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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  [2352 journals]
• Lower Bounds for Haar Projections: Deterministic Examples
• Authors: Andreas Seeger; Tino Ullrich
Pages: 227 - 242
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: 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: 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: 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: 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: 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)

• Meromorphic Solutions of Algebraic Difference Equations
• Authors: Katsuya Ishizaki; Risto Korhonen
Abstract: It is shown that the difference equation 1 \begin{aligned} ({\varDelta }f(z))^2=A(z)(f(z)f(z+1)-B(z)), \end{aligned} where A(z) and B(z) are meromorphic functions, possesses a continuous limit to the differential equation 2 \begin{aligned} (w')^2=A(z)(w^2-1), \end{aligned} which extends to solutions in certain cases. In addition, if (1) possesses two distinct transcendental meromorphic solutions, it is shown that these solutions satisfy an algebraic relation, and that their growth behaviors are almost the same in the sense of Nevanlinna under some conditions. Examples are given to discuss the sharpness of the results obtained. These properties are counterparts of the corresponding results on the algebraic differential equation (2).
PubDate: 2017-10-26
DOI: 10.1007/s00365-017-9401-7

• Conditional Quasi-Greedy Bases in Non-superreflexive Banach Spaces
• Authors: Fernando Albiac; José L. Ansorena; Przemysław Wojtaszczyk
Abstract: For a conditional quasi-greedy basis $$\mathcal {B}$$ in a Banach space, the associated conditionality constants $$k_{m}[\mathcal {B}]$$ verify the estimate $$k_{m}[\mathcal {B}]={\mathcal {O}}(\log m)$$ . Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-greedy bases in some special class of spaces. It is known that every quasi-greedy basis in a superreflexive Banach space verifies $$k_{m}[\mathcal {B}]=\mathcal O((\log m)^{1-\epsilon })$$ for some $$0<\epsilon <1$$ , and this is optimal. Our first goal in this paper will be to fill the gap between the general case and the superreflexive case and investigate the growth of the conditionality constants in nonsuperreflexive spaces. Roughly speaking, the moral will be that we can guarantee optimal bounds only for quasi-greedy bases in superreflexive spaces. We prove that if a Banach space $$\mathbb {X}$$ is not superreflexive, then there is a quasi-greedy basis $$\mathcal {B}$$ in a Banach space $$\mathbb {Y}$$ finitely representable in $$\mathbb {X}$$ with $$k_{m}[\mathcal {B}] \approx \log m$$ . As a consequence, we obtain that for every $$2<q<\infty$$ , there is a Banach space $$\mathbb {X}$$ of type 2 and cotype q possessing a quasi-greedy basis $$\mathcal {B}$$ with $$k_{m}[\mathcal {B}] \approx \log m$$ . We also tackle the corresponding problem for Schauder bases and show that if a space is nonsuperreflexive, then it possesses a basic sequence $$\mathcal {B}$$ with $$k_m[\mathcal {B}]\approx m$$ .
PubDate: 2017-10-26
DOI: 10.1007/s00365-017-9399-x

• Improved Lower Bound for the Mahler Measure of the Fekete Polynomials
• Authors: Tamás Erdélyi
Abstract: We show that there is an absolute constant $$c > 1/2$$ such that the Mahler measure of the Fekete polynomials $$f_p$$ of the form \begin{aligned} f_p(z) := \sum _{k=1}^{p-1}{\left( \frac{k}{p} \right) z^k} \end{aligned} (where the coefficients are the usual Legendre symbols) is at least $$c\sqrt{p}$$ for all sufficiently large primes p. This improves the lower bound $$\left( \frac{1}{2} - \varepsilon \right) \sqrt{p}$$ known before for the Mahler measure of the Fekete polynomials $$f_p$$ for all sufficiently large primes $$p \ge c_{\varepsilon }$$ . Our approach is based on the study of the zeros of the Fekete polynomials on the unit circle.
PubDate: 2017-10-25
DOI: 10.1007/s00365-017-9398-y

