Abstract: Let X be a Banach holomorphic function space on the unit disk. A linear polynomial approximation scheme for X is a sequence of bounded linear operators \(T_n:X\rightarrow X\) with the property that, for each \(f\in X\) , the functions \(T_n(f)\) are polynomials converging to f in the norm of the space. We completely characterize those spaces X that admit a linear polynomial approximation scheme. In particular, we show that it is not sufficient merely that polynomials be dense in X. PubDate: 2019-05-22

Abstract: A logharmonic mapping f is a mapping that is a solution of the nonlinear elliptic partial differential equation \(\dfrac{\overline{f_{ \overline{z}}}}{\overline{f}}=a\dfrac{f_{z}}{f}\) . In this paper we investigate the univalence of logharmonic mappings of the form \( f=zH\overline{G},\) where H and G are analytic on a linearly connected domain. We discuss the relation with the univalence of its analytic counterparts. Stable Univalence and its consequences are also considered. PubDate: 2019-05-22

Abstract: This note surveys recent strategies to estimate the condition number \(CN(T)=\Vert T\Vert \cdot \Vert T^{-1}\Vert \) of complex \(n\times n\) matrices T with given spectrum. More precisely, we present a proof of the fact that if T acts on the Hilbert space \(\mathbb {C}^{n}\) , then the supremum of CN(T) over all contractions T with smallest eigenvalues of modulus \(r>0\) , is equal to \(1/r^{n}\) , and is achieved by an analytic Toeplitz matrix. The same question is treated for n-dimensional Banach spaces. These strategies provide with explicit and constructive solutions to the so-called Halmos and Schäffer’s problems, and are also shown to be effective in a closely related situation, namely considering Kreiss matrices instead of contractions. PubDate: 2019-05-21

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