Unconditional convergence of Fourier series with respect to the haar system in the spaces Λ ω p

Criteria for a Haar system to be a basic system and an unconditional basic system in the spaces $$\Lambda _\omega ^p = \left\{ {f \in L^p :\omega _p \left( {\delta , f} \right) = O\left\{ {\omega \left( \delta \right)} \right\}} \right\}$$ , where 1ω is a modulus of continuity, are proved.     

On Cesáro summability of fourier series with respect to double complete orthonormal systems

Let {ϕn(x), n = 1, 2,...} be an arbitrary complete orthonormal system on the interval I:= [0, 1]which consists of a.e. bounded functions. Suppose that E ₀ ⊂ I ² is any Lebesgue measurable set such that μ₂ E ₀ > 0, and φ, φ(0) = 0, is an increasing continuous function on [0, ∞) with φ(u) = o(u ln u) as u → ∞. Then there exist a function f ∈ L¹(I ²) and a set E ₀ ^′ , ⊂ E ₀, μ₂ E ₀ ^′ > 0, such that

On the relation between Fourier and Leont’ev coefficients with respect to smirnov spaces

Yu. Melnik showed that the Leontev coefficients _f () in the Dirichlet series 2}}$$ " align="middle" border="0"> of a function f E ^p (D), 1 < p < , are the Fourier coefficients of some function F L ^p , ([0, 2]) and that the first modulus of continuity of F can be estimated by the first moduli and majorants in f. In the present paper, we extend his results to moduli of arbitrary order.Published in Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 517–526, April, 2004.     

On the convergence and divergence of multiple Fourier series with respect to the Walsh—Paley system in the spacesC andL

Работа касается вопр осов сходимости и рас ходимости кратных рядов Фурье п о системе Уолша-Пэли в метрикахС и L. Из доказанных тео рем следует, в частности, ч то еишf∈Е (Е=С или E=L) и существует н атуральноеi ₀ (1≦i ₀ ≦N) так ое, что и $$\begin{gathered} \omega (\delta _{i_0 } ;f)_E = o\left( {\frac{1}{{1n^N \frac{1}{{\delta _{i_0 } }}}}} \right) (\delta _{i_0 } \to 0) \hfill \\ \omega (\delta _k ;f)_E = o\left( {\frac{1}{{1n^N \frac{1}{{\delta _k }}}}} \right), k \ne i_0 (\delta _k \to 0), k = 1,2, ...,N, \hfill \\ \end{gathered}$$ тоN-кратный ряд Фурье функцииf по системе У олша-Пэли сходится по Прингсхе йму в смысле метрики пространств аЕ. Доказано также, что вы шеотмеченное утверж дение неусиляемо в метрикеL не только для системы Уолша, но и для некоторого класс а ОНС, ограниченных в совок упности.     

A weak type inequality for the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system

Simon [12] proved that the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space H _p to the space L _p for p > 1/(1 + α). In this paper we prove that this boundedness result does not hold if p ≦ 1/(1 + α). However, in the endpoint case p = 1/(1 + α) the maximal operator σ _* ^α,k is bounded from the martingale Hardy space H _1/(1+α) to the space weak-L _1/(1+α).     

On a general class of trigonometric functions and Fourier series†

We discuss a general class of trigonometric functions whose corresponding Fourier series can be used to calculate several interesting numerical series. Particular cases are presented.     

Summability of generalized Fourier series and Dirichlet's problem inL p(dμ) and weightedH p-spaces (p>1)

Для подсистем тригон ометрической систем ы \(\{ e^{int} \} _{\mathop {|n| \geqq m,}\limits_{n \ne m} } \{ e^{int} \} _{|n| \geqq m} (m = 1,2,{\text{ }} \ldots )\) найдены необходимые и достат очные условия на весо вую функцию У для того, чтобы эти си стемы былиA — базисом вL _[?π,π] ^p (Ψ). Спо мощью этих результат ов решается задача Дири хле в весовых метриках и и сследуются простран ства Н^p(ψ),р>1.     

Localization with respect to a class of maps I — Equivariant localization of diagrams of spaces

Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions) [4, 13, 24, 35]. In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model categories and, more important, will allow for a localization with respect to a class of maps (satisfying some restrictive conditions). We illustrate our technique by applying it to the equivariant model category of diagrams of spaces [12]. This model category is not cofibrantly generated [8]. We give conditions on a class of maps which ensure the existence of the localization functor; these conditions are satisfied by any set of maps and by the classes of maps which induce ordinary localizations on the generalized fixed-points sets. During the preparation of this paper the author was a fellow of Marie Curie Training Site hosted by Centre de Recerca Matemàtica (Barcelona), grant no. HPMT-CT-2000-00075 of the European Commission.     

On basicity of exponentials inL P (dμ) and general prediction problems

Let be a nonnegative measure on the unit circle in the complex plane and 1<p<. It is of interest to find conditions on so that the set of exponentialse ⁱⁿ form a strongM-basis forL ^p (d). Some partial results are proved which can shed some light on this important open question. These results are of fundamental importance in the prediction theory of stochastic processes and other fields of applications. These results is then used to obtain a theorem which reduces some prediction problems to easier ones.To 80^th birthday of Paul ErdsThis research is supported by Office of Naval Research Grant No N00014-89-J-1824.     

Menshov’s “adjustment theorem” with respect to general measures

We prove Menshov’s theorem in the setting of arbitrary Borel measures.     

On the (C, α)-means with respect to the Walsh system

In our main result we prove strong convergence theorems for Cesàro means (C, α) on the Hardy spaces H _1/(1+α), where 0 < α < 1.     

Global smooth solution to a coupled Schrödinger system in atomic Bose-Einstein condensates with two-dimensional spaces

We obtain the global smooth solution of a nonlinear Schrödinger equations in atomic Bose-Einstein condensates with two-dimensional spaces. By using the Galerkin method and a priori estimates, we establish the global existence and uniqueness of the smooth solution.     

A note on the sunouchi operator with respect to the walsh—kaczmarz system

An analogue of the so—called Sunouchi operator with respect to the Walsh—Kaczmarz system will be investigated. We show the boundedness of this operator if we take it as a map from the dyadic Hardy space H ^p to L ^p for all 0<p≤1.. For the proof we consider a multiplier operator and prove its (H ^p H ^p)—boundedness for 0<p≤1. Since the multiplier is obviously bounded from L ² to L ², a theorem on interpolation of operators can be applied to show that our multiplier is of weak type (1,1) and of type (q q) for all 1<q<∞. The same statements follow also for the Sunouchi operator.     

Differential as a harmonic morphism with respect to Cheeger–Gromoll-type metrics

We investigate horizontal conformality of a differential of a map between Riemannian manifolds, where the tangent bundles are equipped with Cheeger–Gromoll-type metrics. As a corollary, we characterize the differential of a map as a harmonic morphism.     

On the summability by Cesàro methods of double series with respect to block-orthonormal systems

The extension of the coefficient test of Menshov and Kaczmarz ensuring the almost everywhere (C, 1, 1)-((C, 1, 0) or (C, 0, 1)) summability of double series with respect to block-orthonormal systems is studied.