• Quantum Ergodic Sequences and Equilibrium Measures
• Authors: Steve Zelditch
Abstract: We generalize the definition of a “quantum ergodic sequence” of sections of ample line bundles $$L \rightarrow M$$ from the case of positively curved Hermitian metrics h on L to general smooth metrics. A choice of smooth Hermitian metric h on L and a Bernstein–Markov measure $$\nu$$ on M induces an inner product on $$H^0(M, L^N)$$ . When $$s_N _{L^2} =1$$ , quantum ergodicity is the condition that $$s_N(z) ^2 d\nu \rightarrow d\mu _{\varphi _{eq}}$$ weakly, where $$d\mu _{\varphi _{eq}}$$ is the equilibrium measure associated with $$(h, \nu )$$ . The main results are that normalized logarithms $$\frac{1}{N} \log s_N ^2$$ of quantum ergodic sections tend to the equilibrium potential, and that random orthonormal bases of $$H^0(M, L^N)$$ are quantum ergodic.
PubDate: 2017-10-24
DOI: 10.1007/s00365-017-9397-z

• Equidistribution of Jellium Energy for Coulomb and Riesz Interactions
• Authors: Mircea Petrache; Simona Rota Nodari
Abstract: For general dimension d, we prove the equidistribution of energy at the micro-scale in $$\mathbb {R}^{d}$$ for the optimal point configurations appearing in Coulomb gases at zero temperature. More precisely, we show that, after blow-up at the scale corresponding to the interparticle distance, the value of the energy in any large enough set is completely determined by the macroscopic density of points. This uses the “jellium energy” which was previously shown to control the next-order term in the large particle number asymptotics of the minimum energy. As a corollary, we obtain sharp error bounds on the discrepancy between the number of points and its expected average of optimal point configurations for Coulomb gases, extending previous results valid only for 2-dimensional log-gases. For Riesz gases with interaction potentials $$g(x)= x ^{-s}, s\in ]\min \{0,d-2\},d[$$ , we prove the same equidistribution result under an extra hypothesis on the decay of the localized energy, which we conjecture to hold for minimizing configurations. In this case, we use the Caffarelli–Silvestre description of the nonlocal fractional Laplacians in $$\mathbb {R}^{d}$$ to render the problem local.
PubDate: 2017-10-24
DOI: 10.1007/s00365-017-9395-1

• A Large Deviation Principle for Weighted Riesz Interactions
• Authors: Tom Bloom; Norman Levenberg; Franck Wielonsky
Abstract: We prove a large deviation principle for the sequence of push-forwards of empirical measures in the setting of Riesz potential interactions on compact subsets K in $$\mathbb {R}^d$$ with continuous external fields. Our results are valid for base measures on K satisfying a strong Bernstein–Markov type property for Riesz potentials. Furthermore, we give sufficient conditions on K (which are satisfied if K is a smooth submanifold) so that a measure on K that satisfies a mass-density condition will also satisfy this strong Bernstein–Markov property.
PubDate: 2017-10-23
DOI: 10.1007/s00365-017-9396-0

• Unique Solvability of a Coupling Problem for Entire Functions
• Authors: Jonathan Eckhardt
Abstract: We establish the unique solvability of a coupling problem for entire functions that arises in inverse spectral theory for singular second-order ordinary differential equations/two-dimensional first-order systems and is also of relevance for the integration of certain nonlinear wave equations.
PubDate: 2017-10-13
DOI: 10.1007/s00365-017-9394-2

• Rahman’s Biorthogonal Rational Functions and Superconformal Indices
• Authors: Hjalmar Rosengren
Abstract: We study biorthogonal functions related to basic hypergeometric integrals with coupled continuous and discrete components. Such integrals appear as superconformal indices for three-dimensional quantum field theories and also in the context of solvable lattice models. We obtain explicit biorthogonal systems given by products of two of Rahman’s biorthogonal rational $${}_{10}W_9$$ -functions or their degenerate cases. We also give new bilateral extensions of the Jackson and q-Saalschütz summation formulas and new continuous and discrete biorthogonality measures for Rahman’s functions.
PubDate: 2017-09-25
DOI: 10.1007/s00365-017-9393-3

• Complex Interpolation of Smoothness Morrey Subspaces
• Authors: Denny Ivanal Hakim; Shohei Nakamura; Yoshihiro Sawano
Abstract: Recently more and more attention has been paid to subspaces of Morrey spaces. The description of interpolation results for many of these spaces is found. However, the ones for smoothness Morrey subspaces are missing. The aim of this paper is to describe the output by the first and the second complex interpolations of these smoothness Morrey subspaces.
PubDate: 2017-09-20
DOI: 10.1007/s00365-017-9392-4

• On Uniform Convergence of Diagonal Multipoint Padé Approximants for
Entire Functions
• Authors: D. S. Lubinsky
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: 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: 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: 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

• Some Properties of the Derivatives on Sierpinski Gasket Type Fractals
• Authors: Shiping Cao; Hua Qiu
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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