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С. С. Волосивец 《Analysis Mathematica》1995,21(1):61-77
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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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. 相似文献
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С. С. Волосивец 《Analysis Mathematica》2007,33(3):227-246
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 相似文献
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С. А. Пичугов 《Analysis Mathematica》1991,17(1):21-33
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-yp,xA,yB} and their diametersd(A)
p, d(B)p, whered(A)
p=sup{x-yp; x,yA}. In particular, it is proved that if in an infinite-demensional spaceL
p we haved
r(A,B)p>2–r+1(dr(A)p+dr(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(dr(A)p+dr(B)p) does not guarantee strict separability. Earlier these results where obtained by V. L. Dol'nikov for the case of Euclidean spaces. 相似文献
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С. А. Теляковскии 《Analysis Mathematica》1992,18(4):307-323
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. 相似文献
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С. Г. Мерзляков 《Analysis Mathematica》1989,15(1):3-16
A=(a
ij)
i
j=1
— k-o ,a
ij
. :
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С. Ю. Лукашенко 《Analysis Mathematica》1986,12(1):23-48
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. 相似文献
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С. А. ТЕЛЯКОВСКИЙ 《Analysis Mathematica》1982,8(4):305-319
Let σ n 2 (f, x) be the Cesàro means of second order of the Fourier expansion of the function f. Upper bounds of the deviationf(x)-σ n 2 (f, x) are studied in the metricC, while f runs over the class \(\bar W^1 C\) , i. e., of the deviation $$F_n^2 (\bar W^1 ,C) = \mathop {\sup }\limits_{f \in \bar W^1 C} \left\| {f(x) - \sigma _n^2 (f,x)} \right\|_c$$ . It is proved that the function $$g^* (x) = \frac{4}{\pi }\mathop \sum \limits_{v = 0}^\infty ( - 1)^v \frac{{\cos (2v + 1)x}}{{(2v + 1)^2 }}$$ , for whichg *′(x)=sign cosx, satisfies the following asymptotic relation: $$F_n^2 (\bar W^1 ,C) = g^* (0) - \sigma _n^2 (g^* ,0) + O\left( {\frac{1}{{n^4 }}} \right)$$ , i.e.g * is close to the extremal function. This makes it possible to find some of the first terms in the asymptotic formula for \(F_n^2 (\bar W^1 ,C)\) asn → ∞. The corresponding problem for approximation in the metricL is also considered. 相似文献
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