pinecone/core-2020-05-10-deduplication huggingface dataset:
Superapproximation and Commutator Properties of Discrete Orthogonal Projections for Continuous Splines

AbstractThis paper builds upon the Lp-stability results for discrete orthogonal projections on the spaces Sh of continuous splines of order r obtained by R. D. Grigorieff and I. H. Sloan in (1998, Bull. Austral. Math. Soc.58, 307–332). Properties of such projections were proved with a minimum of assumptions on the mesh and on the quadrature rule defining the discrete inner product. The present results, which include superapproximation and commutator properties, are similar to those derived by I. H. Sloan and W. Wendland (1999, J. Approx. Theory97, 254–281) for smoothest splines on uniform meshes. They are expected to have applications (as in I. H. Sloan and W. Wendland, Numer. Math. (1999, 83, 497–533)) to qualocation methods for non-constant-coefficient boundary integral equations, as well as to the wide range of other numerical methods in which quadrature is used to evaluate L2-inner products. As a first application, we consider the most basic variable-coefficient boundary integral equation, in which the constant-coefficient operator is the identity. The results are also extended to the case of periodic boundary conditions, in order to allow appplication to boundary integral equations on closed curves
Stone–Weierstrass Theorems in C*(X)

AbstractIn this paper we consider some conditions for a given function f∈C*(X) to belong to the uniform closure of a subset W⊂C*(X). We start with a general theorem which admits a sensible improvement when W, or more generally its uniform closure cl(W), is a lattice. Also, we obtain an approximation result when W or cl(W) is a cone. From this result we can derive one by Blasco and Moltó for linear subspaces and one by Garrido and Montalvo for semi-affine lattices. Finally, using “multipliers,” we extend to C*(X) some other known results for the compact case, such as the Nachbin theorem about the uniform closure of an A-module
On the Construction and Frequency Localization of Finite Orthogonal Quadrature Filters

AbstractWe introduce a new method to construct finite orthogonal quadrature filters using convolution kernels and show that every filter with value 1 at the origin can be obtained from an even nonnegative kernel. We apply the method to estimate the optimal frequency localization of finite filters. The frequency localization γp of a finite filter m0 is given by the distance in Lp-norm between |m0|2 and the Shannon low-pass filter. For each N>0 there is a filter mN0 of length 2N minimizing the value of γp. We prove that for such a minimizing sequence we have γpp(mN0)=O(1/N), 1⩽p⩽2, and this estimate is optimal. We construct several new families of both MRA and non-MRA filters with optimal asymptotic frequency localization
Asymptotic Error Estimates for L2 Best Rational Approximants to Markov Functions

AbstractLet f(z)=∫(t−z)−1dμ(t) be a Markov function, where μ is a positive measure with compact support in R. We assume that supp(μ)⊂(−1, 1), and investigate the best rational approximants to f in the Hardy space H02(V), where V≔{z∈C∣|z|>1} and H02(V) is the subset of functions f∈H2(V) with f(∞)=0. The central topic of the paper is to obtain asymptotic error estimates for these approximants. The results are presented in three groups. In the first one no specific assumptions are made with respect to the defining measure μ of the function f. In the second group it is assumed that the measure μ is not too thin anywhere on its support so that the polynomials pn, orthonormal with respect to the measure μ, have a regular nth root asymptotic behavior. In the third group the defining measure μ is assumed to belong to the Szegő class. For each of the three groups, asymptotic error estimates are proved in the L2-norm on the unit circle and in a pointwise fashion. Also the asymptotic distribution of poles, zeros, and interpolation points of the best L2 approximants are studied
Uniqueness of Minimal Projections in Smooth Matrix Spaces

AbstractLet X=(M(n, m), ‖·‖), where ‖·‖ fulfills Condition 0.3 and W=M(n, 1)+M(1, m). A formula for a minimal projection from X onto W is given in (E. W. Cheney and W. A. Light, 1985, “Approximation Theory in Tensor Product Spaces,” Lecture Notes in Mathematics, Springer-Verlag, Berlin; E. J. Halton and W. A. Light, 1985, Math. Proc. Cambridge Philos. Soc.97, 127–136; and W. A. Light, 1986, Math. Z.191, 633–643). We will show that this projection is the unique minimal projection (see Theorem 2.1)
Strong and Plancherel–Rotach Asymptotics of Non-diagonal Laguerre–Sobolev Orthogonal Polynomials

