discrete and unbounded. For the indeterminate moment problem this measure is N-extremal. Some results on the zeros of the basic Bessel functions, both as functions of the order and of the argument are obtained. Precise asymptotic behaviour of the zeros of the1ϕ1q-Bessel function is obtained
On the Positivity of the Fundamental Polynomials for Generalized Hermite–Fejér Interpolation on the Chebyshev Nodes

AbstractIt is shown that the fundamental polynomials for (0,1,…,2m+1) Hermite–Fejér interpolation on the zeros of the Chebyshev polynomials of the first kind are non-negative for −1⩽x⩽1, thereby generalising a well-known property of the original Hermite–Fejér interpolation method. As an application of the result, Korovkin's 10theorem on monotone operators is used to present a new proof that the (0,1,…,2m+1) Hermite–Fejér interpolation polynomials off∈C[−1,1], based onnChebyshev nodes, converge uniformly tofasn→∞
On the Construction of Optimal Monotone Cubic Spline Interpolations

AbstractIn this paper we derive necessary optimality conditions for an interpolating spline function which minimizes the Holladay approximation of the energy functional and which stays monotone if the given interpolation data are monotone. To this end optimal control theory for state-restricted optimal control problems is applied. The necessary conditions yield a complete characterization of the optimal spline. In the case of two or three interpolation knots, which we call thelocalcase, the optimality conditions are treated analytically. They reduce to polynomial equations which can very easily be solved numerically. These results are used for the construction of a numerical algorithm for the optimal monotone spline in the general (global) case via Newton's method. Here, the local optimal spline serves as a favourable initial estimation for the additional grid points of the optimal spline. Some numerical examples are presented which are constructed by FORTRAN and MATLAB programs
Stability and Independence for Multivariate Refinable Distributions

AbstractDue to their so-called time-frequency localization properties, wavelets have become a powerful tool in signal analysis and image processing. Typical constructions of wavelets depend on the stability of the shifts of an underlying refinable function. In this paper, we derive necessary and sufficient conditions for the stability of the shifts of certain compactly supported refinable functions. These conditions are in terms of the zeros of the refinement mask. Our results are actually applicable to more general distributions which are not of function type, if we generalize the notion of stability appropriately. We also provide a similar characterization of the (global) linear independence of the shifts. We present several examples illustrating our results, as well as one example in which known results on box splines are derived using the theorems of this paper
Greedy Algorithms andM-Term Approximation with Regard to Redundant Dictionaries

AbstractWe study the efficiency of greedy type algorithms with regard to redundant dictionaries in Hilbert space and we prove a general result which gives a sufficient condition on a dictionary to guarantee that the pure greedy algorithm is near best in the sense of power decay of error of approximation. We discuss also some important examples. It is already known (see DeVore and Temlyakov,Adv. Comput. Math.5(1996), 173–187) that the Pure Greedy Algorithm for some dictionaries has a saturation property. We construct an example which shows that a natural generalization of the Pure Greedy Algorithm also has a saturation property. Next we discuss some new phenomena which occur in approximation by a greedy type algorithm with regards to a highly redundant dictionary
Weighted Moduli of Smoothness and Spline Spaces

AbstractIn this paper we study relations between moduli of smoothness with the step-weight functionϕand the best approximation by splines with knots uniformly distributed according to the measure with density 1/ϕ(x). The direct and converse results are obtained for a class of step-weight functions, containingϕ(x)=x(1−x); it is well known that the modulus of smoothness corresponding to thisϕis related to the best polynomial approximation. As a consequence, we obtain relations between the best polynomial and spline approximations
Stability of the Cauchy Type Equations in the Class of Differentiable Functions

AbstractLetXbe a real normed space,Ya real Banach space, and letCn(X, Y) denote the space ofn-times continuously differentiable functionsf:X→Y. We prove that the classCnhas the double difference property, that is if Cf(x, y)≔f(x+y)−f(x)−f(y) belongs to the spaceCn(X×X, Y) then there exists an additive functionA:X→Ysuch thatf−A∈Cn(X, Y). Similar result is also obtained for the Jensen equation. As an application we show that the Cauchy and Jensen equations are stable with respect to large class of seminorms defined by means of derivatives
P. L. Chebyshev (1821–1894) A Guide to his Life and Work

AbstractThe aim of this paper is to outline the life and work of Chebyshev, creator in St. Petersburg of the largest prerevolutionary school of mathematics in Russia, who permitted himself to be equated only with Archimedes. Chebyshev, who was regularly in Paris, at the latest by 1852, if not already by 1842, a friend of Liouville and Hermite, was the author of ca 80 publications, covering approximation theory, probability theory, number theory, theory of mechanisms, as well as many problems of analysis and practical mathematics. He was also proud to be a constructor of various mechanisms, including an arithmomètre. Although the paper is intended for an approximation theorist readership, an attempt is made to give proportionate coverage of the broad spectrum of Chebyshev's achievements, emphasis being placed upon their background. The paper is based in part upon the authors' studies during 1985–199
On the Approximation of Functional Classes Equipped with a Uniform Measure Using Ridge Functions

