Authors:Jesús Carnicer; Tomas Sauer Pages: 373 - 389 Abstract: In contrast to the univariate case, interpolation with polynomials of a given maximal total degree is not always possible even if the number of interpolation points and the space dimension coincide. Due to that, numerous constructions for interpolation sets have been devised, the most popular ones being based on intersections of lines. In this paper, we study algebraic properties of some such interpolation configurations, namely the approaches by Radon–Berzolari and Chung–Yao. By means of proper H-bases for the vanishing ideal of the configuration, we derive properties of the matrix of first syzygies of this ideal that allow us to draw conclusions on the geometry of the point configuration. PubDate: 2018-06-01 DOI: 10.1007/s00365-017-9380-8 Issue No:Vol. 47, No. 3 (2018)

Authors:Byoung Jin Choi; Un Cig Ji; Yongdo Lim Pages: 391 - 405 Abstract: We establish the asymptotic regularity and the \(\varDelta \) -convergence of the sequence constructed by alternating projections onto closed convex sets in a \(\mathrm{CAT}(\kappa )\) space with \(\kappa > 0\) . Furthermore, the strong convergence of the alternating von Neumann sequence is presented under certain regularity or compactness conditons. PubDate: 2018-06-01 DOI: 10.1007/s00365-017-9382-6 Issue No:Vol. 47, No. 3 (2018)

Authors:Árpád Baricz; Tivadar Danka Pages: 407 - 435 Abstract: In this paper, we study the local zero behavior of orthogonal polynomials around an algebraic singularity, that is, when the measure of orthogonality is supported on \( [-1,1] \) and behaves like \( h(x) x - x_0 ^\lambda dx \) for some \( x_0 \in (-1,1) \) , where h(x) is strictly positive and analytic. We shall sharpen the theorem of Yoram Last and Barry Simon and show that the so-called fine zero spacing (which is known for \( \lambda = 0\) ) unravels in the general case, and the asymptotic behavior of neighbouring zeros around the singularity can be described with the zeros of the function \( c J_{\frac{\lambda - 1}{2}}(x) + d J_{\frac{\lambda + 1}{2}}(x) \) , where \( J_a(x) \) denotes the Bessel function of the first kind and order a. Moreover, using Sturm–Liouville theory, we study the behavior of this linear combination of Bessel functions, thus providing estimates for the zeros in question. PubDate: 2018-06-01 DOI: 10.1007/s00365-017-9411-5 Issue No:Vol. 47, No. 3 (2018)

Authors:Vladimir V. Andrievskii Pages: 437 - 452 Abstract: We establish the exact (up to the constants) double inequality for the Christoffel function for a measure supported on a Jordan domain bounded by a quasiconformal curve. We show that this quasiconformality of the boundary cannot be omitted. PubDate: 2018-06-01 DOI: 10.1007/s00365-017-9404-4 Issue No:Vol. 47, No. 3 (2018)

Authors:Fernando Cobos; Óscar Domínguez; Thomas Kühn Pages: 453 - 486 Abstract: We consider general linear approximation spaces \(X^b_q\) based on a quasi-Banach space X, and we analyze the degree of compactness of the embedding \(X^b_q \hookrightarrow X\) . Applications are given to periodic Besov spaces on the d-torus, including spaces of generalized and logarithmic smoothness. In particular, we obtain the exact asymptotic behavior of approximation and entropy numbers of embeddings of such Besov spaces in Lebesgue spaces and in Besov spaces of logarithmic smoothness. PubDate: 2018-06-01 DOI: 10.1007/s00365-017-9383-5 Issue No:Vol. 47, No. 3 (2018)

Authors:Dang-Zheng Liu Pages: 487 - 528 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: 2018-06-01 DOI: 10.1007/s00365-017-9389-z Issue No:Vol. 47, No. 3 (2018)

Authors:Hjalmar Rosengren Pages: 529 - 552 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: 2018-06-01 DOI: 10.1007/s00365-017-9393-3 Issue No:Vol. 47, No. 3 (2018)

