Optimal methods for the approximate calculation of functional on classesW r L ∞

Пусть T_n(f)={L₁(f), ..., L_n(f)} — набор линейных функционал ов, заданных на простран стве \(C_{(r - 1)} (\parallel f\parallel _{C_{(r - 1)} } = \mathop {\max }\limits_{0 \leqq i \leqq r - 1} \parallel f^{(i)} \parallel _C );A_{n,r}\) — множество всех так их наборов функцио налов; С_{2n, 2} — множество всех н аборов из 2n функциона лов вида $$T_{2n} (f) = \{ f(x_1 ), \ldots ,f(x_n ),f'(x_1 ), \ldots ,f'(x_n )\}$$ и s: Е_n→Е₁. Доказано, что е слиW _∞ ^r множество всех 2π-периодических функ цийfεW∞0, 2π_r, то приr=1,2,3,... ирε(1, ∞) и $$\begin{gathered} \mathop {\inf }\limits_{T_{2n} \in A_{2n,r} } \parallel \mathop {\inf }\limits_s \mathop {\sup }\limits_{f \in W_\infty ^r } |f( \cdot ) - s(T_{2n} ,f, \cdot )|\parallel _p = \parallel \varphi _{n,r} \parallel _p \hfill \\ \mathop {\inf }\limits_{T_{2n} \in C_{2n,2} } \parallel \mathop {\inf }\limits_s \mathop {\sup }\limits_{f \in W_\infty ^r } |f( \cdot ) - s(T_{2n} ,f, \cdot )|\parallel _p = \parallel \parallel \varphi _{n,r} \parallel _\infty - \varphi _{n,r} \parallel _p , \hfill \\ \end{gathered}$$ где ?_n,r —r-й периодичес кий интеграл, в средне м равный нулю на периоде, от фун кции ?_n, 0t=sign sinnt. При этом указан ы оптимальные методы приближенного вычис ления.     

On linear summation methods of Fourier series

Получены новые оценк иL-нормы тригонометр ических полиномов $$T_n (t) = \frac{{\lambda _0 }}{2} + \mathop \sum \limits_{k = 1}^n \lambda _k \cos kt$$ в терминах коэффицие нтовλ _k и их разностейΔλ _k=λ _k?λ _k?1: (1) $$\mathop \smallint \limits_{ - \pi }^\pi |T_n (t)|dt \leqq \frac{c}{n}\mathop \sum \limits_{k = 0}^n |\lambda _\kappa | + c\left\{ {x(n,\varphi )\mathop \sum \limits_{k = 0}^n \Delta \lambda _\kappa \mathop \sum \limits_{l = 0}^n \Delta \lambda _l \delta _{\kappa ,l} (\varphi )} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,$$ где $$\kappa (n,\varphi ) = \mathop \smallint \limits_{1/n}^\pi [t^2 \varphi (t)]^{ - 1} dt, \delta _{k,1} (\varphi ) = \mathop \smallint \limits_0^\infty \varphi (t)\sin \left( {k + \frac{1}{2}} \right)t \sin \left( {l + \frac{1}{2}} \right)t dt,$$ a ?(t) — произвольная фун кция ≧0, для которой опр еделены соответствующие инт егралы. Из (1) следует, что методы $$\tau _n (f;t) = (N + 1)^{ - 1} \mathop \sum \limits_{k = 0}^{\rm N} S_{[2^{k^\varepsilon } ]} (f;t), n = [2^{N\varepsilon } ],$$ являются регулярным и для всех 0<ε≦1/2. ЗдесьS _m (f, x) частные суммы ряда Фу рье функцииf(x). В статье исследуется многомерный случай. П оказано, что метод суммирования (о бобщенный метод Рисса) с коэффиц иентами $$\lambda _{\kappa ,l} = (R^v - k^\alpha - l^\beta )^\delta R^{ - v\delta } (0 \leqq k^\alpha + l^\beta \leqq R^v ;\alpha \geqq 1,\beta \geqq 1,v< 0)$$ является регулярным, когда δ > 1.     