AbstractWe introduce a construction of a uniform measure over a functional class Br which is similar to a Besov class with smoothness index r. We then consider the problem of approximating Br using a manifold Mn which consists of all linear manifolds spanned by n ridge functions, i.e., Mn={∑ni=1gi(ai·x):ai∈Sd−1, gi∈L2([−1, 1])}, x∈Bd. It is proved that for some subset A⊂Br of probabilistic measure 1−δ, for all f∈A the degree of approximation of Mn behaves asymptotically as 1/nr/(d−1). As a direct consequence the probabilistic (n, δ)-width for nonlinear approximation denoted as dn, δ(Br, μ, Mn), where μ is a uniform measure over Br, is similarly bounded. The lower bound holds also for the specific case of approximation using a manifold of one hidden layer neural networks with n hidden units
Piecewise Smooth Spaces in Duality: Application to Blossoming

AbstractGiven an (n+1)-dimensional space U of piecewise smooth functions in which each basis has a non-vanishing Wronskian, and its dual space U*, a canonical bilinear form is defined on U×U*, which provides a simple characterization of a contact of order r⩽n. An intrinsic reproducing function is introduced, leading to Marsden-type identities. In the case of Chebyshev spaces connected with totally positive matrices, the bilinear form yields a general notion of blossom which can be extended to Chebyshev splines
Optimal Interpolating Spaces Preserving Shape

AbstractIn this paper we study the existence and characterization of spaces which are images of minimal-norm projections that are required to interpolate at given functionals and satisfy additional shape-preserving requirements. We will call such spaces optimal interpolating spaces preserving shape. This investigation leads to concrete solutions in classical settings and, as examples, Πn will be determined to be such spaces with regard to certain interpolation and shape-preserving requirements on the projections. Restated, the theory of this paper gives rist to an n-dimensional Hahn–Banach extension theorem, where the minimal-norm extension is required to keep invariant a fixed cone
Strong Unicity of Best Uniform Approximations from Periodic Spline Spaces

AbstractIn this paper we give a complete characterization of the strongly unique best uniform approximations from periodic spline spaces. We distinguish between even-dimensional and odd-dimensional periodic spline spaces. These spaces are weak Chebyshev if and only if their dimension is odd. We show that the strongly unique best approximation from periodic spline spaces of odd dimension can be characterized alone by alternation properties of the error. This is not the case for even dimension. In this case an additional interpolation condition arises in our characterization
Zero location and n-th root asymptotics of Sobolev orthogonal polynomials

14 pages, no figures.-- MSC2000 codes: 42C05, 33C45.MR#: MR1696589 (2000g:42035)Zbl#: Zbl 0949.42020For a wide class of Sobolev orthogonal polynomials, it is proved that their zeros are contained in a compact subset of the complex plane and the asymptotic zero distribution is obtained. With this information, the nth root asymptotic behavior outside the compact set containing all the zeros is given.Research by the first author (G.L.L.) was partially supported by Dirección General de Enseñanza Superior under Grant PB 96-0120-C03-01 and by INTAS under Grant 93-0219 EXT. Research by the second author (H.P.) carried out while following a Doctoral Program at Universidad Carlos III de Madrid under grant from Agencia Española de Cooperación Internacional.Publicad
Weighted Polynomial Approximation for Weights with Slowly Varying Extremal Density

AbstractPolynomial approximation by weighted polynomials of the form wn(x)Pn(x) is investigated on closed subsets of the real line. It is known that the possibility of approximation is closely related to the density of an extremal measure associated with w via a weighted energy problem. It is also known that if in a neighborhood of a point x0 this density is continuous and positive, then, in that neighborhood, any continuous function can be approximated. The aim of the present paper is twofold. On the one hand it is shown that the same approximation theorem is true if in a neighborhood of x0 the density is slowly varying and is bounded away from 0. This allows singularities of logarithmic types. On the other hand, we also show that under some mild conditions, if the density at x0 is slowly varying, then approximation is still possible even if the density vanishes at x0 . This is the first positive result for approximation with a vanishing density
Asymptotic Properties of Balanced Extremal Sobolev Polynomials: Coherent Case

