Recent zbMATH articles in MSC 30E10https://www.zbmath.org/atom/cc/30E102022-05-16T20:40:13.078697ZWerkzeugThe discrete case of the mixed joint universality for a class of certain partial zeta-functionshttps://www.zbmath.org/1483.111912022-05-16T20:40:13.078697Z"Kačinskaitė, Roma"https://www.zbmath.org/authors/?q=ai:kacinskaite.roma"Matsumoto, Kohji"https://www.zbmath.org/authors/?q=ai:matsumoto.kohjiAuthors' abstract: We give a new type of mixed discrete joint universality properties, which is satisfied by a wide class of zeta-functions. We study the universality for a certain modification of Matsumoto zeta-functions \(\varphi_h(s)\) and a collection of periodic Hurwitz zeta-functions \(\zeta (s;\alpha;\mathfrak B)\) under the condition that the common difference of arithmetical progression \(h > 0\) is such that \(\exp \{ \frac{2\pi}h\}\) is a rational number and parameter \(\alpha\) is a transcendental number.
Reviewer: Anatoly N. Kochubei (Kyïv)Nonlinear conditions for ultradifferentiabilityhttps://www.zbmath.org/1483.260232022-05-16T20:40:13.078697Z"Nenning, David Nicolas"https://www.zbmath.org/authors/?q=ai:nenning.david-nicolas"Rainer, Armin"https://www.zbmath.org/authors/?q=ai:rainer.armin"Schindl, Gerhard"https://www.zbmath.org/authors/?q=ai:schindl.gerhardSummary: A remarkable theorem of Joris states that a function \(f\) is \(C^{\infty}\) if two relatively prime powers of \(f\) are \(C^{\infty}\). Recently, Thilliez showed that an analogous theorem holds in Denjoy-Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris's result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.Weighted Chebyshev polynomials on compact subsets of the complex planehttps://www.zbmath.org/1483.300232022-05-16T20:40:13.078697Z"Novello, Galen"https://www.zbmath.org/authors/?q=ai:novello.galen"Schiefermayr, Klaus"https://www.zbmath.org/authors/?q=ai:schiefermayr.klaus"Zinchenko, Maxim"https://www.zbmath.org/authors/?q=ai:zinchenko.maximSummary: We study weighted Chebyshev polynomials on compact subsets of the complex plane with respect to a bounded weight function. We establish existence and uniqueness of weighted Chebyshev polynomials and derive weighted analogs of Kolmogorov's criterion, the alternation theorem, and a characterization due to Rivlin and Shapiro. We derive invariance of the Widom factors of weighted Chebyshev polynomials under polynomial pre-images and a comparison result for the norms of Chebyshev polynomials corresponding to different weights. Finally, we obtain a lower bound for the Widom factors in terms of the Szegő integral of the weight function and discuss its sharpness.
For the entire collection see [Zbl 1479.47003].Approximate properties of the \(p\)-Bieberbach polynomials in regions with simultaneously exterior and interior zero angleshttps://www.zbmath.org/1483.300312022-05-16T20:40:13.078697Z"Abdullayev, F. G."https://www.zbmath.org/authors/?q=ai:abdullayev.fahreddin-g"Imashkyzy, M."https://www.zbmath.org/authors/?q=ai:imashkyzy.meerim"Özkartepe, P."https://www.zbmath.org/authors/?q=ai:ozkartepe.naciye-pelinSummary: In this paper, we study the uniform convergence of \(p\)-Bieberbach polynomials in regions with a finite number of both interior and exterior zero angles at the boundary.Approximation of functions and all derivatives on compact setshttps://www.zbmath.org/1483.300682022-05-16T20:40:13.078697Z"Armeniakos, Sotiris"https://www.zbmath.org/authors/?q=ai:armeniakos.sotiris"Kotsovolis, Giorgos"https://www.zbmath.org/authors/?q=ai:kotsovolis.giorgos"Nestoridis, Vassili"https://www.zbmath.org/authors/?q=ai:nestoridis.vassiliSummary: In Mergelyan type approximation we uniformly approximate functions on compact sets \(K\) by polynomials or rational functions or holomorphic functions on varying open sets containing \(K\). In the present paper we consider analogous approximation, where uniform convergence on \(K\) is replaced by uniform approximation on \(K\) of all order derivatives.Induced fields in isolated elliptical inhomogeneities due to imposed polynomial fields at infinityhttps://www.zbmath.org/1483.300692022-05-16T20:40:13.078697Z"Calvo-Jurado, Carmen"https://www.zbmath.org/authors/?q=ai:calvo-jurado.carmen"Parnell, William J."https://www.zbmath.org/authors/?q=ai:parnell.william-jSummary: The Eshelby inhomogeneity problem plays a crucial role in the micromechanical analysis of the effective mechanical behaviour of inhomogeneous media since it provides a mechanism to predict interior fields associated with ellipsoidal inhomogeneities. In the context of linear elasticity, Eshelby showed that given an isolated elliptical (two dimensions) or ellipsoidal (three dimensions) inhomogeneity embedded in a homogeneous material of infinite extent, then for any uniform strain or traction imposed in the far field, the induced strain inside the inhomogeneity is also uniform. In the case of non-uniform far-field conditions, Eshelby showed that if the loading is a polynomial of order \(n\), the associated interior field