Abstract: In this paper, we aim to replace in the definitions of covariance and correlation the usual trace Tr by a tracial positive map between unital \(C^*\) -algebras and to replace the functions \(x^{\alpha }\) and \(x^{1- \alpha }\) by functions f and g satisfying some mild conditions. These allow us to define the generalized covariance, the generalized variance, the generalized correlation and the generalized Wigner–Yanase–Dyson skew information related to the tracial positive maps and functions f and g. We persent a generalization of Heisenberg’s uncertainty relation in the noncommutative framework. We extend some inequalities and properties for the generalized correlation and the generalized Wigner–Yanase–Dyson skew information. Furthermore, we extend some inequalities for the generalized skew information such as uncertainty relation and the relation between the generalized variance and the generalized skew information. PubDate: 2019-05-15

Abstract: We extend a classical result about weighted averages of harmonic functions to solutions of second-order strongly elliptic systems of PDE with constant coefficients in disks in the complex plane. It is well known that a non-tangential cluster set of the (harmonic) Poisson integral with a given piecewise continuous boundary function f at every point \(\zeta \) in the unit circle is the segment joining the left- and right-hand side limits of f at \(\zeta \) being taken along the unit circle. Using the recently obtained Poisson-type integral representation formula for solutions of aforementioned systems, we establish an analogous result about weighted averages for solutions of such systems. Furthermore, we illustrate the nature of the obtained results by presenting some special mappings of the unit disk by solutions with piecewise constant boundary data. PubDate: 2019-05-13

Abstract: We prove local solvability and stability for the inverse problem of recovering a complex-valued square integrable potential in the Sturm–Liouville equation on a finite interval from spectra of two boundary value problems with one common boundary condition. For this purpose we generalize classical Borg’s method to the case of multiple spectra. PubDate: 2019-05-09

Abstract: We give an overview of some results on the class of functions with subharmonic behaviour and their invariance properties under conformal and quasiconformal mappings. While many of the results we present will be related to author’s own work, we shall present also some other results and examples about this class of functions. PubDate: 2019-05-09

Abstract: In this addendum to the paper On Bernstein’s inequality for polynomials [Anal. Math. Phys. online 20 March 2019], we rectify the beginning of Section 5 where we mentioned a proof of Mahler’s result, i.e. the case p=0 in Bernstein’s inequality, using subharmonicity. In particular, we take into account a reference that we previously missed, and that Paul Nevai, whom we thank, has very recently brought to our attention. PubDate: 2019-05-08

Abstract: We show that the positive and negative parts \( u_{k}^{\pm }\) of any frame in a real \( L^{2}\) space with respect to a continuous measure have both “infinite \( l^{2}\) masses”: (1) always, \( \sum _{k}u_{k}^{\pm }(x)^{2}=\infty \) almost everywhere (in particular, there exist no positive frames, nor Riesz bases), but (2) \( \sum _{k=1}^{n}(u_{k}^{+}(x)-u_{k}^{-}(x))^{2}\) can grow “locally” as slow as we wish (for \( n\longrightarrow \infty \) ), and (3) it can happen that \( \sum _{k=1}^{n}u_{k}^{-}(x)^{2}= o(\sum _{k=1}^{n}u_{k}^{+}(x)^{2})\) , and vice versa, as \( n\longrightarrow \infty \) on a set of positive measure. Property (1) for the case of an orthonormal basis in \( L^{2}(0,1)\) was settled earlier (V. Ya. Kozlov, 1948) using completely different (and more involved) arguments. Our elementary treatment includes also the case of unconditional bases in a variety of Banach spaces, as well as the case of complex valued spaces and frames. For property (2), we show that, moreover, whatever is a monotone sequence \( \epsilon _{k}>0\) satisfying \( \sum _{k}\epsilon ^{2}_{k}= \infty \) there exists an orthonormal basis \( (u_{k})_{k}\) in \( L^{2}\) such that \( \vert u_{k}(x)\vert \le A(x)\epsilon _{k}\) , \( 0<A(x)< \infty \) . PubDate: 2019-05-06

Abstract: In this article we give a survey on different methods to estimate the values of functionals in the coefficients of Bloch functions. PubDate: 2019-04-26