AbstractFor eachn∈N andλn⩾0,Qn, λnis the monic polynomial of degreenthat minimizes the norm ‖q‖2=∫|q|2dμ0+λn∫|q′|2dμ1in the class of all monic polynomials of degreen. Asymptotic properties of {Qn, λn} asn→∞ are studied under additional assumption that (μ0, μ1) is a coherent pair of measures on [−1, 1] and the sequence {λn} is regularly decreasing and satisfies limnn2λn=L∈[0, +∞]. The behavior of the norms and zeros of these polynomials is also studied. We show that in some cases the sequence {Qn, λn} asymptotically behaves as the monic orthogonal polynomials sequence corresponding to a new measure constructed as a combination ofμ0andμ1; we conjecture that this result is valid in a more general setting
Interpolation Formulas for Harmonic Functions

AbstractIt is known that a real-valued entire harmonic functionuof exponential type less thanπis uniquely determined by its values at the pointsnandneiα,n=0,±1,±2,…, unlessαis a rational multiple ofπ. Forα=π/2, which belongs to the exceptional cases, Ching has proved thatuis uniquely determined by its values at these points ifuis in addition an odd function. In the present paper we shall extend this result to the caseα=(2k+1)π/(2l), wherekandl≠0 are arbitrary integers. Furthermore, we shall present formulas which allow a reconstruction of real-valued entire harmonic functions of exponential typeπby their samples at the pointsnandneiα,n=0,±1,±2,…, whenα=(2k+1)π/(2l) or whenα/πis irrational and algebraic
The Asymptotic Zero Distribution of Orthogonal Polynomials with Varying Recurrence Coefficients

AbstractWe study the zeros of orthogonal polynomials pn, N, n=0, 1, …, that are generated by recurrence coefficients an, N and bn, N depending on a parameter N. Assuming that the recurrence coefficients converge whenever n, N tend to infinity in such a way that the ratio n/N converges, we show that the polynomials pn, N have an asymptotic zero distribution as n/N tends to t>0 and we present an explicit formula for the limiting measure. This formula contains the asymptotic zero distri- butions for various special classes of orthogonal polynomials that were found earlier by different methods, such as Jacobi polynomials with varying parameters, discrete Chebyshev polynomials, Krawtchouk polynomials, and Tricomi–Carlitz polynomials. We also give new results on zero distributions of Charlier polynomials, Stieltjes–Wigert polynomials, and Lommel polynomials
Uniform Approximation of Functions with Discrete Approximation Functionals

AbstractHoffman, Kouri, and collaborators have calculated nonrelativistic quantum scattering amplitudes by numerically evaluating Feynman path integrals. They observed that the errors introduced by their numerical scheme were uniform in coordinate space, implying that their scheme accurately reproduces both the shape and the phase of functions. Furthermore, they observed that the size and the uniform nature of the errors were preserved when the functions were allowed to evolve in time under the action of the kinetic energy operator. In this paper it is established that these observed properties of the errors are not numerical artifacts but follow from analytical properties of a general class of approximations that include those of Hoffman, Kouri, and collaborators as a special case
Comonotone Jackson's Inequality

AbstractLet 2s points yi=−π⩽y2s<…<y1<π be given. Using these points, we define the points yi for all integer indices i by the equality yi=yi+2s+2π. We shall write f∈Δ(1)(Y) if f is a 2π-periodic continuous function and f does not decrease on [yi, yi−1], if i is odd; and f does not increase on [yi, yi−1], if i is even. In this article the following Theorem 1—the comonotone analogue of Jackson's inequality—is proved. Theorem1.If f∈Δ(1)(Y), then for each nonnegative integer n there is a trigonometric polynomial τn(x) of order ⩽n such that τn∈Δ(1)(Y), and |f(x)−πn(x)|⩽c(s)ω(f;1/(n+1)), x∈R, where ω(f;t) is the modulus of continuity of f, c(s)=const. Depending only on s
Generalized Bernstein–Durrmeyer Operators and the Associated Limit Semigroup

AbstractThe aim of this paper is the study of a new sequence of positive linear approximation operators Mn, λ on C([0, 1]) which generalize the classical Bernstein–Durrmeyer operators. After proving a Voronovskaja-type result, we show that there exists a strongly continuous positive contraction semigroup on C([0, 1]) which may be expressed in terms of powers of these operators. As a direct consequence, we are able to represent explicitly the solutions of the Cauchy problems associated with a particular class of second order differential operators
Best Harmonic and Superharmonic L1-Approximants in Strips

AbstractLet Ω denote the open strip (−1, 1)×Rn−1, where n⩾2. We completely solve the problem of characterizing a best harmonic L1-approximant to a subharmonic function s on Ω (all functions are assumed to be continuous and integrable on Ω). This characterization was previously known only under highly restrictive hypotheses on s. The approach of this paper is based, in part, on ideas used recently to solve the corresponding problem for the unit ball. However, the unboundedness of Ω presents difficulties which require the use of new techniques and recent results from other branches of harmonic approximation theory. Superharmonic L1-approximation of subharmonic functions is also treated
Local Accuracy for Radial Basis Function Interpolation on Finite Uniform Grids

