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 Constructive Approximation   [SJR: 1.029]   [H-I: 35]   [0 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1432-0940 - ISSN (Online) 0176-4276    Published by Springer-Verlag  [2353 journals]
• Lower Bounds for Haar Projections: Deterministic Examples
• Authors: Andreas Seeger; Tino Ullrich
Pages: 227 - 242
Abstract: Abstract In a previous paper by the authors, the existence of Haar projections with growing norms in Sobolev–Triebel–Lizorkin spaces has been shown via a probabilistic argument. This existence was sufficient to determine the precise range of Triebel–Lizorkin spaces for which the Haar system is an unconditional basis. The aim of the present paper is to give simple deterministic examples of Haar projections that show this growth behavior in the respective range of parameters.
PubDate: 2017-10-01
DOI: 10.1007/s00365-016-9353-3
Issue No: Vol. 46, No. 2 (2017)

• Best Rational Approximation of Functions with Logarithmic Singularities
• Authors: Alexander Pushnitski; Dmitri Yafaev
Pages: 243 - 269
Abstract: Abstract We consider functions $$\omega$$ on the unit circle $${\mathbb T}$$ with a finite number of logarithmic singularities. We study the approximation of $$\omega$$ by rational functions and find an asymptotic formula for the distance in the $${{\mathrm{BMO}}}$$ -norm between $$\omega$$ and the set of rational functions of degree n as $$n\rightarrow \infty$$ . Our approach relies on the Adamyan–Arov–Krein theorem and on the study of the asymptotic behavior of singular values of Hankel operators.
PubDate: 2017-10-01
DOI: 10.1007/s00365-016-9347-1
Issue No: Vol. 46, No. 2 (2017)

• Strictly Positive Definite Kernels on the Torus
• Authors: J. Guella; V. A. Menegatto
Pages: 271 - 284
Abstract: Abstract We determine a necessary and sufficient condition for the strict positive definiteness of a continuous and positive definite kernel on the torus.
PubDate: 2017-10-01
DOI: 10.1007/s00365-016-9354-2
Issue No: Vol. 46, No. 2 (2017)

• Orthogonal Polynomials on the Circle for the Weight w Satisfying
Conditions $$w,w^{-1}\in \mathrm{BMO}$$ w , w - 1 ∈ BMO
• Authors: Sergey Denisov; Keith Rush
Pages: 285 - 303
Abstract: Abstract For the weight w satisfying $$w,w^{-1}\in \mathrm{BMO}({\mathbb {T}})$$ , we prove the asymptotics of $$\{\Phi _n(e^{i\theta },w)\}$$ in $$L^p[-\pi ,\pi ], 2\leqslant p<p_0$$ , where $$\{\Phi _n(z,w)\}$$ are monic polynomials orthogonal with respect to w on the unit circle $${\mathbb {T}}$$ . Immediate applications include the estimates on the uniform norm and asymptotics of the polynomial entropies. The estimates on higher-order commutators between the Calderon–Zygmund operators and BMO functions play the key role in the proofs of main results.
PubDate: 2017-10-01
DOI: 10.1007/s00365-016-9350-6
Issue No: Vol. 46, No. 2 (2017)

• On the Zeros and Critical Points of Polynomials with Nonnegative
Coefficients: A Nonconvex Analogue of the Gauss–Lucas Theorem
• Authors: Blagovest Sendov; Hristo Sendov
Pages: 305 - 317
Abstract: Abstract In this work, we present a nonconvex analogue of the classical Gauss–Lucas theorem stating that the critical points of a polynomial with complex coefficients are in the convex hull of its zeros. We show that if the polynomial p(z) of degree n has nonnegative coefficients and zeros in the sector $$\{z \in \mathcal C: \arg (z) \ge \varphi \}$$ , for some $$\varphi \in [0,\pi ]$$ , then the critical points of p(z) are also in that sector. Clearly, when $$\varphi \in [\pi /2,\pi ]$$ , our result follows from the classical Gauss–Lucas theorem. But when $$\varphi \in [0,\pi /2)$$ , we obtain a nonconvex analogue.
PubDate: 2017-10-01
DOI: 10.1007/s00365-017-9374-6
Issue No: Vol. 46, No. 2 (2017)

