scholarly journals Characterization of the Schur class in terms of the coefficients of a series on the Laguerre basis

2020 ◽  
pp. 9-15
Author(s):  
V.V. Savchuk ◽  
◽  
M.V. Savchuk ◽  
Keyword(s):  
Power Series ◽  
Unit Disk ◽  
Orthogonal Series ◽  
Schur Function ◽  
Laguerre Basis ◽  
Complex Coefficients ◽  
Schur Class

The classical Schur criterion answers the question of whether a function f given by its power series f(x)=∑k=0∞CkZk is a Schur function that is, holomorphic in a unit disk D and such that supz∈D | f (z) | ≤ 1. Regarding this criterion, there are a large number of completed results devoted to its generalizations and various applications, but, as it seems to us, there is no criterion for a complete description of the Schur class in terms of coefficients of orthogonal series on arbitrary complete orthonormal systems. In this paper, we formulate such criterion for a formal orthogonal series with complex coefficients based on the Laguerre system.

scholarly journals Optimal cycles in ultrametric dynamics and minimally ramified power series

2015 ◽  
Vol 152 (1) ◽  
pp. 187-222 ◽  
Author(s):  
Karl-Olof Lindahl ◽  
Juan Rivera-Letelier
Keyword(s):  
Fixed Point ◽  
Power Series ◽  
Unit Disk ◽  
Periodic Points ◽  
Quadratic Polynomial ◽  
Minimal Period ◽  
Degree Polynomial

We study ultrametric germs in one variable having an irrationally indifferent fixed point at the origin with a prescribed multiplier. We show that for many values of the multiplier, the cycles in the unit disk of the corresponding monic quadratic polynomial are ‘optimal’ in the following sense: they minimize the distance to the origin among cycles of the same minimal period of normalized germs having an irrationally indifferent fixed point at the origin with the same multiplier. We also give examples of multipliers for which the corresponding quadratic polynomial does not have optimal cycles. In those cases we exhibit a higher-degree polynomial such that all of its cycles are optimal. The proof of these results reveals a connection between the geometric location of periodic points of ultrametric power series and the lower ramification numbers of wildly ramified field automorphisms. We also give an extension of Sen’s theorem on wildly ramified field automorphisms, and a characterization of minimally ramified power series in terms of the iterative residue.

scholarly journals Crystals and Schur $P$-Positive Expansions

10.37236/7557 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Seung-Il Choi ◽  
Jae-Hoon Kwon
Keyword(s):  
Crystal Structures ◽  
Schur Function ◽  
Skew Schur Function

We give a new characterization of Littlewood-Richardson-Stembridge tableaux for Schur $P$-functions by using the theory of $\mathfrak{q}(n)$-crystals. We also give alternate proofs of the Schur $P$-expansion of a skew Schur function due to Ardila and Serrano, and the Schur expansion of a Schur $P$-function due to Stembridge using the associated crystal structures.

The characterization of orders for regular functions

1992 ◽  
Vol 111 (2) ◽  
pp. 299-307 ◽  
Author(s):  
C. N. Linden
Keyword(s):  
Unit Disk ◽  
Regular Functions ◽  
Logarithmic Means ◽  
Image Position

For a given function f, regular in the unit disk D(0, 1), logarithmic means of order p may be defined in terms of integrals by the formulaefor 0 < p < , and bywhen 0 < r < 1. Associated p-orders are subsequently defined byfor 0 < p .

scholarly journals Inequalities Of Lipschitz Type For Power Series In Banach Algebras

2015 ◽  
Vol 29 (1) ◽  
pp. 61-83
Author(s):  
Sever S. Dragomir
Keyword(s):  
Power Series ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Open Disk ◽  
Complex Coefficients

AbstractLet $f(z) = \sum\nolimits_{n = 0}^\infty {\alpha _n z^n }$ be a function defined by power series with complex coefficients and convergent on the open disk D (0, R) ⊂ ℂ, R > 0. For any x, y ∈ ℬ, a Banach algebra, with ‖x‖, ‖y‖ < R we show among others that $$\left\| {f(y) - f(x)} \right\| \le \left\| {y - x} \right\|\int_0^1 {f_a^\prime } (\left\| {(1 - t)x + ty} \right\|)dt$$ where $f_a (z) = \sum\nolimits_{n = 0}^\infty {|\alpha _n |} \;z^n$ . Inequalities for the commutator such as $$\left\| {f(x)f(y) - f(y)f(x)} \right\| \le 2f_a (M)f_a^\prime (M)\left\| {y - x} \right\|,$$ if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.

scholarly journals A NEW CHARACTERIZATION OF HOLOMORPHIC FUNCTIONS IN THE UNIT DISK

2018 ◽  
Vol 25 (1) ◽  
pp. 134-147
Author(s):  
Valerii V. Volchkov ◽  
Vitalii V. Volchkov
Keyword(s):  
Unit Disk ◽  
Holomorphic Functions
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