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.     

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.     

Pointwise convergence of expansions with respect to certain product systems

- , - _n A_n-,. , . , ¦_n¦=1 (n=1,2, ), . , — — , .     

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<∞.     

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.     

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.     

Neighborhoods and convergence with respect to a closure operator

We study neighborhoods with respect to a categorical closure operator. In particular, we discuss separation and compactness obtained from neighborhoods in a natural way and compare them with the usual closure separation and closure compactness. We also introduce a concept of convergence based on using centered systems of subobjects, which naturally generalizes the classical filter convergence in topological spaces. We investigate behavior of the convergence introduced and show, among others, that it relates to the separation and compactness in natural ways.     

The error of polytopal approximation with respect to the symmetric difference metric and theL p metric

LetM be a convex body in ℝ ^d withC ₊ ³ boundary. Polytopal approximation ofM with respect to the symmetric difference metric (or theL _p metric) is considered, if the approximating polytope has at mostn facets (or at mostn vertices). The asymptotic behavior of the distance of the best approximating polytope is well-known; it is of order . This paper provides an estimate of order for the error term. Supported by OTKA, Hungary. The paper was written during a visit at University College London.     

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.     

Differential operators symmetric with respect to an indefinite metric

We describe classes of second-order differential operators on a circle that are symmetric with respect to an indefinite metric with a finite rank of indefiniteness.     

Almost everywhere convergence of series with respect to a price system

Moscow. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 47–52, September–October, 1992.     

Robustness with respect to small delays for exponential stability of non-autonomous systems

Robustness of stability with respect to small delays, e.g., motivated by feedback systems in control theory, is of great theoretical and practical important, but this property does not hold for many systems. In this paper, we introduce the conception of robustness with respect to small time-varying delays for exponential stability of the non-autonomous linear systems. Sufficient conditions are given for the non-autonomous systems to be robust, and examples are provided to illustrate that the conditions are satisfied for a large class of the non-autonomous parabolic systems.     

Conformal vector fields with respect to the Sasaki metric tensor

The purpose of this article is to characterize conformal vector fields with respect to the Sasaki metric tensor field on the tangent bundle of a Riemannian manifold of dimension at least three. In particular, if the manifold in question is compact, it is found that the only conformal vector fields are Killing vector fields.     

Matrix rounding with respect to small submatrices

We show that any real valued matrix A can be rounded to an integer one B such that the error in all 2 × 2 (geometric) submatrices is less than 1.5, that is, we have |a_ij ? b_ij| < 1 and for all i,j. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Well-posedness and generalized metric subregularity with respect to an admissible function

Science China Mathematics - In the framework of complete metric spaces, this paper provides several sufficient conditions for the well-posedness with respect to an admissible function, which...     

Some classes of Kenmotsu manifolds with respect to semi-symmetric metric connection

In this paper, we study conharmonic curvature tensor in Kenmotsu manifolds with respect to semi-symmetric metric connection and also characterize conharmonically flat, conharmonically semi-symmetric and φ-conharmonically flat Kenmotsu manifolds with respect to semi-symmetric metric connection.     

A test of convergence of Fourier series with respect to multiplicative systems,analogous to the Jordan test

, {p _n} _n=0 (p₀=1, _n2 n2). : x f(t) V(G)