Λ(Ψ)-Fluctuation and the Fourier-Walsh series of bounded functions

We consider the questions of convergence of Fourier-Walsh series in Lorentz spaces. Some condition is given on a function ? sufficient for its Fourier-Walsh series to converge in the Lorentz spaces “near” L _∞. We show that this result is sharp.     

Multiplicative convolutions of functions from Lorentz spaces and convergence of series of Fourier–Vilenkin coefficients

Let f and g be functions from different Lorentz spaces L ^{p, q} [0, 1), h be theirmultiplicative convolution and xxxx be Fourier coefficients of h with respect to a multiplicative system with bounded generating sequence. We estimate the remainder of the series of xxxx with multiplicators of type k ^b in terms of the best approximations of f and g in the corresponding Lorentz spaces. We establish sharpness of this result and of its corollaries for the Lebesgue spaces.     

The Fixed Points of Contractions of <Emphasis Type="Italic">f</Emphasis>-Quasimetric Spaces

The recent articles of Arutyunov and Greshnov extend the Banach and Hadler Fixed-Point Theorems and the Arutyunov Coincidence-Point Theorem to the mappings of (q₁, q₂)-quasimetric spaces. This article addresses similar questions for f-quasimetric spaces.Given a function f: R ₊² → R₊ with f(r₁, r₂) → 0 as (r₁, r₂) → (0, 0), an f-quasimetric space is a nonempty set X with a possibly asymmetric distance function ρ: X² → R+ satisfying the f-triangle inequality: ρ(x, z) ≤ f(ρ(x, y), ρ(y, z)) for x, y, z ∈ X. We extend the Banach Contraction Mapping Principle, as well as Krasnoselskii’s and Browder’s Theorems on generalized contractions, to mappings of f-quasimetric spaces.     

The Schwarzian derivative and the Wiman-Valiron property

Let f be a transcendental meromorphic function in the plane such that f has finitely many critical values, the multiple points of f have bounded multiplicities, and the inverse function of f has finitely many transcendental singularities. Using the Wiman-Valiron method, it is shown that if the Schwarzian derivative S_f of f is transcendental, then f has infinitely many multiple points, the inverse function of S_f does not have a direct transcendental singularity over ∞, and ∞ is not a Borel exceptional value of S_f. The first of these conclusions was proved by Nevanlinna and Elfving via a fundamentally different method.     

Invariant <Emphasis Type="Italic">f</Emphasis>-structures on naturally reductive homogeneous spaces

We study invariant metric f-structures on naturally reductive homogeneous spaces and establish their relation to generalized Hermitian geometry. We prove a series of criteria characterizing geometric and algebraic properties of important classes of metric f-structures: nearly Kähler, Hermitian, Kähler, and Killing structures. It is shown that canonical f-structures on homogeneous Φ-spaces of order k (homogeneous k-symmetric spaces) play remarkable part in this line of investigation. In particular, we present the final results concerning canonical f-structures on naturally reductive homogeneous Φ-spaces of order 4 and 5.     

Detecting fixed points of nonexpansive maps by illuminating the unit ball

We give necessary and sufficient conditions for a nonexpansive map on a finite-dimensional normed space to have a nonempty, bounded set of fixed points. Among other results we show that if f: V → V is a nonexpansive map on a finite-dimensional normed space V, then the fixed point set of f is nonempty and bounded if and only if there exist w₁,..., w _m in V such that {f(w _i ) ? w _i : i = 1,..., m} illuminates the unit ball. This yields a numerical procedure for detecting fixed points of nonexpansive maps on finite-dimensional spaces. We also discuss applications of this procedure to certain nonlinear eigenvalue problems arising in game theory and mathematical biology.     

On behavior of Fourier coefficients by Walsh system

The paper proves that for any ε > 0 there exists ameasurable set E ? [0, 1] with measure |E| > 1 ? ε such that for each f ∈ L¹[0, 1] there is a function \(\tilde f \in {L^1}\left[ {0,1} \right]\) coinciding with f on E whose Fourier-Walsh series converges to \(\tilde f\) in L¹[0, 1]-norm, and the sequence \(\left\{ {\left| {{c_k}\left( {\tilde f} \right)} \right|} \right\}_{n = 0}^\infty \) is monotonically decreasing, where \(\left\{ {{c_k}\left( {\tilde f} \right)} \right\}\) is the sequence of Fourier-Walsh coefficients of \(\left\{ {\left| {{c_k}\left( {\tilde f} \right)} \right|} \right\}_{n = 0}^\infty \).     

Equality of Some Classical Lorentz Spaces

We prove that although the class of M _p -weights of Muckenhoupt is strictly smaller than class of B _p -weights of Ariño and Muckenhoupt, both classes produce the same classical Lorentz spaces. An analogous result is obtained for other classes of weights.     

Frames and non linear approximations in Hilbert spaces

A simple proof of the proposition, stated in ([2], p. 346), asserting that in Hilbert spaces a Riesz basis is greedy, is given. Also, greedy approximant for frames in Hilbert spaces is defined and it is shown that frames satisfy the quasi greedy and almost greedy conditions. Finally, we give the characterizations of approximation spaces A^s(Ψ), A_q^s(Ψ) by means of weak-l_p and Lorentz sequence spaces for frames.     

Kreĭn space representation and Lorentz groups of analytic Hilbert modules

This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H²(D²). A closed subspace M in H²(D²) is called a submodule if z _i M ? M (i = 1, 2). An associated integral operator (defect operator) C _M captures much information about M. Using a Kre?n space indefinite metric on the range of C _M , this paper gives a representation of M. Then it studies the group (called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup (called little Lorentz group) which turns out to be a finer invariant for M.     

