Authors:Yacin Ameur; Seong-Mi Seo Pages: 3 - 37 Abstract: Abstract We extend the method of rescaled Ward identities of Ameur, Kang, and Makarov to study the distribution of eigenvalues close to a bulk singularity, i.e., a point in the interior of the droplet where the density of the classical equilibrium measure vanishes. We prove results to the effect that a certain “dominant part” of the Taylor expansion determines the microscopic properties near a bulk singularity. A description of the distribution is given in terms of the Bergman kernel of a certain Fock-type space of entire functions. PubDate: 2018-02-01 DOI: 10.1007/s00365-017-9368-4 Issue No:Vol. 47, No. 1 (2018)

Authors:Laurent Bétermin; Etienne Sandier Pages: 39 - 74 Abstract: Abstract We study the Hamiltonian of a two-dimensional log-gas with a confining potential V satisfying the weak growth assumption—V is of the same order as \(2\log \Vert x\Vert \) near infinity—considered by Hardy and Kuijlaars [J Approx Theory 170:44–58, 2013]. We prove an asymptotic expansion, as the number n of points goes to infinity, for the minimum of this Hamiltonian using the gamma-convergence method of Sandier and Serfaty [Ann Probab 43(4):2026–2083, 2015]. We show that the asymptotic expansion as \(n\rightarrow +\infty \) of the minimal logarithmic energy of n points on the unit sphere in \(\mathbb {R}^3\) has a term of order n, thus proving a long-standing conjecture of Rakhmanov et al. [Math Res Lett 1:647–662, 1994]. Finally, we prove the equivalence between the conjecture of Brauchart Brauchart, Hardin and Saff [Contemp. Math., 578:31–61, 2012] about the value of this term and the conjecture of Sandier and Serfaty [Commun Math Phys. 313(3):635–743, 2012] about the minimality of the triangular lattice for a “renormalized energy” W among configurations of fixed asymptotic density. PubDate: 2018-02-01 DOI: 10.1007/s00365-016-9357-z Issue No:Vol. 47, No. 1 (2018)

Authors:Jordi Marzo; Joaquim Ortega-Cerdà Pages: 75 - 88 Abstract: Abstract We compute the expected Riesz energy of random points on flat tori drawn from certain translation invariant determinantal processes and determine the process in the family providing the optimal asymptotic expected Riesz energy. PubDate: 2018-02-01 DOI: 10.1007/s00365-017-9386-2 Issue No:Vol. 47, No. 1 (2018)

Authors:Tom Bloom; Norman Levenberg; Franck Wielonsky Pages: 119 - 140 Abstract: 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: 2018-02-01 DOI: 10.1007/s00365-017-9396-0 Issue No:Vol. 47, No. 1 (2018)

Authors:Mircea Petrache; Simona Rota Nodari Pages: 163 - 210 Abstract: 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: 2018-02-01 DOI: 10.1007/s00365-017-9395-1 Issue No:Vol. 47, No. 1 (2018)

Authors:Árpád Baricz; Tivadar Danka Abstract: 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: 2018-01-11 DOI: 10.1007/s00365-017-9411-5

Authors:Dmitriy Bilyk; Feng Dai; Ryan Matzke Abstract: Abstract The classical Stolarsky invariance principle connects the spherical cap \(L^2\) discrepancy of a finite point set on the sphere to the pairwise sum of Euclidean distances between the points. In this paper, we further explore and extend this phenomenon. In addition to a new elementary proof of this fact, we establish several new analogs, which relate various notions of discrepancy to different discrete energies. In particular, we find that the hemisphere discrepancy is related to the sum of geodesic distances. We also extend these results to arbitrary measures on the sphere and arbitrary notions of discrepancy and apply them to problems of energy optimization and combinatorial geometry and find that, surprisingly, the geodesic distance energy behaves differently than its Euclidean counterpart. PubDate: 2018-01-09 DOI: 10.1007/s00365-017-9412-4