AbstractWe study properties of the monic polynomials {Qn}n∈N orthogonal with respect to the Sobolev inner product(p, q)S=∫∞0(p, p′)1μμλqq′xαe−xdx,where λ−μ2>0 and α>−1. This inner product can be expressed as(p, q)S=∫∞0p(x)q(x)((μ+1)x−αμ)xα−1e−xdx+λ∫∞0p′q′xαe−xdx,when α>0. In this way, the measure which appears in the first integral is not positive on [0, ∞) for μ∈R\[−1, 0]. The aim of this paper is the study of analytic properties of the polynomials Qn. First we give an explicit representation for Qn using an algebraic relation between Sobolev and Laguerre polynomials together with a recursive relation for kn=(Qn, Qn)S. Then we consider analytic aspects. We first establish the strong asymptotics of Qn on C\[0, ∞) when μ∈R and we also obtain an asymptotic expression on the oscillatory region, that is, on (0, ∞). Then we study the Plancherel–Rotach asymptotics for the Sobolev polynomials Qn(nx) on C\[0, 4] when μ∈(−1, 0]. As a consequence of these results we obtain the accumulation sets of zeros and of the scaled zeros of Qn. We also give a Mehler–Heine type formula for the Sobolev polynomials which is valid on compact subsets of C when μ∈(−1, 0], and hence in this situation we obtain a more precise result about the asymptotic behaviour of the small zeros of Qn. This result is illustrated with three numerical examples
Fekete Potentials and Polynomials for Continua

AbstractFor planar continua, upper and lower bounds are given for the growth of the associated Fekete potentials, polynomials and energies. The main result is that for continua K of capacity 1 whose outer boundary is an analytic Jordan curve, the family of Fekete polynomials is bounded on K. Our work makes use of precise results of Pommerenke on the growth of the discriminant and on the distribution of the Fekete points. We also use potential theory, including the exterior Green function with pole at infinity. The Lipschitz character of this function determines the separation of the Fekete points
Orthogonal Expansion of Real Polynomials, Location of Zeros, and an L2 Inequality

AbstractLet f(z)=a0φ0(z)+a1φ1(z)+…+anφn(z) be a polynomial of degree n, given as an orthogonal expansion with real coefficients. We study the location of the zeros of f relative to an interval and in terms of some of the coefficients. Our main theorem generalizes or refines results due to Turán and Specht. In particular, it includes a best possible criterion for the occurrence of real zeros. Our approach also allows us to establish a weighted L2 inequality giving a lower estimate for the product of two polynomials
On the Rates of Approximation of Bernstein Type Operators

AbstractAsymptotic behavior of two Bernstein-type operators is studied in this paper. In the first case, the rate of convergence of a Bernstein operator for a bounded function f is studied at points x where f(x+) and f(x−) exist. In the second case, the rate of convergence of a Szász operator for a function f whose derivative is of bounded variation is studied at points x where f(x+) and f(x−) exist. Estimates of the rate of convergence are obtained for both cases and the estimates are the best possible for continuous points
An Extension of E. Bishop's Localization Theorem

AbstractWe prove that if f is a function belonging to Baire first class on a compact set K⊂C and each point of K has a (closed) neighborhood where f is the pointwise limit of some sequence of uniformly bounded rational functions, then f on the whole of K is the pointwise limit of a sequence of rational functions uniformly bounded on K. This is an extension of Bishop's localization theorem. As an application we establish a “pointwise” version of Mergelyan's classical theorem on uniform approximation by rational functions on compact sets for which the components of its complement have diameters greater than a fixed positive number
Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

AbstractWe derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle
Approximation Properties and Construction of Hermite Interpolants and Biorthogonal Multiwavelets