• Approximation and Orthogonality in Sobolev Spaces on a Triangle
• Authors: Yuan Xu
Pages: 349 - 434
Abstract: Abstract Approximation by polynomials on a triangle is studied in the Sobolev space $$W_2^r$$ that consists of functions whose derivatives of up to r-th order have bounded $$L^2$$ norm. The first part aims at understanding the orthogonal structure in the Sobolev space on the triangle, which requires explicit construction of an inner product that involves derivatives and its associated orthogonal polynomials, so that the projection operators of the corresponding Fourier orthogonal expansion commute with partial derivatives. The second part establishes the sharp estimate for the error of polynomial approximation in $$W_2^r$$ , when  $$r = 1$$ and $$r=2$$ , where the polynomials of approximation are the partial sums of the Fourier expansions in orthogonal polynomials of the Sobolev space.
PubDate: 2017-10-01
DOI: 10.1007/s00365-017-9377-3
Issue No: Vol. 46, No. 2 (2017)

• Spectrally Optimized Pointset Configurations
• Authors: Braxton Osting; Jeremy Marzuola
Pages: 1 - 35
Abstract: Abstract The search for optimal configurations of pointsets, the most notable examples being the problems of Kepler and Thompson, have an extremely rich history with diverse applications in physics, chemistry, communication theory, and scientific computing. In this paper, we introduce and study a new optimality criteria for pointset configurations. Namely, we consider a certain weighted graph associated with a pointset configuration and seek configurations that minimize certain spectral properties of the adjacency matrix or graph Laplacian defined on this graph, subject to geometric constraints on the pointset configuration. This problem can be motivated by solar cell design and swarming models, and we consider several spectral functions with interesting interpretations such as spectral radius, algebraic connectivity, effective resistance, and condition number. We prove that the regular simplex extremizes several spectral invariants on the sphere. We also consider pointset configurations on flat tori via (i) the analogous problem on lattices and (ii) through a variety of computational experiments. For many of the objectives considered (but not all), the triangular lattice is extremal.
PubDate: 2017-08-01
DOI: 10.1007/s00365-017-9365-7
Issue No: Vol. 46, No. 1 (2017)

• How Many Zolotarëv Fractions are There'
• Authors: A. B. Bogatyrëv
Pages: 37 - 45
Abstract: Abstract Known properties of the Chebyshev polynomials are the following: they have simple critical points and only two (finite) critical values. Those properties uniquely determine the polynomials modulo affine transformations of dependent and independent variables. Zolotarëv fractions have similar properties: simple critical points with only four distinct critical values. These properties determine many classes of rational functions modulo projective transformations of the dependent and independent variables. In this paper, we explicity describe these classes.
PubDate: 2017-08-01
DOI: 10.1007/s00365-017-9366-6
Issue No: Vol. 46, No. 1 (2017)

• On the Order Derivatives of Bessel Functions
• Authors: T. M. Dunster
Pages: 47 - 68
Abstract: Abstract The derivatives with respect to order for the Bessel functions $$J_{\nu }(x)$$ and $$Y_{\nu }(x)$$ , where $$\nu >0$$ and $$x\ne 0$$ (real or complex), are studied. Representations are derived in terms of integrals that involve the products pairs of Bessel functions, and in turn series expansions are obtained for these integrals. From the new integral representations for $$\partial J_{\nu }(x)/\partial \nu$$ and $$\partial Y_{\nu }(x)/\partial \nu$$ , asymptotic approximations involving Airy functions are constructed for the case $$\nu$$ large, which are uniformly valid for $$0<x<\infty$$ .
PubDate: 2017-08-01
DOI: 10.1007/s00365-016-9355-1
Issue No: Vol. 46, No. 1 (2017)