AbstractWe consider interpolation on a finite uniform grid by means of one of the radial basis functions (RBF) φ(r)=rγ for γ>0, γ∉2N or φ(r)=rγlnr for γ∈2N+. For each positive integer N, let h=N−1 and let {xi:i =1, 2, …, (N+1)d} be the set of vertices of the uniform grid of mesh-size h on the unit d-dimensional cube [0, 1]d. Given f:[0, 1]d→R, let sh be its unique RBF interpolant at the grid vertices: sh(xi)=f(xi), i=1, 2, …, (N+1)d. For h→0, we show that the uniform norm of the error f−sh on a compact subset K of the interior of [0, 1]d enjoys the same rate of convergence to zero as the error of RBF interpolation on the infinite uniform grid hZd, provided that f is a data function whose partial derivatives in the interior of [0, 1]d up to a certain order can be extended to Lipschitz functions on [0, 1]d
On the Convergence of Bounded J-Fractions on the Resolvent Set of the Corresponding Second Order Difference Operator

AbstractWe study connections between continued fractions of type J and spectral properties of second order difference operators with complex coefficients. It is known that the convergents of a bounded J-fraction are diagonal Padé approximants of the Weyl function of the corresponding difference operator and that a bounded J-fraction converges uniformly to the Weyl function in some neighborhood of infinity. In this paper we establish convergence in capacity in the unbounded connected component of the resolvent set of the difference operator and specify the rate of convergence. Furthermore, we show that the absence of poles of Padé approximants in some subdomain implies already local uniform convergence. This enables us to verify the Baker–Gammel–Wills conjecture for a subclass of Weyl functions. For establishing these convergence results, we study the ratio and the nth root asymptotic behavior of Padé denominators of bounded J-fractions and give relations with the Green function of the unbounded connected component of the resolvent set. In addition, we show that the number of “spurious” Padé poles in this set may be bounded
Some Positive Results and Counterexamples in Comonotone Approximation, II

AbstractLetfbe a continuous function on [−1, 1], which changes its monotonicity finitely many times in the interval, saystimes. In the first part of this paper we have discussed the validity of Jackson type estimates for the approximation offby algebraic polynomials that are comonotone with it. We have proved the validity of a Jackson type estimate involving the Ditzian–Totik (first) modulus of continuity and a constant which depends only ons, and we have shown by counterexamples that in many cases the Jackson estimates involving the DT-moduli do not hold when there are certain relations betweens, the number of changes of monotonicity, andr, the number of derivatives of the approximated function. Here we deal with all other cases and we obtain Jackson type estimates involving modified DT-moduli. We also provide counterexamples to complete the picture. Our technique for the positive results involves a two-tier approach. We first approximate the given function by comonotone piecewise polynomials which yield good approximation and then we replace the latter by polynomials
Biorthogonal Rational Functions and the Generalized Eigenvalue Problem

AbstractWe present some general results concerning so-called biorthogonal polynomials of RII type introduced by M. Ismail and D. Masson. These polynomials give rise to a pair of rational functions which are biorthogonal with respect to a linear functional. It is shown that these rational functions naturally appear as eigenvectors of the generalized eigenvalue problem for two arbitrary tri-diagonal matrices. We study spectral transformations of these functions leading to a rational modification of the linear functional. An analogue of the Christoffel–Darboux formula is obtained
Notes on Inequalities with Doubling Weights

AbstractVarious important weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., inequalities, have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most of the cases this minimal assumption is the doubling condition. Sometimes, however, as in the weighted Nikolskii inequality, the slightly strongerA∞condition is used. Throughout their paper theLpnorm is studied under the assumption 1⩽p<∞. In this note we show that their proofs can be modified so that many of their inequalities hold even if 0<p<1. The crucial tool is an estimate for quadrature sums for thepth power (0<p<∞ is arbitrary) of trigonometric polynomials established by Lubinsky, Máté, and Nevai. For technical reasons we discuss only the trigonometric cases
A New Class of Modified Bernstein Operators

AbstractThe left Bernstein quasi-interpolant operator introduced by Sablonnière is a kind of modified Bernstein operator that has good stability and convergence rate properties. However, we recently found that it is not very convenient for practical applications. Fortunately, we showed in a previous paper that there exist many operators having stability and convergence rate properties similar to those of Sablonnière's operator. In this paper, we introduce another specific class of such operators generated from the operator introduced by Stancu. We present detailed results about this class, some of which can be applied to numerical quadrature. Finally, we clarify its advantages and assert that it is more natural and more convenient, both theoretically and practically, than that of Sablonnière. Our paper, at the same time, provides several new results regarding Stancu's operator