Monotone polynomial approximation inL p space

We prove that for monotone functionf(x)∈L _p [?1, 1], 1≤p<∞, there exists a monotone algebraic polynomialL _n (f,x) of degreen such that \(\left\| {f(x) - L_n (f,x)} \right\|_p \leqslant C\omega _{2,\varphi } \left( {f,\frac{1}{n}} \right)_p \) , where \(\varphi (x) = \sqrt {1 - x^2 } \) and ω_2,?, is the second-order modulus of smoothness which weighs differently the behavior off(x) in the middle of the interval and near the endpoints. This estimate improves the known result of Shvedov.     

Steckin-marchaud-type inequalities in connection with bernstein polynomials

The purpose of this paper is to derive the estimate (0≤α≤2,n∈N,?(x)=[x(1?x)]^1/2) $$\omega _\alpha (n^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,f) \leqslant M_\alpha n^{ - 1} \sum\limits_{k = 1}^n {\left\| {\varphi ^{ - \alpha } (B_k f - f)} \right\|} c$$ in terms of the modulus of continuity (of second order) $$\omega _\alpha (t,f): = \sup \{ \varphi ^{ - \alpha } (x)|\Delta _{h\varphi (x)}^ * f(x)|:x,x \pm h\varphi (x) \in [0,1],0< h \leqslant t\} $$ and the Bernstein polynomial B_nf for ?^?αf∈C[0,1].     

On rational approximations of functions of complex variable integrable over plane domains

пУстьE — ИжМЕРИМОЕ пО лЕБЕгУ ОгРАНИЧЕННОЕ МНОжЕстВО пОлОжИтЕльНОИ плОЩА ДИ mes₂ E кОМплЕксНОИ плОск ОстИ с. кАк ОБыЧНО, пРИp≧1 ОБОжНАЧИМ ЧЕРЕжL ^p (E) БА НАхОВО пРОстРАНстВО ИжМЕРИ Мых пО лЕБЕгУ НАE кОМплЕксНОжНАЧНых Ф УНкцИИf с сУММИРУЕМО Иp—стЕпЕНьУ Их МОДУль И ОБыЧНОИ НОРМОИ \(\left\| \cdot \right\|_p = \left\| \cdot \right\|_{L_p (E)}\) . ЧЕР ЕжL ^p R _n (f,E) ОБОжНАЧИМ НАИМЕН ьшЕЕ УклОНЕНИЕf?L ^p (E) От РАц ИОНАльНых ФУНкцИИ ст ЕпЕНИ ≦n кОМплЕксНОгО пЕРЕМЕ ННОгОz пО НОРМЕ ∥ · ∥. пОлОжИМf(z)=0 Дльz?¯CE,E _δ —δ-ОкРЕстНОсть МНО жЕстВАE (δ>0), И $$\omega _p (\delta ,f) = \mathop {\sup {\mathbf{ }}}\limits_{\left| h \right|< \delta } \{ \int\limits_{E_\sigma } {\int {{\mathbf{ }}|f(z + h) - f(z)|^p } d\sigma } \} ^{1/p} .$$ тЕОРЕМА.пУсть 1≦p<2,f?L _p (E),n≧4.тОгДА $$\begin{array}{*{20}c} {L^p R_n (f,E) \leqq 12\omega _p \left( {\frac{{\delta + \ln n}}{{\sqrt n }},f} \right){\mathbf{ }}npu{\mathbf{ }}p = 1,} \\ {L^p R_n (f,E) \leqq \frac{{24}}{{(p - 1)(2 - p)}}\omega _p (n^{(p - 2)/2p} ,f){\mathbf{ }}npu{\mathbf{ }}1< p< 2,} \\ {L^1 R_n (\bar z,[0,1] \times [0,1]) \geqq \frac{1}{{32\sqrt n }}.} \\ \end{array} $$ .     