• Change of Variable in Spaces of Mixed Smoothness and Numerical Integration
of Multivariate Functions on the Unit Cube
• Authors: Van Kien Nguyen; Mario Ullrich; Tino Ullrich
Pages: 69 - 108
Abstract: Abstract In a recent article, two of the present authors studied Frolov’s cubature formulae and their optimality in Besov–Triebel–Lizorkin spaces of functions with dominating mixed smoothness supported in the unit cube. In this paper, we give a general result that the asymptotic order of the minimal worst-case integration error is not affected by boundary conditions in the above mentioned spaces. In fact, we propose two tailored modifications of Frolov’s cubature formulae suitable for functions supported on the cube (not in the cube) that yield the same order of convergence up to a constant. This constant involves the norms of a “change of variable” and a “pointwise multiplication” mapping, respectively, between the function spaces of interest. We complement, extend, and improve classical results on the boundedness of change of variable mappings in Besov–Sobolev spaces of mixed smoothness. The second modification, suitable for classes of periodic functions, is based on a pointwise multiplication and is therefore most likely more suitable for applications than the (traditional) “change of variable” approach. These new theoretical insights are expected to be useful for the design of new (and robust) cubature rules for multivariate functions on the cube.
PubDate: 2017-08-01
DOI: 10.1007/s00365-017-9371-9
Issue No: Vol. 46, No. 1 (2017)

• Orthogonal Polynomials for a Class of Measures with Discrete Rotational
Symmetries in the Complex Plane
• Authors: F. Balogh; T. Grava; D. Merzi
Pages: 109 - 169
Abstract: Abstract We obtain the strong asymptotics of polynomials $$p_n(\lambda )$$ , $$\lambda \in {\mathbb {C}}$$ , orthogonal with respect to measures in the complex plane of the form \begin{aligned} \hbox {e}^{-N( \lambda ^{2s}-t\lambda ^s-\overline{t}\overline{\lambda }^s)}\hbox {d}A(\lambda ), \end{aligned} where s is a positive integer, t is a complex parameter, and $$\hbox {d}A$$ stands for the area measure in the plane. This problem has its origin in normal matrix models. We study the asymptotic behavior of $$p_n(\lambda )$$ in the limit $$n,N\rightarrow \infty$$ in such a way that $$n/N\rightarrow T$$ constant. Such asymptotic behavior has two distinguished regimes according to the topology of the limiting support of the eigenvalues distribution of the normal matrix model. If $$0< t ^2<T/s$$ , the eigenvalue distribution support is a simply connected compact set of the complex plane, while for $$t ^2>T/s$$ , the eigenvalue distribution support consists of s connected components. Correspondingly, the support of the limiting zero distribution of the orthogonal polynomials consists of a closed contour contained in each connected component. Our asymptotic analysis is obtained by reducing the planar orthogonality conditions of the polynomials to equivalent contour integral orthogonality conditions. The strong asymptotics for the orthogonal polynomials is obtained from the corresponding Riemann–Hilbert problem by the Deift–Zhou nonlinear steepest descent method.
PubDate: 2017-08-01
DOI: 10.1007/s00365-016-9356-0
Issue No: Vol. 46, No. 1 (2017)

• Orthogonal Polynomial Projection Error Measured in Sobolev Norms in the
Unit Disk
• Authors: Leonardo E. Figueroa
Pages: 171 - 197
Abstract: Abstract We study approximation properties of weighted $$L^2$$ -orthogonal projectors onto the space of polynomials of degree less than or equal to N on the unit disk where the weight is of the generalized Gegenbauer form $$x \mapsto (1-\left x\right ^2)^\alpha$$ . The approximation properties are measured in Sobolev-type norms involving canonical weak derivatives, all measured in the same weighted $$L^2$$ norm. Our basic tool consists in the analysis of orthogonal expansions with respect to Zernike polynomials. The sharpness of the main result is proved in some cases. A number of auxiliary results of independent interest are obtained including some properties of the uniformly and nonuniformly weighted Sobolev spaces involved, connection coefficients between Zernike polynomials, an inverse inequality, and relations between the Fourier–Zernike expansions of a function and its derivatives.
PubDate: 2017-08-01
DOI: 10.1007/s00365-016-9358-y
Issue No: Vol. 46, No. 1 (2017)