Authors:Ronald DeVore; Simon Foucart; Guergana Petrova; Przemyslaw Wojtaszczyk Abstract: Scientific problems often feature observational data received in the form \(w_1=l_1(f),\ldots \) , \(w_m=l_m(f)\) of known linear functionals applied to an unknown function f from some Banach space \(\mathcal {X}\) , and it is required to either approximate f (the full approximation problem) or to estimate a quantity of interest Q(f). In typical examples, the quantities of interest can be the maximum/minimum of f or some averaged quantity such as the integral of f, while the observational data consists of point evaluations. To obtain meaningful results about such problems, it is necessary to possess additional information about f, usually as an assumption that f belongs to a certain model class \(\mathcal {K}\) contained in \(\mathcal {X}\) . This is precisely the framework of optimal recovery, which produced substantial investigations when the model class is a ball in a smoothness space, e.g., when it is a unit ball in Lipschitz, Sobolev, or Besov spaces. This paper is concerned with other model classes described by approximation processes, as studied in DeVore et al. [Data assimilation in Banach spaces, (To Appear)]. Its main contributions are: (1) designing implementable optimal or near-optimal algorithms for the estimation of quantities of interest, (2) constructing linear optimal or near-optimal algorithms for the full approximation of an unknown function using its point evaluations. While the existence of linear optimal algorithms for the approximation of linear functionals Q(f) is a classical result established by Smolyak, a numerically friendly procedure that performs this approximation is not generally available. In this paper, we show that in classical recovery settings, such linear optimal algorithms can be produced by constrained minimization methods. We illustrate these techniques on several examples involving the computation of integrals using point evaluation data. In addition, we show that linearization of optimal algorithms can be achieved for the full approximation problem in the important situation where the \(l_j\) are point evaluations and \(\mathcal {X}\) is a space of continuous functions equipped with the uniform norm. It is also revealed how quasi-interpolation theory enables the construction of linear near-optimal algorithms for the recovery of the underlying function. PubDate: 2018-06-01 DOI: 10.1007/s00365-018-9433-7

Authors:Daniel Pohl; Oliver Roth Abstract: We prove Runge-type theorems and universality results for locally univalent holomorphic and meromorphic functions. Refining a result of M. Heins, we also show that there is a universal bounded locally univalent function on the unit disk. These results are used to prove that on any hyperbolic simply connected plane domain there exist universal conformal metrics with prescribed constant curvature. PubDate: 2018-05-31 DOI: 10.1007/s00365-018-9434-6

Authors:Douglas P. Hardin; Thomas Leblé; Edward B. Saff; Sylvia Serfaty Abstract: We study N-particle systems in \(\mathbb {R}^d\) whose interactions are governed by a hypersingular Riesz potential \( x-y ^{-s}\) , \(s>d\) , and subject to an external field. We provide both macroscopic results as well as microscopic results in the limit as \(N\rightarrow \infty \) for random point configurations with respect to the associated Gibbs measure at scaled inverse temperature \(\beta \) . We show that a large deviation principle holds with a rate function of the form ‘ \(\beta \) -Energy + Entropy’, yielding that the microscopic behavior (on the scale \(N^{-1/d}\) ) of such N-point systems is asymptotically determined by the minimizers of this rate function. In contrast to the asymptotic behavior in the integrable case \(s<d\) , where on the macroscopic scale N-point empirical measures have limiting density independent of \(\beta \) , the limiting density for \(s>d\) is strongly \(\beta \) -dependent. PubDate: 2018-05-17 DOI: 10.1007/s00365-018-9431-9

Authors:Johann S. Brauchart; Peter J. Grabner; Wöden Kusner Abstract: The notion of hyperuniformity originally introduced as a measure of regularity of infinite point sets in Euclidean space is generalized and extended to sequences of finite point sets on the sphere. It is shown that hyperuniformity implies uniform distribution. Furthermore, it is shown that Quasi-Monte Carlo design sequences with strength at least \(\frac{d+1}{2}\) and especially sequences of spherical designs of optimal growth order are hyperuniform. PubDate: 2018-05-08 DOI: 10.1007/s00365-018-9432-8

Authors:Ryszard Szwarc Abstract: We study Jacobi matrices on trees with one end at infinity. We show that the defect indices cannot be greater than 1 and give criteria for essential self-adjointness. PubDate: 2018-05-07 DOI: 10.1007/s00365-018-9430-x