Abstract: A method for the decomposition of data functions sampled on a finite fragment of rectangular lattice is described. The symmetry of a square lattice in a 2-dimensional real Euclidean space is either given by the semisimple Lie group \(SU(2)\times SU(2)\) or equivalently by the Lie algebra \(A_1\times A_1\) , or by the simple Lie group O(5) or its Lie algebra called \(C_2\) or equivalently \(B_2\) . In this paper we consider the first of these possibilities which is applied to data which is given in 2 orthogonal directions—hence the method is a concatenation of two 1-dimensional cases. The asymmetry we underline here is a different density of discrete data points in the two orthogonal directions which cannot be studied with the simple Lie group symmetry. PubDate: 2019-04-26

Abstract: For \(b\in L_{\mathrm{loc}}({\mathbb {R}}^n)\) and \(0<\alpha <1\) , we use fractional differentiation to define a new type of commutator of the Littlewood-Paley g-function operator, namely $$\begin{aligned} g_{\Omega ,\alpha ;b}(f )(x) =\bigg (\int _0^\infty \bigg \frac{1}{t} \int _{ x-y \le t}\frac{\Omega (x-y)}{ x-y ^{n+\alpha -1}}(b(x)-b(y))f(y)\,dy\bigg ^2\frac{dt}{t}\bigg )^{1/2}. \end{aligned}$$ Here, we obtain the necessary and sufficient conditions for the function b to guarantee that \(g_{\Omega ,\alpha ;b}\) is a bounded operator on \(L^2({\mathbb {R}}^n)\) . More precisely, if \(\Omega \in L(\log ^+ L)^{1/2}{(S^{n-1})}\) and \(b\in I_{\alpha }(BMO)\) , then \(g_{\Omega ,\alpha ;b}\) is bounded on \(L^2({\mathbb {R}}^n)\) . Conversely, if \(g_{\Omega ,\alpha ;b}\) is bounded on \(L^2({\mathbb {R}}^n)\) , then \(b \in Lip_\alpha ({\mathbb {R}}^n)\) for \(0<\alpha < 1\) . PubDate: 2019-04-26

Abstract: The famous Koebe \(\frac{1}{4}\) theorem deals with univalent (i.e., injective) analytic functions f on the unit disk \({\mathbb {D}}\) . It states that if f is normalized so that \(f(0)=0\) and \(f'(0)=1\) , then the image \(f({\mathbb {D}})\) contains the disk of radius \(\frac{1}{4}\) about the origin, the value \(\frac{1}{4}\) being best possible. Now suppose f is only allowed to range over the univalent polynomials of some fixed degree. What is the optimal radius in the Koebe-type theorem that arises' And for which polynomials is it attained' A plausible conjecture is stated, and the case of small degrees is settled. PubDate: 2019-04-25

Abstract: We investigate the minimization of the energy per point \(E_f\) among d-dimensional Bravais lattices, depending on the choice of pairwise potential equal to a radially symmetric function \(f( x ^2)\) . We formulate criteria for minimality and non-minimality of some lattices for \(E_f\) at fixed scale based on the sign of the inverse Laplace transform of f when f is a superposition of exponentials, beyond the class of completely monotone functions. We also construct a family of non-completely monotone functions having the triangular lattice as the unique minimizer of \(E_f\) at any scale. For Lennard-Jones type potentials, we reduce the minimization problem among all Bravais lattices to a minimization over the smaller space of unit-density lattices and we establish a link to the maximum kissing problem. New numerical evidence for the optimality of particular lattices for all the exponents are also given. We finally design one-well potentials f such that the square lattice has lower energy \(E_f\) than the triangular one. Many open questions are also presented. PubDate: 2019-04-19

Abstract: Let X be a finite connected graph, possibly with multiple edges. We provide each edge of the graph by two possible orientations. An automorphism group of a graph acts harmonically if it acts freely on the set of directed edges of the graph. Following M. Baker and S. Norine define a genus g of the graph X to be the rank of the first homology group. A finite group acting harmonically on a graph of genus g is a natural discrete analogue of a finite group of automorphisms acting on a Riemann surface of genus g. In the present paper, we give a sharp upper bound for the size of cyclic group acting harmonically on a graph of genus \(g\ge 2\) with a given number of fixed points. Similar results, for closed orientable surfaces, were obtained earlier by T. Szemberg, I. Farkas and H. M. Kra. PubDate: 2019-04-16

Abstract: We consider extremal problems for the energy of the logarithmic potential with external fields closely related with the inverse spectral problem method. The method is based on the relations between the external field and the supports of the equilibrium measures which were discovered in the pioneering papers of Rakhmanov, Saff, Mhaskar and Buyarov (RSMB-method). We propose a generalization of the RSMB-method for the vector of measures with matrix of interaction between components. PubDate: 2019-03-30