• Bounds on Order of Indeterminate Moment Sequences
• Authors: Raphael Pruckner; Roman Romanov; Harald Woracek
Pages: 199 - 225
Abstract: Abstract We investigate the order $$\rho$$ of the four entire functions in the Nevanlinna matrix of an indeterminate Hamburger moment sequence. We give an upper estimate for $$\rho$$ which is explicit in terms of the parameters of the canonical system associated with the moment sequence via its three-term recurrence. Under a weak regularity assumption, this estimate coincides with a lower estimate, and hence $$\rho$$ becomes computable. Dropping the regularity assumption leads to examples where upper and lower bounds do not coincide and differ from the order. In particular, we provide examples for which the order is different from its lower estimate due to M.S. Livšic.
PubDate: 2017-08-01
DOI: 10.1007/s00365-016-9351-5
Issue No: Vol. 46, No. 1 (2017)

• On Uniform Convergence of Diagonal Multipoint Padé Approximants for
Entire Functions
• Authors: D. S. Lubinsky
Abstract: Abstract We prove that for most entire functions f in the sense of category, a strong form of the Baker–Gammel–Wills conjecture holds. More precisely, there is an infinite sequence $${\mathcal {S}}$$ of positive integers n, such that given any $$r>0$$ , and multipoint Padé approximants $$R_{n}$$ to f with interpolation points in $$\left\{ z:\left z\right \le r\right\}$$ , $$\left\{ R_{n}\right\} _{n\in S}$$ converges locally uniformly to f in the plane. The sequence $${\mathcal {S}}$$ does not depend on r, or on the interpolation points. For entire functions with smooth rapidly decreasing coefficients, full diagonal sequences of multipoint Padé approximants converge.
PubDate: 2017-09-05
DOI: 10.1007/s00365-017-9391-5

• Singular Values for Products of Two Coupled Random Matrices: Hard Edge
Phase Transition
• Authors: Dang-Zheng Liu
Abstract: Abstract Consider the product GX of two rectangular complex random matrices coupled by a constant matrix $$\Omega$$ , where G can be thought to be a Gaussian matrix and X is a bi-invariant polynomial ensemble. We prove that the squared singular values form a biorthogonal ensemble in Borodin’s sense, and further that for X being Gaussian, the correlation kernel can be expressed as a double contour integral. When all but finitely many eigenvalues of $$\Omega ^{} \Omega ^{*}$$ are equal, the corresponding correlation kernel is shown to admit a phase transition phenomenon at the hard edge in four different regimes as the coupling matrix changes. Specifically, the four limiting kernels in turn are the Meijer G-kernel for products of two independent Gaussian matrices, a new critical and interpolating kernel, the perturbed Bessel kernel, and the finite coupled product kernel associated with GX. In the special case when X is also a Gaussian matrix and $$\Omega$$ is scalar, such a product has been recently investigated by Akemann and Strahov. We also propose a Jacobi-type product and prove the same transition.
PubDate: 2017-08-22
DOI: 10.1007/s00365-017-9389-z

• Metric Entropy of Classes of Sets with Positive Reach
• Authors: James Cockreham; Fuchang Gao
Abstract: Abstract Sharp metric entropy estimates under Hausdorff distance are obtained for classes of bounded sets with positive reach, extending a well-known result of Bronshtein for the class of bounded convex sets.
PubDate: 2017-08-18
DOI: 10.1007/s00365-017-9388-0

