Абсолютная сходимость кратных рядов по мультипликативным системам

Certain sufficient conditions are obtained for the absolute convergence of multiple Fourier-Vilenkin series. The authors prove that the generalized condition of Bernstein and Ste?kin as well as certain other conditions are unimprovable in the usual sense.     

Приближение функции ограниченнойp-флукту адии полиномами по мульти пликативным система м

A new discrete modulus of continuity is introduced for functions of boundedp-fluctuation, and direct and converse theorems are proved on the approximation of these functions by polynomials with respect to multiplicative systems. Sufficient conditions for the convergence of Fourier series with respect to multiplicative systems are also obtained and these are the best possible in a certain sense.

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Об отделимости множе ств гиперплоскостью вL p

LetA and be two arbitrary sets in the real spaceL _p, 1p<. Sufficient conditions are obtained for their strict separability by a hyperplane, in terms of the distance between the setsd(A,B) _p=inf{x-y_p,xA,yB} and their diametersd(A) _p, d(B)_p, whered(A) _p=sup{x-y_p; x,yA}. In particular, it is proved that if in an infinite-demensional spaceL _p we haved ^r(A,B)_p>2^–r+1(d^r(A)_p+d^r(B)_p), r=min{p, p(p–1)^–1}, then there is a hyperplane which separatesA andB. On the other hand, the conditiond ^r(A,B)_p=2^–r+1(d^r(A)_p+d^r(B)_p) does not guarantee strict separability. Earlier these results where obtained by V. L. Dol'nikov for the case of Euclidean spaces.     

Некоторые свойства р ядов по синусам с моно тонными коэффициентами

We consider the sine series $$\mathop \sum \limits_{k = 1}^\infty a_k \sin kx$$ with monotone coefficients tending to zero and denote byg(x) its sum. We establish estimates of the integral ∝¦g¦dx over a given subinterval of (0,π]. These estimates are uniform with respect to the coefficientsa _k and the endpoints of the subinterval. In the particular case wheng is not integrable over the period, we get an asymptotic estimate of the growth order of the integral over [∈, π] as∈↓+0. It is of the same form as in the case of series with convex coefficients. We compare the estimates of the integrals ofg with those of the corresponding integrals of the majorant of the partial sums of series (1). We obtain also estimates of the integral modulus of continuity of order s of the functiong, which are uniform with respect to all parameters.     

О наилучшем приближении периодических функций тригонометрическими полиномами и точных эначениях поперечников функциональных классов в L 2

Exact inequalities of Jackson-Stechkin type have been obtained for the average moduli of continuity of mth order (m ?? ?) with the weight function. The exact values of any n-widths were calculated for the function classes, which are defined by the majorant and the indicated average magnitudes.     

Выделение меченых соединений методом препаративной хроматографии

Mica Track Microfilters (MTM) are successful applied to separate two different and strongly mixed liquid phases (emulsion). The organic and the inorganic phase of the emulsion were 30% tributyl phosphate (TBP) in kerosene and 2 N HNO₃ (or H₂O), respectively. Mica filters with sharp edges (i.e. etched with H₂F₂) and having pore sizes k of 8 μm > k > 2 μm are used to get under some well defined conditions a rather complete extraction of the aqueous phase out of the emulsion. The system was calibrated with the pure phases (water as well as TBP in kerosene), and the liquid throughput is shown as a function of the pore size of the filter. The throughput depends mainly on the pore size and not that much on the kind of liquid, it is slightly larger for water as compared to that of TBP in kerosene.     

О некоторых условиях в теории рядов по мультипликативным системам

Реэюме В данной статье иэучаются ряды по мультипликативным системам с условиями трех типов на их козффициенты. Получены некоторые реэультаты об А-интегрируемости и весовой интегрируемости сумм зтих рядов. Эти реэультаты являются аналогами и обобшениями теорем П. Л. Ульянова, Л. Лейндлера и Г. К. Лебедя в тригонометрическом случае. Посвяшается памяти П. Л. Улянова     

О множествах расходи мости рядов по некото рым ортогональным систе мам функций

The main results concern the structure of divergence sets of power series on the unit circle. A class of setsF ¹,F ¹F , is characterized which are not divergence sets for power series on the unit circle. It is also emphasized that, the divergence sets of power series can be of sufficiently complicated structure. Divergence sets of power series on the unit circle are studied, whose partial sums satisfy lim sup ¦s _n(x)¦< outside a set of first category. Analogous results hold for trigonometric series on [0,2] and also for series with respect to Vilenkin sets on corresponding zero measure compact Abelian groups. Nonsummability sets for Abel's method on the unit circle are also studied for power and trigonometric series.