Authors:Albert Borbély; Michael J. Johnson Abstract: Abstract Given points \(P_1,P_2,\ldots ,P_m\) in the complex plane, we are concerned with the problem of finding an interpolating curve with minimal bending energy (i.e., an optimal interpolating curve). It was shown previously that existence is assured if one requires that the pieces of the interpolating curve be s-curves. In the present article, we also impose the restriction that these s-curves have chord angles not exceeding \(\pi /2\) in magnitude. With this setup, we have identified a sufficient condition for the curvature continuity of optimal interpolating curves. This sufficient condition relates to the stencil angles \(\{\psi _j\}\) , where \(\psi _j\) is defined as the angular change in direction from segment \([P_{j-1},P_j]\) to segment \([P_j,P_{j+1}]\) . An angle \(\Psi \) ( \(\approx 37^\circ \) ) is identified, and we show that if the stencil angles satisfy \(\vert {\psi _j} \vert <\Psi \) , then optimal interpolating curves are curvature continuous. We also prove that the angle \(\Psi \) is sharp. As with the previous article (Borbély and Johnson in Constr Approx 40:189–218, 2014), much of our effort is concerned with the geometric Hermite interpolation problem of finding an optimal s-curve \(c_1(\alpha ,\beta )\) that connects \(0+i0\) to \(1+i0\) with prescribed chord angles \((\alpha ,\beta )\) . Whereas existence was previously shown, and sometimes uniqueness, the present article begins by establishing uniqueness when \(\vert {\alpha } \vert ,\vert {\beta } \vert \le \pi /2\) and \(\vert {\alpha -\beta } \vert <\pi \) . We also prove two fundamental identities involving the initial and terminal signed curvatures of \(c_1(\alpha ,\beta )\) and partial derivatives, with respect to \(\alpha \) or \(\beta \) , of the bending energy of \(c_1(\alpha ,\beta )\) . PubDate: 2018-01-09 DOI: 10.1007/s00365-017-9414-2

Authors:Pavel Gumenyuk Pages: 435 - 458 Abstract: Abstract A classical result in the theory of Loewner’s parametric representation states that the semigroup \({{\mathfrak {U}}}_0\) of all conformal self-maps \(\varphi \) of the unit disk \({\mathbb {D}}\) normalized by \(\varphi (0) = 0\) and \(\varphi '(0) > 0\) can be obtained as the reachable set of the Loewner–Kufarev control system $$\begin{aligned} \frac{\mathrm {d}w_t}{\mathrm {d}t}=G_t\circ w_t,\quad t\geqslant 0,\qquad w_0=\mathsf{id}_{\mathbb {D}}, \end{aligned}$$ where the control functions \(t\mapsto G_t\in \mathsf{Hol}({\mathbb {D}},{\mathbb {C}})\) form a convex cone. We extend this result to semigroups \({{\mathfrak {U}}}[F]\) formed by all conformal self-maps of \({\mathbb {D}}\) with the prescribed finite set F of boundary regular fixed points and to their counterparts \({{\mathfrak {U}}}_{\tau }[F]\) for the case of self-maps having the Denjoy–Wolff point at \(\tau \in {\overline{{\mathbb {D}}}}{\setminus } F\) . PubDate: 2017-12-01 DOI: 10.1007/s00365-017-9376-4 Issue No:Vol. 46, No. 3 (2017)

Authors:Erik Koelink; Ana M. de los Ríos; Pablo Román Pages: 459 - 487 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: 2017-12-01 DOI: 10.1007/s00365-017-9384-4 Issue No:Vol. 46, No. 3 (2017)

Authors:V. Temlyakov; S. Tikhonov Pages: 593 - 615 Abstract: Abstract In this paper, we study the Remez-type and Nikol’skii-type inequalities for trigonometric polynomials with harmonics from hyperbolic crosses. The interrelation between these inequalities for individual functions and its applications are discussed. PubDate: 2017-12-01 DOI: 10.1007/s00365-017-9370-x Issue No:Vol. 46, No. 3 (2017)

Authors:T. M. Dunster; A. Gil; J. Segura Pages: 645 - 675 Abstract: Abstract Linear second-order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function and its derivative, along with two infinite series, the coefficients of which are usually difficult to compute. By considering the series as asymptotic expansions for two explicitly defined analytic functions, Cauchy’s integral formula is employed to compute the coefficient functions to a high order of accuracy. The method employs a certain exponential form of Liouville–Green expansions for solutions of the differential equation, as well as for the Airy function. We illustrate the use of the method with the high accuracy computation of Airy-type expansions of Bessel functions of complex argument. PubDate: 2017-12-01 DOI: 10.1007/s00365-017-9372-8 Issue No:Vol. 46, No. 3 (2017)

Authors:Adrien Hardy Abstract: Abstract Polynomial ensembles are determinantal point processes associated with (not necessarily orthogonal) projections onto polynomial subspaces. The aim of this survey article is to put forward the use of recurrence coefficients to obtain the global asymptotic behavior of such ensembles in a rather simple way. We provide a unified approach to recover well-known convergence results for real OP ensembles. We study the mutual convergence of the polynomial ensemble and the zeros of its average characteristic polynomial; we discuss in particular the complex setting. We also control the variance of linear statistics of polynomial ensembles and derive comparison results, as well as asymptotic formulas for real OP ensembles. Finally, we reinterpret the classical algorithm to sample determinantal point processes so as to cover the setting of nonorthogonal projection kernels. A few open problems are also suggested. PubDate: 2017-12-19 DOI: 10.1007/s00365-017-9413-3

