OnL p -norm convergence of series with respect to product systems

Изучается поточечна я сходимость операто ров \(\sum\limits_{n = 1}^\infty {T_n } \) Т_n, где оператор связан с операторомЕ _n условного математического ожи дания относительно некоторой σ-алгебры $$T_n \circ (E_n - E_{n - 1} ) = (E_n - E_{n - 1} ) \circ T_n = T_n .$$ соотношением_no(Е_n?Е_n?1)=(Е_n?Е_n?1)oТ_n=Т_n. Доказано, что огранич енностьТ _n влечет пот очечную сходимость ряда \(\sum\limits_{n = 1}^\infty {T_n } \) По лученные результаты находят п рименения в исследов ании сходимости в метрикеL ^p рядов по системам произвoдени й некоторых систем. В ч астности, доказано, чтокаждый ряд Фурье по ортогональной сис теме Виленкина сходи тся в метрикеL ^p для 1<р<∞.     

Pointwise convergence of eigenfunction expansions associated with ordinary differential operators

With an ordinary differential expression L = ∑ⁿ_k=0P_kD^k on an open interval I?$\text{r}$ is associated a selfadjoint operator H in a Hilbert space, possibly beyond $\text{K}$=L²(l). The set $\text{D}$H∩$\text{K}$ only depends on the generalized spectral family associated with H. It is shown that the (differentiated) eigenfunction expansion given by H converges uniformly on compact subintervals of l for functions in $\text{D}$(H)∩$\text{L}$ In case H is a semibounded selfadjoint operator in $\text{K}$=L²T, a similar result is proved for functions in $\text{D}$|H|, which is the set of all $\text{K}$∈$\text{K}$ for which there exists a sequence f_n∈(H) such that f_n→f in $\text{H}$ and (H(f_n ? f_m), f_n ? f_m → 0 as n, m → ∞.     

Pointwise convergence of a class of non-orthogonal wavelet expansions

Non-orthogonal wavelet expansions associated with a class of mother wavelets is considered. This class of wavelets comprises mother wavelets that are not necessarily integrable over the whole real line, such as Shannon's wavelet. The pointwise convergence of these wavelet expansions is investigated. It is shown that, unlike other wavelet expansions, the ones under consideration do not necessarily converge pointwise to the functions at points of continuity, unless a more stringent condition, such as bounded variation, is imposed.

     

Absolute convergence of Fourier series with respect to bounded systems

In this paper we prove that for the piecewise-linear unit jump approximation the sum of the moduli of the Fourier coefficients with respect to an arbitrary complete orthonormal system, which is totally bounded, has, when averaged over sections, a lower bound of order log N, where N^?1 is the approximation step.     

Unconditional convergence of Fourier expansions in systems of product bases in Orlicz spaces

We study unconditional convergence of Fourier expansions in systems of product bases in Orlicz spaces.     

Multipliers of convergence in norm of series with respect to multiplicative systems

We prove analogs of the Telyakovskii-Pochuev criteria for multipliers of uniform convergence and of convergence in the integral metric for multiplicative systems with bounded generating sequence.     

The convergence of Fourier series with respect to systems of polynomial kind

We establish sufficient conditions for the convergence of the Fourier expansions of functions from L ^p (p1) in terms of the order of growth of the system { _n(t)}, of polynomial kind, orthonormal with respect to the measure(t) on [a, b] and containing a constant. The convergence is considered either in a given point of the orthogonality interval or inside the interval [c, d] [a, b]. In connection with this we obtain estimates for the Lebesgue functions of the system { _n(t)}, and we consider the localization problem of the convergence conditions.Translated from Matematicheskie Zametki, Vol. 14, No. 5, pp. 633–644, November, 1973.     

On the convergence and summability of series with respect to block-orthonormal systems

Statements connected with the so-called block-orthonormalized systems are given. The convergence and summability almost everywhere by the (c, 1) method with respect to such systems are considered. In particular, the well-known theorems of Menshov-Rademacher and Kacmarz on the convergence and (c, 1)-summability almost everywhere of orthogonal series are generalized.     

Pointwise convergence fails to be strict

It is known that the ring of all Baire functions carrying the pointwise convergence yields a sequential completion of the ring of all continuous functions. We investigate various sequential convergences related to the pointwise convergence and the process of completion of . In particular, we prove that the pointwise convergence fails to be strict and prove the existence of the categorical ring completion of which differs from .     

The Fourier coefficients of continuous functions with respect to certain orthonormal systems

The properties of the Fourier coefficients for some classes of functions with respect to both the Haar system and general orthonormal systems are studied. It is established that the results obtained in the paper cannot be strengthened.     

Recursive expansions with respect to a chain of subspaces

In this work, recursive expansions in Hilbert space H = L ₂[0, 1] are considered. We discuss a related notion of frames in finite-dimensional spaces. We also suggest a constructive approach to extend an arbitrary basis to obtain a tight frame. The algorithm of extending is applied to bases of a special form, whose Gram matrix is circulant. A construction of a chain of nested subspaces { Vⁿ }_{n = 1}^￥ \left\{ {{V^n}} \right\}_{n = 1}^\infty is given, and in its foundation lies an example of a function that can be expressed as a linear combination of its contractions and translations. The main result of the paper is the theorem that provides the uniform convergence of recursive Fourier series with respect to the chain { Vⁿ }_{n = 1}^￥ \left\{ {{V^n}} \right\}_{n = 1}^\infty for continuous functions.     

Generalized absolute convergence of single and double Fourier series with respect to multiplicative systems

Several sufficient conditions for generalized absolute convergence of single and double Vilenkin-Fourier series of bounded type are obtained. These conditions give multiplicative analogues of results due to L. Gogoladze and R. Meskhia, F. Móricz and A. Veres, M. Izumi and S. Izumi. Some of them are sharp in a certain sense.     

Functional conditions for the convergence of Fourier series with respect to general orthonormal systems

In this paper we establish some functional conditions for the convergence of the Fourier series for functions from the class C(0, 1) with respect to general orthonormal systems.     

Pointwise convergence of Fourier-Jacobi series

The purpose of this paper is to study the pointwise and almost everywhere convergence of the Cesàro means (C,δ) of Fourier-Jacobi expansions, the main term of the Lebesgue constant of the (C,δ) means for −1<δ≤α+1/2 is obtained. With the aid of the generalized translation in terms of Jacobi polynomials, pointwise convergence theorems of the (C,δ) means for δ>α+1/2 and equiconvergence theorems for −1<δ≤α+1/2 are proved. The analogues of the Lebesgue, Salem and Young theorems of the Cesàro means at the critical index δ=α+1/2 are established. Supported by the Natural Science Foundation of Beijing