• Norm Estimates for the Bergman and Cauchy–Szegö Projections Over the
Siegel Upper Half-Space
• Authors: Congwen Liu
Abstract: Abstract We obtain several estimates for the $$L^p$$ operator norms of the Bergman and Cauchy–Szegö projections over the the Siegel upper half-space. As a by-product, we also determine the precise value of the $$L^p$$ operator norm of a family of integral operators over the Siegel upper half-space.
PubDate: 2017-08-18
DOI: 10.1007/s00365-017-9390-6

• Entropy of Convex Functions on $$\mathbb {R}^d$$ R d
• Authors: Fuchang Gao; Jon A. Wellner
Abstract: Abstract Let $$\varOmega$$ be a bounded closed convex set in $$\mathbb {R}^d$$ with nonempty interior, and let $${\mathcal C}_r(\varOmega )$$ be the class of convex functions on $$\varOmega$$ with $$L^r$$ -norm bounded by 1. We obtain sharp estimates of the $$\varepsilon$$ -entropy of $${\mathcal C}_r(\varOmega )$$ under $$L^p(\varOmega )$$ metrics, $$1\le p<r\le \infty$$ . In particular, the results imply that the universal lower bound $$\varepsilon ^{-d/2}$$ is also an upper bound for all d-polytopes, and the universal upper bound of $$\varepsilon ^{-\frac{(d-1)}{2}\cdot \frac{pr}{r-p}}$$ for $$p>\frac{dr}{d+(d-1)r}$$ is attained by the closed unit ball. While a general convex body can be approximated by inscribed polytopes, the entropy rate does not carry over to the limiting body. Our results have applications to questions concerning rates of convergence of nonparametric estimators of high-dimensional shape-constrained functions.
PubDate: 2017-08-17
DOI: 10.1007/s00365-017-9387-1

• Matrix-Valued Gegenbauer-Type Polynomials
• Authors: Erik Koelink; Ana M. de los Ríos; Pablo Román
Abstract: Abstract We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter $$\nu >0$$ . The LDU-decomposition of the weight is explicitly given in terms of Gegenbauer polynomials. We establish a matrix-valued Pearson equation for these matrix weights leading to explicit shift operators relating the weights for parameters $$\nu$$ and $$\nu +1$$ . The matrix coefficients of the Pearson equation are obtained using a special matrix-valued differential operator in a commutative algebra of symmetric differential operators. The corresponding orthogonal polynomials are the matrix-valued Gegenbauer-type polynomials which are eigenfunctions of the symmetric matrix-valued differential operators. Using the shift operators, we find the squared norm, and we establish a simple Rodrigues formula. The three-term recurrence relation is obtained explicitly using the shift operators as well. We give an explicit nontrivial expression for the matrix entries of the matrix-valued Gegenbauer-type polynomials in terms of scalar-valued Gegenbauer and Racah polynomials using the LDU-decomposition and differential operators. The case $$\nu =1$$ reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations.
PubDate: 2017-07-20
DOI: 10.1007/s00365-017-9384-4

• Some Properties of the Derivatives on Sierpinski Gasket Type Fractals
• Authors: Shiping Cao; Hua Qiu
Abstract: Abstract In this paper, we focus on Strichartz’s derivatives, a family of derivatives including the normal derivative, on post critically finite fractals, which are defined at vertices in the graphs that approximate the fractal. We obtain a weak continuity property of the derivatives for functions in the domain of the Laplacian. For a function with zero normal derivative at any fixed vertex, the derivatives, including the normal derivatives, of the neighboring vertices will decay to zero. The rates of approximations are described, and several nontrivial examples are provided to illustrate that our estimates are optimal. We also study the boundedness property of derivatives for functions in the domain of the Laplacian. A necessary condition for a function having a weak tangent of order one at a vertex is provided. Furthermore, we give a counterexample of a conjecture of Strichartz on the existence of higher-order weak tangents.
PubDate: 2017-07-17
DOI: 10.1007/s00365-017-9385-3

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