AbstractMultiwavelets are generated from refinable function vectors by using multiresolution analysis. In this paper we investigate the approximation properties of a multivariate refinable function vector associated with a general dilation matrix in terms of both the subdivision operator and the order of sum rules satisfied by the matrix refinement mask. Based on a fact about the sum rules of biorthogonal multiwavelets, a coset by coset (CBC) algorithm is presented to construct biorthogonal multiwavelets with arbitrary order of vanishing moments. More precisely, to obtain biorthogonal multiwavelets, we have to construct primal and dual masks. Given any primal matrix mask a and a general dilation matrix M, the proposed CBC algorithm reduces the construction of all dual masks of a, which satisfy the sum rules of arbitrary order, to a problem of solving a well organized system of linear equations. We prove in a constructive way that for any given primal mask a with a dilation matrix M and for any positive integer k, we can always construct a dual mask ã of a such that ã satisfies the sum rules of order k. In addition, we provide a general way for the construction of Hermite interpolatory matrix masks in the univariate setting with any dilation factors. From such Hermite interpolatory masks, smooth Hermite interpolants, including the well known cubic Hermite splines as a special case, are obtained and are used to construct biorthogonal multiwavelets. As an example, a C3 Hermite interpolant with support [−3, 3] is presented. Then we shall apply the CBC algorithm to such Hermite interpolatory masks to construct biorthogonal multiwavelets. Several examples of biorthogonal multiwavelets are provided to illustrate the general theory. In particular, a C1 dual function vector with support [−4, 4] of the cubic Hermite splines is given
Zeros of the Hypergeometric Polynomials F(−n, b; −2n; z)

AbstractWe investigate the location of the zeros of the hypergeometric polynomial F(−n, b; −2n; z) for b real. The Hilbert–Klein formulas are used to specify the number of real zeros in the intervals (−∞, 0), (0, 1), or (1, ∞). For b>0 we obtain the equation of the Cassini curve which the zeros of wnF(−n, b; −2n; 1/w) approach as n→∞ and thereby prove a special case of a conjecture made by Martı́nez-Finkelshtein, Martı́nez-González, and Orive. We also present some numerical evidence linking the zeros of F with more general Cassini curves
A Unified Calculus Using the Generalized Bernoulli Polynomials

AbstractWe introduce an α-calculus with the help of the generalized Bernoulli polynomials. The parameter α is the order of a Bessel function of the first kind. The differential α-calculus can be put in a general context where the concept of supporting function is an important tool for practical purposes. Our somewhat more restrictive point of view has the advantage of permitting a consistent definition of an α-integral with several interesting properties. It results in the possibility of expressing a remainder, in the aforementioned context, in a completely new form in our case
The Approximation Functional and Interpolation with Perturbed Continuity

AbstractThe continuity conditions at the endpoints of interpolation theorems, ‖Ta‖Bj⩽Mj‖a‖Aj for j=0, 1, can be written with the help of the approximation functional: ‖E(t, Ta; B1, B0)‖L∞⩽M0‖a‖A0 and ‖E(t, Ta; B0, B1)‖L∞⩽M1‖a‖A1. As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the L∞ norms can be replaced by BMO(R+) norms. This leads to a strong version of the Stein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition f∈L0, i.e., f∗∈L∞, where f∗(γ)=μ{|f|>γ} is the distribution function of f, can be replaced in interpolation with L(p, q) spaces by the weaker f∗∈BMO(R+)
Modes of Convergence: Interpolation Methods I

AbstractIn the present paper we explore an approximation theoretic approach to some classical convergence theorems of real analysis. The background of this paper is the intuition that some of the usual compactness theorems on various modes of convergence in classical analysis are based on suitable ways of obtaining good decompositions of functions to exploit rates of approximation, cancellations, or appropriate control of sizes that can be controlled by the basic functionals of real interpolation
Properties of Multivariate Homogeneous Orthogonal Polynomials

AbstractIt is well-known that the denominators of Padé approximants can be considered as orthogonal polynomials with respect to a linear functional. This is usually shown by defining Padé-type approximants from so-called generating polynomials and then improving the order of approximation by imposing orthogonality conditions on the generating polynomials. In the multivariate case, a similar construction is possible when dealing with the multivariate homogeneous Padé approximants introduced by the second author. Moreover it is shown here, that several well-known properties of the zeroes of classical univariate orthogonal polynomials, in the case of a definite linear functional, generalize to the multivariate homogeneous case. For the multivariate homogeneous orthogonal polynomials, the absence of common zeroes is translated to the absence of common factors
On Quasi-Interpolation with Non-uniformly Distributed Centers on Domains and Manifolds

AbstractThe paper studies quasi-interpolation by scaled shifts of a smooth and rapidly decaying function. The centers are images of a smooth mapping of the hZn-lattice in Rs, s⩾n, and the scaling parameters are proportional to h. We show that for a large class of generating functions the quasi-interpolants provide high order approximations up to some prescribed accuracy. Although in general the approximants do not converge as h tends to zero, the remaining saturation error is negligible in numerical computations if a scalar parameter is suitably chosen. The lack of convergence is compensated for by a greater flexibility in the choice of generating functions used in numerical methods for solving operator equations
Relative Asymptotics for Orthogonal Matrix Polynomials with Convergent Recurrence Coefficients