Authors:Emil Horozov Abstract: Classical orthogonal polynomial systems of Jacobi, Hermite, Laguerre, and Bessel have the property that the polynomials of each system are eigenfunctions of a second-order ordinary differential operator. According to a classical theorem by Bochner, they are the only systems with this property. Similarly, the polynomials of Charlier, Meixner, Kravchuk, and Hahn are both orthogonal and are eigenfunctions of a suitable difference operator of second order. We recall that according to the famous theorem of Favard–Shohat, the condition of orthogonality is equivalent to the 3-term recurrence relation. Vector orthogonal polynomials (VOP) satisfy finite-term recurrence relation with more terms, according to a theorem by J. Van Iseghem, and this characterizes them. Motivated by Bochner’s theorem, we are looking for VOP that are also eigenfunctions of a differential (difference) operator. We call these simultaneous conditions Bochner’s property. The goal of this paper is to introduce methods for construction of VOP which have Bochner’s property. The methods are purely algebraic and are based on automorphisms of noncommutative algebras. They also use ideas from the so-called bispectral problem. Applications of the abstract methods include broad generalizations of the classical orthogonal polynomials, both continuous and discrete. Other results connect different families of VOP, including the classical ones, by linear transforms of purely algebraic origin, despite of the fact that, when interpreted analytically, they are integral transformations. PubDate: 2017-12-18 DOI: 10.1007/s00365-017-9410-6

Authors:Boris Shapiro; František Štampach Abstract: Abstract We introduce and investigate a class of complex semi-infinite banded Toeplitz matrices satisfying the condition that the spectra of their principal submatrices accumulate onto a real interval when the size of the submatrix grows to \(\infty \) . We prove that a banded Toeplitz matrix belongs to this class if and only if its symbol has real values on a Jordan curve located in \({{\mathbb {C}}}{\setminus }\{0\}\) . Surprisingly, it turns out that, if such a Jordan curve is present, the spectra of all the principal submatrices have to be real. The latter claim is also proved for matrices given by a more general symbol. The special role of the Jordan curve is further demonstrated by a new formula for the limiting density of the asymptotic eigenvalue distribution for banded Toeplitz matrices from the studied class. Certain connections between the problem under investigation, Jacobi operators, and the Hamburger moment problem are also discussed. The main results are illustrated by several concrete examples; some of them allow an explicit analytic treatment, while some are only treated numerically. PubDate: 2017-12-07 DOI: 10.1007/s00365-017-9408-0

Authors:Janne Gröhn Abstract: Abstract Behavior of solutions of \(f''+Af=0\) is discussed under the assumption that A is analytic in \(\mathbb {D}\) and \(\sup _{z\in \mathbb {D}}(1- z ^2)^2 A(z) <\infty \) , where \(\mathbb {D}\) is the unit disc of the complex plane. As a main result, it is shown that such differential equation may admit a nontrivial solution whose zero-sequence does not satisfy the Blaschke condition. This gives an answer to an open question in the literature. It is also proved that \(\varLambda \subset \mathbb {D}\) is the zero-sequence of a nontrivial solution of \(f''+Af=0\) , where \( A(z) ^2(1- z ^2)^3\, \hbox {d}m(z)\) is a Carleson measure if and only if \(\varLambda \) is uniformly separated. As an application, an old result, according to which there exists a non-normal function that is uniformly locally univalent, is improved. PubDate: 2017-12-04 DOI: 10.1007/s00365-017-9409-z

Authors:Pavel Bleher; Karl Liechty Abstract: Abstract We obtain asymptotic formulas for the partition function of the six-vertex model with domain wall boundary conditions and half-turn symmetry in each of the phase regions. The proof is based on the Izergin–Korepin–Kuperberg determinantal formula for the partition function and its reduction to orthogonal polynomials, and on an asymptotic analysis of the orthogonal polynomials under consideration in the framework of the Riemann–Hilbert approach. PubDate: 2017-11-13 DOI: 10.1007/s00365-017-9405-3

Authors:Steve Zelditch Abstract: 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

Authors:Denny Ivanal Hakim; Shohei Nakamura; Yoshihiro Sawano Abstract: 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