Authors:Antonio R. Vargas Abstract: We show that the partial sums of the power series for a certain class of entire functions possess scaling limits in various directions in the complex plane. In doing so, we obtain information about the zeros of the partial sums. We will only assume that these entire functions have a certain asymptotic behavior at infinity. With this information, we will partially verify for this class of functions a conjecture on the location of the zeros of their partial sums known as the Saff–Varga width conjecture. Numerical results and figures are included to illustrate the results obtained for several well-known functions including the Airy functions and the parabolic cylinder functions. PubDate: 2018-04-13 DOI: 10.1007/s00365-018-9422-x

Authors:Anne-Maria Ernvall-Hytönen; Tapani Matala-aho; Louna Seppälä Abstract: We present a completely explicit transcendence measure for e. This is a continuation and an improvement to the works of Borel, Mahler, and Hata on the topic. Furthermore, we also prove a transcendence measure for an arbitrary positive integer power of e. The results are based on Hermite–Padé approximations and on careful analysis of common factors in the footsteps of Hata. PubDate: 2018-04-09 DOI: 10.1007/s00365-018-9429-3

Authors:David Krieg Abstract: We construct Monte Carlo methods for the \(L^2\) -approximation in Hilbert spaces of multivariate functions sampling not more than n function values of the target function. Their errors catch up with the rate of convergence and the preasymptotic behavior of the error of any algorithm sampling n pieces of arbitrary linear information, including function values. PubDate: 2018-04-06 DOI: 10.1007/s00365-018-9428-4

Authors:Carlos Beltrán; Ujué Etayo Abstract: We consider a determinantal point process on the complex projective space that reduces to the so-called spherical ensemble for complex dimension 1 under identification of the 2-sphere with the Riemann sphere. Through this determinantal point process, we propose a new point processs in odd-dimensional spheres that produces fairly well-distributed points, in the sense that the expected value of the Riesz 2-energy for these collections of points is smaller than all previously known bounds. PubDate: 2018-04-04 DOI: 10.1007/s00365-018-9426-6

Authors:Michael S. Floater; Espen Sande Abstract: In this paper we show that, with respect to the \(L^2\) norm, three classes of functions in \(H^r(0,1)\) , defined by certain boundary conditions, admit optimal spline spaces of all degrees \(\ge r-1\) , and all these spline spaces have uniform knots. PubDate: 2018-04-04 DOI: 10.1007/s00365-018-9427-5

Authors:Yacin Ameur; Nam-Gyu Kang; Nikolai Makarov Abstract: We study spacing distribution for the eigenvalues of a random normal matrix, in particular at points on the boundary of the spectrum. Our approach uses Ward’s (or the “rescaled loop”) equation—an identity satisfied by all sequential limits of the rescaled one-point functions. PubDate: 2018-04-02 DOI: 10.1007/s00365-018-9423-9

Authors:A. M. Ishkhanyan Abstract: Starting from a second-order Fuchsian differential equation having five regular singular points, an equation obeyed by a function proportional to the first derivative of the solution of the Heun equation, we construct several expansions of the solutions of the general Heun equation in terms of Appell generalized hypergeometric functions of two variables of the first kind. Several cases when the expansions reduce to those written in terms of simpler mathematical functions such as the incomplete Beta function or the Gauss hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed. In general, the coefficients of the expansions obey four-term recurrence relations; however, there exist certain choices of parameters for which the recurrence relations involve only two terms, though not necessarily successive. For such cases, the coefficients of the expansions are explicitly calculated and the general solution of the Heun equation is constructed in terms of the Gauss hypergeometric functions. PubDate: 2018-04-02 DOI: 10.1007/s00365-018-9424-8

Authors:George Costakis; Andreas Jung; Jürgen Müller Abstract: It is known that, generically, Taylor series of functions holomorphic in the unit disk turn out to be “maximally divergent” outside of the disk. For functions in classical Banach spaces of holomorphic functions, as for example, Hardy spaces or the disk algebra, the situation is more delicate. In this paper, it is shown that the behavior of the partial sums on sets outside the open unit disk sensitively depends on the way the sets touch the unit circle. As main tools, results on simultaneous approximation by polynomials are proved. PubDate: 2018-04-02 DOI: 10.1007/s00365-018-9425-7