AbstractThe asymptotic behavior of γn(dβ)γn(dα)−1 and Pn(x, dβ)P−1n(x, dα) is studied. Here (γn(.))n are the leading coefficients of the orthonormal matrix polynomials Pn(x, .) with respect to the matrix measures dβ and dα which are related by dβ(u)=dα(u)+∑Nk=1Mkδ(u−ck), where Mk are positive definite matrices, δ is the Dirac measure and ck lies outside the support of dα for k=1, …, N. Finally, we deduce the asymptotic behavior of Pn(c, dβ)MP*n(c, dα) when dβ(u)=dα(u)+Mδ(u−c), with M a positive definite matrix and c outside the support of dα
Maximal Inequalities and Lebesgue's Differentiation Theorem for Best Approximant by Constant over Balls

AbstractFor f∈Lp(Rn), with 1⩽p<∞, ε>0 and x∈Rn we denote by Tε(f)(x) the set of every best constant approximant to f in the ball B(x, ε). In this paper we extend the operators Tεp to the space Lp−1(Rn)+L∞(Rn), where L0 is the set of every measurable functions finite almost everywhere. Moreover we consider the maximal operators associated to the operators Tεp and we prove maximal inequalities for them. As a consequence of these inequalities we obtain a generalization of Lebesgue's Differentiation Theorem
A Counter-Example to the General Convergence of Partially Greedy Algorithms

AbstractIn a separable Hilbert space H, greedy algorithms iteratively define m-term approximants to a given vector from a complete redundant dictionary D. With very large dictionaries, the pure greedy algorithm cannot be implemented and must be replaced with a weak greedy algorithm. In numerical applications, partially greedy algorithms have been introduced to reduce the numerical complexity. A conjecture about their convergence arises naturally from the observation of numerical experiments. We introduce, study and disprove this conjecture
Ridge Functions and Orthonormal Ridgelets

AbstractOrthonormal ridgelets provide an orthonormal basis for L2(R2) built from special angularly-integrated ridge functions. In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x1cosθ+x2sinθ). We derive a formula for the ridgelet coefficients of a ridge function in terms of the 1-D wavelet coefficients of the ridge profile r(t). The formula shows that the ridgelet coefficients of a ridge function are heavily concentrated in ridge parameter space near the underlying scale, direction, and location of the ridge function. It also shows that the rearranged weighted ridgelet coefficients of a ridge function decay at essentially the same rate as the rearranged weighted 1-D wavelet coefficients of the 1-D ridge profile r(t). In short, the full ridgelet expansion of a ridge function is in a certain sense equally as sparse as the 1-D wavelet expansion of the ridge profile. It follows that partial ridgelet expansions can give good approximations to objects which are countable superpositions of well-behaved ridge functions. We study the nonlinear approximation operator which “kills” coefficients below certain thresholds (depending on angular- and ridge-scale); we show that for approximating objects which are countable superpositions of ridge functions with 1-D ridge profiles in the Besov space B1/pp, p(R), 0<p<1, the thresholded ridgelet approximation achieves optimal rates of N-term approximation. This implies that appropriate thresholding in the ridgelet basis is equally as good, for certain purposes, as an ideally-adapted N-term nonlinear ridge approximation, based on perfect choice of N-directions
Universal Polynomial Majorants on Convex Bodies

AbstractLet K be a convex body in Rd (d⩾2), and denote by Bn(K) the set of all polynomials pn in Rd of total degree ⩽n such that |pn|⩽1 on K. In this paper we consider the following question: does there exist a p*n∈Bn(K) which majorates every element of Bn(K) outside of K? In other words can we find a minimal γ⩾1 and p*n∈Bn(K) so that |pn(x)|⩽γ|p*n(x)| for every pn∈Bn(K) and x∈Rd\K? We discuss the magnitude of γ and construct the universal majorants p*n for evenn. It is shown that γ can be 1 only on ellipsoids. Moreover, γ=O(1) on polytopes and has at most polynomial growth with respect to n, in general, for every convex body K
Strong Asymptotic Behavior and Weak Convergence of Polynomials Orthogonal on an Arc of the Unit Circle