Nearly Kähler and Hermitian <Emphasis Type="Italic">f</Emphasis>-structures on homogeneous Φ-spaces of order <Emphasis Type="Italic">k</Emphasis> with special metrics

We consider arbitrary homogeneous Φ-spaces of order k ≥ 3 of semisimple compact Lie groups G in the case of a series of special metrics. We give formulas for the Nomizu function of the Levi-Civita connection of these metrics. Using these formulas and other relations for Φ-spaces of order k, we prove necessary and sufficient conditions for the canonical f-structures on these spaces to lie in some generalized Hermitian geometry classes of f-structures: nearly Kähler (NKf-structures) and Hermitian (Hf-structures).     

Inductive and projective limits of Banach spaces of measurable functions with order unities with respect to power parameter

We prove that a measurable function f is bounded and invertible if and only if there exist at least two equivalent norms by order unit spaces with order unities f^α and f^β with α > β > 0. We show that it is natural to understand the limit of ordered vector spaces with order unities f^α (α approaches to infinity) as a direct sum of one inductive and one projective limits. We also obtain some properties for the corresponding limit topologies.     

Decomposition theorems for<Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> spaces

We study the Möbius invariant spacesQ _p andQ _{p, 0} of analytic functions. These scales of spaces include BMOA=Q₁, VMOA=Q_{1, 0} and the Dirichlet space=Q₀. Using the Bergman metric, we establish decomposition theorems for these spaces. We obtain also a fractional derivative characterization for bothQ _p andQ _{p, 0} .     

Inverse functions of polynomials and its applications to initialize the search of solutions of polynomials and polynomial systems

In this paper we present a new algorithm for solving polynomial equations based on the Taylor series of the inverse function of a polynomial, f _P (y). The foundations of the computing of such series have been previously developed by the authors in some recent papers, proceeding as follows: given a polynomial function \(y=P(x)=a_0+a_1x+\cdots+a_mx^m\), with \(a_i \in \mathcal{R}, 0 \leq i \leq m\), and a real number u so that P′(u)?≠?0, we have got an analytic function f _P (y) that satisfies x?=?f _P (P(x)) around x?=?u. Besides, we also introduce a new proof (completely different) of the theorems involves in the construction of f _P (y), which provide a better radius of convergence of its Taylor series, and a more general perspective that could allow its application to other kinds of equations, not only polynomials. Finally, we illustrate with some examples how f _P (y) could be used for solving polynomial systems. This question has been already treated by the authors in preceding works in a very complex and hard way, that we want to overcome by using the introduced algorithm in this paper.     

Equiconvergence of the trigonometric Fourier series and the expansion in eigenfunctions of the Sturm-Liouville operator with a distribution potential

We consider the Sturm-Liouville operator L = ?d ²/dx ² + q(x) with the Dirichlet boundary conditions in the space L ₂[0, π] under the assumption that the potential q(x) belongs to W ₂ ^?1 [0, π]. We study the problem of uniform equiconvergence on the interval [0, π] of the expansion of a function f(x) in the system of eigenfunctions and associated functions of the operator L and its Fourier sine series expansion. We obtain sufficient conditions on the potential under which this equiconvergence holds for any function f(x) of class L ₁. We also consider the case of potentials belonging to the scale of Sobolev spaces W ₂ ^?θ [0, π] with ½ < θ ≤ 1. We show that if the antiderivative u(x) of the potential belongs to some space W ₂ ^θ [0, π] with 0 < θ < 1/2, then, for any function in the space L ₂[0, π], the rate of equiconvergence can be estimated uniformly in a ball lying in the corresponding space and containing u(x). We also give an explicit estimate for the rate of equiconvergence.     

When a smooth self-map of a semi-simple Lie group can realize the least number of periodic points

There are two algebraic lower bounds of the number of n-periodic points of a self-map f : M → M of a compact smooth manifold of dimension at least 3: NF_n(f) = min{#Fix(g~n); g ～ f; g continuous} and NJD_n(f) = min{#Fix(g~n); g ～ f; g smooth}. In general, NJD_n(f) may be much greater than NF_n(f). We show that for a self-map of a semi-simple Lie group, inducing the identity fundamental group homomorphism,the equality NF_n(f) = NJD_n(f) holds for all n ? all eigenvalues of a quotient cohomology homomorphism induced by f have moduli 1.     

On real solutions of systems of equations

Systems of equations f ₁ = ··· = f _n?1 = 0 in ? ⁿ = {x} having the solution x = 0 are considered under the assumption that the quasi-homogeneous truncations of the smooth functions f ₁,..., f _n?1 are independent at x ≠ 0. It is shown that, for n ≠ 2 and n ≠ 4, such a system has a smooth solution which passes through x = 0 and has nonzero Maclaurin series.     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

A Midpoint Method for Generalized Equations Under Mild Differentiability Condition

The aim of this study is the approximation of a solution x ^? of the generalized equation 0∈f(x)+F(x) in Banach spaces, where f is a single function whose second order Fréchet derivative ?² f verifies an Hölder condition, and F stands for a set-valued map with closed graph. Using a fixed point theorem and proceeding by induction under the pseudo-Lipschitz property of F, we obtain a sequence defined by a midpoint formula whose convergence to x ^? is superquadratic. Taking a weaker condition, we present the result obtained when ?² f satisfies a center-Hölder conditioning.