Abstract: We obtain several new sharp necessary and sufficient \({{\,\mathrm{\textit{Lip}}\,}}^m\) -continuity conditions for operators of harmonic reflection of functions over boundaries of simple Carathéodory domains in \({\mathbb {R}}^N\) . These results are based on our \({{\,\mathrm{\textit{Lip}}\,}}^m\) -continuity criterion for the Poisson operator in the aforementioned domains. PubDate: 2019-03-23

Abstract: Bernstein’s classical inequality asserts that given a trigonometric polynomial T of degree \(n\ge 1\) , the sup-norm of the derivative of T does not exceed n times the sup-norm of T. We present various approaches to prove this inequality and some of its natural extensions/variants, especially when it comes to replacing the sup-norm with the \(L^p-{\textit{norm}}\) . PubDate: 2019-03-20

Abstract: In this paper, we investigate the boundedness of maximal operator and its commutators in generalized Orlicz–Morrey spaces on the spaces of homogeneous type. As an application of this boundedness, we give necessary and sufficient condition for the boundedness of fractional integral and its commutators in these spaces. We also discuss criteria for the boundedness of these operators in Orlicz spaces. PubDate: 2019-03-20

Abstract: We introduce and analyze the concept of infinitesimal relative position vector field between “infinitesimally nearby” observers, showing the equivalence between different definitions. Through the Fermi–Walker derivative of infinitesimal relative position vector fields along an observer in a reference frame, we characterize spacetimes admitting an umbilic foliation. Sufficient and necessary conditions for those spacetimes to be a conformally stationary spacetime are given. Finally, the important class of cosmological models known as generalized Robertson–Walker spacetimes is characterized. PubDate: 2019-03-13

Abstract: In this paper, we study certain classes of analytic functions which satisfy a Ma–Minda type subordination condition and are associated with the crescent-shaped region. We first give the extremal functions of these function classes and related to them, we present various characteristic properties. It is shown that for certain range of coefficients, a bilinear transformation belongs to a certain class (defined below). Also, for this and related classes, we present certain results and give some specific examples. Further, we obtain results on coefficient inequalities and coefficient estimates. Finally, using the subordination theory, we derive various results and corollaries for functions belonging to the classes studied in this paper. PubDate: 2019-03-12

Abstract: By using the inner diameter distance condition we define and investigate new, in such a generality, class \({\mathcal {F}}\) of homeomorphisms between domains in metric spaces and show that, under additional assumptions on domains, \({\mathcal {F}}\) contains (quasi)conformal, bi-Lipschitz and quasisymmetric mappings as illustrated by examples. Moreover, we employ a prime ends theory in metric spaces and provide conditions allowing continuous and homeomorphic extensions of mappings in \({\mathcal {F}}\) to topological closures of domains, as well as homeomorphic extensions to the prime end boundary. Domains satisfying the bounded turning condition, locally and finitely connected at the boundary and the structure of prime end boundaries for such domains play a crucial role in our investigations. We apply our results to show the Koebe theorem on arcwise limits for mappings in \({\mathcal {F}}\) . Furthermore, relations between the Royden boundary and the prime end boundary are presented. Our work generalizes results due to Carathéodory, Näkki, Väisälä and Zorič. PubDate: 2019-03-08

Abstract: For a bounded function \(\varphi \) on the unit circle \({\mathbb {T}}\) , let \(T_{\varphi }\) be the associated Toeplitz operator on the Hardy space \(H^2\) . Assume that the kernel $$\begin{aligned} K_2(\varphi ):=\left\{ f\in H^2:\,T_{\varphi } f=0\right\} \end{aligned}$$ is nontrivial. Given a unit-norm function f in \(K_2(\varphi )\) , we ask whether an identity of the form \( f ^2=\frac{1}{2}\left( f_1 ^2+ f_2 ^2\right) \) may hold a.e. on \({\mathbb {T}}\) for some \(f_1,f_2\in K_2(\varphi )\) , both of norm 1 and such that \( f_1 \ne f_2 \) on a set of positive measure. We then show that such a decomposition is possible if and only if either f or \(\overline{z\varphi f}\) has a nonconstant inner factor. The proof relies on an intrinsic characterization of the moduli of functions in \(K_2(\varphi )\) , a result which we also extend to \(K_p(\varphi )\) (the kernel of \(T_{\varphi }\) in \(H^p\) ) with \(1\le p\le \infty \) . PubDate: 2019-03-04