Recent zbMATH articles in MSC 30E10https://zbmath.org/atom/cc/30E102021-05-28T16:06:00+00:00WerkzeugExtremal and approximative properties of simple partial fractions.https://zbmath.org/1459.410152021-05-28T16:06:00+00:00"Danchenko, V. I."https://zbmath.org/authors/?q=ai:danchenko.vladimir-ilich"Komarov, M. A."https://zbmath.org/authors/?q=ai:komarov.mikhail-a"Chunaev, P. V."https://zbmath.org/authors/?q=ai:chunaev.p-vIn this large survey article approximation with logarithmic derivatives of complex polynomials, i.e., with simple partial fractions (SPFs) and their modifications are considered. The analogy of metric problems of best approximation and interpolation in complex areas and real intervals systematically is studied. In particular, Gorin-Chebyshev's problem, Gorin-Gelfond problems, analogs of Zolotarev's problem, Jackson-Nikol'ski problem, the problem of interpolation by SPFs are tackled. As well Markov-Berstein's inequalities for derivatives of SPFs on the real axis, on circles and segments. The problem of interpolation of analytic functions by SPFs, the problem of generalized interpolation with simple nodes, generalized multiple interpolation by SPFs, interpolation of real constants, Pade interpolation are treated. An analog of Mergelyan's theorem is proved, the analog of the classical Jacson-Korneychuk theorem on estimating the deviations through the module of continuity is obtained.
According to Remez algorithm in the theory of polynomial approximations the best polynomial approximation and the corresponding alternance of $n+2$ points can be constructed. Some analog of this algorithm for construction of the SPF of the best approximation of constants is proved. The analogy of criterions of the best approximation for real-valued functions on a segment by real-valued SPFs is proved. The analog of the Kolmogorov criterion of the best approximation for real-valued functions on a segment, Chebyshev's systems of functions, connected with SPFs, as well an alternance theorem and the theorem of Vallee-Poussin type are considered.
Reviewer: David Zarnadze (Tbilisi)On the boundedness of poles of generalized Padé approximants.https://zbmath.org/1459.300052021-05-28T16:06:00+00:00"Bosuwan, Nattapong"https://zbmath.org/authors/?q=ai:bosuwan.nattapongSummary: Given a function \(F\) holomorphic on a neighborhood of some compact subset of the complex plane, we prove that if the zeros of the denominators of generalized Padé approximants (orthogonal Padé approximants and Padé-Faber approximants) for some row sequence remain uniformly bounded, then either \(F\) is a polynomial or \(F\) has a singularity in the complex plane. This result extends the known one for classical Padé approximants. Its proof relies, on the one hand, on difference equations where their coefficients relate to the coefficients of denominators of these generalized Padé approximants and, on the other hand, on a curious property of complex numbers.Joint universality of zeta functions with periodic coefficients. II.https://zbmath.org/1459.111682021-05-28T16:06:00+00:00"Laurinčikas, A."https://zbmath.org/authors/?q=ai:laurincikas.antanasSummary: Considering periodic \(\zeta \)-functions and periodic Hurwitz \(\zeta \)-functions, we obtain joint universality for approximation to a collection of analytic functions by generalized shifts of the \(\zeta \)-functions.
For Part I, see [the author, Šiauliai Math. Semin. 2(10), 53--65 (2007; Zbl 1138.11040)].