Analysis Mathematica

т. шЕИл-сМЕлл ВыскАжАл пРЕДпОлОжЕНИЕ, ЧтО Дл ьn=1,2, ... $$\mathop {\lim }\limits_{r \to 0} \frac{{(1 - r)^{n + 1} M^{n + 1} (r,f)}}{{M(r,f)}} = \alpha (\alpha + 1) \ldots (\alpha + n),$$ гДЕf-жВЕжДООБРАжНАь ФУНкцИь ИM(r,f)=max{¦f(z)¦: [z]=r}. Мы пОкАжыВАЕМ Дль слУЧА ьn=2, ЧтО ЁтО НЕВЕРНО.

Given f ∈ L loc 1 (R +), we define $$s\left( t \right): = \int\limits_0^t {f\left( x \right)} {\text{ }}dx{\text{ and }}\sigma \left( t \right):\frac{1}{t}{\text{ }}\int\limits_0^t {s\left( u \right)} {\text{ }}du{\text{ for }}t >0$$ Our permanent assumption is that σ(t) → A as t → ∞, where A is a finite number. First, we consider real-valued functions, and prove that s(t) → A as t → ∞ if and only if two one-sided Tauberian conditions are satisfied. In particular, these two conditions are satisfied if s(t) is slowly decreasing (or increasing) in the sense of R. Schmidt; in particular, if f(x) obeys a Landau type one-sided Tauberian condition. Second, we extend these results for complex-valued functions by giving a two-sided Tauberian condition, being necessary and sufficient in order that σ(t) → A imply s(t) → A as t → ∞. In particular, this condition is satisfied if s(t) is slowly oscillating; in particular if f(x) obeys a Landau type two-sided Tauberian condition.

Let $$W_1 (G) = \{ x = G*y:y\varepsilon L_1 ,\parallel y\parallel _1 \leqq 1\} (L_1 = L_1 (R^1 ))$$ and $$W_1^0 (G) = \{ x = G*y\varepsilon W_1 (G), y \bot 1\} .$$ For each even functionG(t) (¦G(v)(t)¦≦/(1+t2), v=0,1,...; t∃R1) such that its Fourier transformg(t) is 4 times monotonous on [λ, ∞) and tends to zero ast→∞, exact estimates of the best one-sided approximations by entire functions of exponential type≦σ (σ≧λ) are calculated for the classesW 1 (G) andW 1 0 (G) inL 1-metric.

The factorization of inequalities for infinite numerical series is a new research field. A monograph giving a systematic treatment of this subject is due to BENNETT. Here two theorems of this type are proved. Our new results extend previous theorems to the so-called Mulholland functions.

We introduce the vector-valued Sobolev spaces W m,p(x) (Ω;E 0,E) with variable exponent associated with two Banach spaces E 0 and E. The most regular space E α is found such that the differential operator D α is bounded and compact from W m,p(x)(Ω;E 0,E) to L q(x)(Ω;E α ), where E α are interpolation spaces between E 0 and E is depending on α = (α 1, α 2,..., α n ) and the positive integer m, where Ω ⊂ ℝ n is a region such that there exists a bounded linear extension operator from W m,p(x) (Ω;E 0,E) to W m,p(x) (ℝ n ;E (A), E). The function p(x) is Lipschitz continuous on Ω and q(x) is a measurable function such that $$1 < p(x) \leqslant q(x) \leqslant \tfrac{{np(x)}} {{n - mp(x)}}$$ for a.e. $$x \in \bar \Omega $$ . Ehrling–Nirenberg–Gagilardo type sharp estimates for mixed derivatives are obtained. Then, by using this embedding result, the separability properties of abstract differential equations are established.

В работе изучаются вопросы приближения линейными методами классов B , Ω , 1 ≤ p ≤ ∞, периодических функций многих переменных в пространстве L ∞.