AbstractLet σ be a finite positive Borel measure supported on an arc γ of the unit circle, such that σ′>0 a.e. on γ. We obtain a theorem about the weak convergence of the corresponding sequence of orthonormal polynomials. Moreover, we prove an analogue of the Szegő–Geronimus theorem on strong asymptotics of the orthogonal polynomials on the complement of γ, which completes to its full extent a result of N. I. Akhiezer. The key tool in the proofs is the use of orthogonality with respect to varying measures
On the Semi-norm of Radial Basis Function Interpolants

AbstractRadial basis function interpolation has attracted a lot of interest in recent years. For popular choices, for example thin plate splines, this problem has a variational formulation, i.e. the interpolant minimizes a semi-norm on a certain space of radial functions. This gives rise to a function space, called the native space. Every function in this space has the property that the semi-norm of an arbitrary interpolant to this function is uniformly bounded. In applications it is of interest whether a sufficiently smooth function belongs to the native space. In this paper we give sufficient conditions on the differentiability of a function with compact support, in the case of cubic, linear and thin plate splines. In the case of multiquadrics and Gaussian functions, it is shown that the only compactly supported function that satisfies these conditions is identically zero
Asymptotic Properties of Zeros of Hypergeometric Polynomials

AbstractIn a paper by K. Driver and P. Duren (1999, Numer. Algorithms21, 147–156) a theorem of Borwein and Chen was used to show that for each k∈N the zeros of the hypergeometric polynomials F(−n, kn+1; kn+2; z) cluster on the loop of the lemniscate {z:|zk(1−z)|=kk/(k+1)k+1}, with Re{z}>k/(k+1) as n→∞. We now supply a direct proof which generalizes this result to arbitrary k>0, while showing that every point of the curve is a cluster point of zeros. Examples generated by computer graphics suggest some finer asymptotic properties of the zeros
Rational Approximation to the Exponential Function with Complex Conjugate Interpolation Points

AbstractIn this paper, we study asymptotic properties of rational functions that interpolate the exponential function. The interpolation is performed with respect to a triangular scheme of complex conjugate points lying in bounded rectangular domains included in the horizontal strip |Imz|<2π. Moreover, the height of these domains cannot exceed some upper bound which depends on the type of rational functions. We obtain different convergence results and precise estimates for the error function in compact sets of C that generalize the classical properties of Padé approximants to the exponential function. The proofs rely on, among others, Walsh's theorem on the location of the zeros of linear combinations of derivatives of a polynomial and on Rolle's theorem for real exponential polynomials in the complex domain
Markov-Type Inequalities for Products of Müntz Polynomials

AbstractLet Λ≔(λj)∞j=0 be a sequence of distinct real numbers. The span of {xλ0, xλ1, …, xλn} over R is denoted by Mn(Λ)≔span{xλ0, xλ1, …, xλn}. Elements of Mn(Λ) are called Müntz polynomials. The principal result of this paper is the following Markov-type inequality for products of Müntz polynomials. Theorem2.1.LetΛ≔(λj)∞j=0andΓ≔(γj)∞j=0be increasing sequences of nonnegative real numbers. LetK(Mn(Λ), Mm(Γ))≔sup‖x(pq)′(x)‖[0, 1]‖pq‖[0, 1]:p∈Mn(Λ),q∈Mm(Γ).Then13((m+1)λn+(n+1)γm)⩽K(Mn(Λ), Mm(Γ))⩽18(n+m+1)(λn+γm).In particular ,23(n+1)λn⩽K(Mn(Λ), Mn(Λ))⩽36(2n+1)λn. Under some necessary extra assumptions, an analog of the above Markov-type inequality is extended to the cases when the factor x is dropped, and when the interval [0, 1] is replaced by [a, b]⊂(0, ∞)
Interpolation by Radial Basis Functions on Sobolev Space

AbstractInterpolation by translates of suitable radial basis functions is an important approach towards solving the scattered data problem. However, for a large class of smooth basis functions (including multiquadrics φ(x)=(|x|2+λ2)m−d/2, m>d/2, 2m−d∉2Z), the existing theories guarantee the interpolant to approximate well only for a very small class of very smooth approximands. The approximands f need to be extremely smooth. Hence, the purpose of this paper is to study the behavior of interpolation by smooth radial basis functions on larger spaces, especially on the homogeneous Sobolev spaces