is characterized by a polynomial of the same order. This is often called `Eshelby's polynomial conservation theorem'. Since then, the problem has been studied by many, but in most cases for the uniform loading scenario, i.e. when strains or tractions in the far field are uniform. However, in many applications, e.g. permittivity, conductivity, elasticity, etc., the case of non-uniform conditions is also of interest and furthermore, methods to deal with non-elliptical and non-ellipsoidal inhomogeneities are required. In this work, for prescribed non-uniform polynomial far-field conditions, we introduce a method to approximate interior fields for isolated inhomogeneities of elliptical shape. This subproblem is relevant for approximating effective properties of numerous composites since constituent inhomogeneities are often of this form, or limiting forms, e.g. layered and fibre reinforced composites. We verify that the obtained results agree with the polynomial conservation property and with results determined using conformal mappings or the classical circle inclusion theorem. We close with a discussion of how the method can be straightforwardly extended to the case of non-elliptical inhomogeneities.Bounded extremal problems in Bergman and Bergman-Vekua spaceshttps://www.zbmath.org/1483.300892022-05-16T20:40:13.078697Z"Delgado, Briceyda B."https://www.zbmath.org/authors/?q=ai:delgado.briceyda-b"Leblond, Juliette"https://www.zbmath.org/authors/?q=ai:leblond.julietteSummary: We analyze Bergman spaces \(A_f^p(\mathbb{D})\) of generalized analytic functions of solutions to the Vekua equation \(\bar{\partial}w = (\bar{\partial}f/f)\bar{w}\) in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions \(f\) and \(1<p<\infty\). We consider a family of bounded extremal problems (best constrained approximation) in the Bergman space \(A^p(\mathbb{D})\) and in its generalized version \(A^p_f(\mathbb{D})\), that consists in approximating a function in subsets of \(\mathbb{D}\) by the restriction of a function belonging to \(A^p(\mathbb{D})\) or \(A^p_f(\mathbb{D})\) subject to a norm constraint. Preliminary constructive results are provided for \(p = 2\).Escaping Fatou components of transcendental self-maps of the punctured planehttps://www.zbmath.org/1483.370542022-05-16T20:40:13.078697Z"Martí-Pete, David"https://www.zbmath.org/authors/?q=ai:marti-pete.davidHolomorphic self-maps of the Riemann sphere are the most well-studied families of systems in complex dynamics (rational dynamics), followed by self-maps of the once punctured plane (transcendental dynamics). This paper concerns the study of holomorphic self-maps of the twice punctured Riemann sphere, a subject still in its early days.
In transcendental dynamics, an important role is played by the escaping set, which consists of those points which iterate to the essential singularity of the function. In the present setting, however, there are two essential singularities. The main result is that any possible way of escaping is possible for a wandering Fatou component. The author's main technique is approximation theory.
Reviewer: Kirill Lazebnik (New Haven)Complex best \(r\)-term approximations almost always exist in finite dimensionshttps://www.zbmath.org/1483.410102022-05-16T20:40:13.078697Z"Qi, Yang"https://www.zbmath.org/authors/?q=ai:qi.yang"Michałek, Mateusz"https://www.zbmath.org/authors/?q=ai:michalek.mateusz"Lim, Lek-Heng"https://www.zbmath.org/authors/?q=ai:lim.lek-hengSummary: We show that in finite-dimensional nonlinear approximations, the best \(r\)-term approximant of a function \(f\) almost always exists over \(\mathbb{C}\) but that the same is not true over \(\mathbb{R}\), i.e., the infimum \(\inf_{f_1, \dots, f_r \in D} \| f - f_1 - \dots - f_r \|\) is almost always attainable by complex-valued functions \(f_1, \ldots, f_r\) in \(D\), a set (dictionary) of functions (atoms) with some desired structures. Our result extends to functions that possess properties like symmetry or skew-symmetry under permutations of arguments. When \(D\) is the set of separable functions, this is the best rank-\(r\) tensor approximation problem. We show that over \(\mathbb{C}\), any tensor almost always has a unique best rank-\(r\) approximation. This extends to other notions of ranks such as symmetric and alternating ranks, to best \(r\)-block-terms approximations, and to best approximations by tensor networks. Applied to sparse-plus-low-rank approximations, we obtain that for any given \(r\) and \(k\), a general tensor has a unique best approximation by a sum of a rank-\(r\) tensor and a \(k\)-sparse tensor with a fixed sparsity pattern; a problem arising in covariance estimation of Gaussian model with \(k\) observed variables conditionally independent given \(r\) hidden variables. The existential (but not uniqueness) part of our result also applies to best approximations by a sum of a rank-\(r\) tensor and a \(k\)-sparse tensor with no fixed sparsity pattern, and to tensor completion problems.