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

Certain positive linear operators

Properties (including the approximating ones) are investigated of positive linear operators L_n(f; x) for which the relation $$L_n \left( {\left( {t - x} \right)f\left( t \right); x} \right) = \frac{{\varphi \left( x \right)}}{n}L'_n \left( {f\left( t \right); x} \right)$$ is fulfilled, as well as the properties of operators L_n ^(m)(f;x). The results are applicable, in particular, to Bernstein polynomials, to the operators of Mirak'yan, Baskakov, and others.     

The completeness of systems of functions of the Mittag-Leffler type for weighted uniform approximation in a complex domain

For a givenρ(1/2 <ρ < + ∞) let us set L _ρ = {z: |arg z| = π/(2ρ)} and assume that a real valued measurable function ?(t) such that ?(t) ≥ 1(t ∈ L _ρ ) and \(\mathop {\lim }\limits_{|t| \to + \infty } \varphi (t) = + \infty (t \in L_\rho )\) is defined on L _ρ . Let C _? (L _ρ ) denote the space of continuous functionsf(t) on L _ρ such that \(\lim \tfrac{{f(t)}}{{\varphi (t)}} = 0\) , where the norm of an elementf is defined as: \(\parallel f\parallel = \mathop {\sup }\limits_{t \in L_\rho } \tfrac{{|f(t)|}}{{\varphi (t)}}\) . In this note we pose the question about the completeness of the system of functions of the Mittag-Leffler type {Eρ(ut; μ)} (μ ≥ 1, 0 ≤ u ≤a) or, what is the same thing, of the system of functions \(p(t) = \int_0^a {E_\rho (ut;\mu )d\sigma (u)} \) in C _? (L _ρ ). The following theorem is proved: The system of functions of the Mittag-Leffler type is complete in C _? (L _ρ ) if and only if sup |p(z)| ≡ +∞, z ∈ L _ρ , where the supremum is taken over the set of functions p(t) such that ∥p(t) (t + 1)^?1 ∥ ≤ 1.     

L p-approximation by Kantorovič operators

Оператор Канторович а дляf∈L _p(I), I=[0,1], определяе тся соотношением $$P_n (f,x) = (n + 1)\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} x^k (1 - x)^{n - 1} \int\limits_{I_k } {f(t)dt,} $$ гдеI _k=[k/(n}+1),(k+1)/(n+ 1)],n∈N. Доказывается, что есл ир>1 иf∈W _p ² (I), т.е.f абсол ютно непрерывна наI иf″∈L _p(I), то $$\left\| {P_n f - f} \right\|_p = O(n^{ - 1} ).$$ Далее, установлено, чт о еслиf∈L _p(I),p>1 и ∥P _n f-f∥_р=О(n ^?1), тоf∈S, гдеS={f∶f аб-солютно непрерывна наI, x(1?x)f′(x)=∝ ₀ ^x h(t)dt, гдеh∈L _p(I) и ∝ ₀ ¹ h(t)dt=0}. Если жеf∈L_p(I),p>1, то из условия ∥P _n(f)?f∥_pL=o(n^?1) вытекает, чтоf постоянна почти всюду.     

Limit ultraspherical series and their approximation properties

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x ₂)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S _n ^?1 (f) = S _n ^?1 (f, x) is the projection onto the subspace of algebraic polynomials p _n = p _n (x) of degree at most n, i.e., S _n (p _n ) = p _n ; in addition, S _n ^?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S _n ^?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ _n (x) of the partial sums S _n ^?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

Best L 1 Approximation of Truncated Functions,Whitney-Type and Borh–Favard-Type Inequalities

We present upper and lower estimates for the best constant C _n such that, if \({f \in W^{n+1}_1}\) [?1, 1], then $$E_n(f)_1 \leqq C_n||\varphi^{n+1}f^{(n+1)}||_1,$$ where \({\varphi(x) =\sqrt{1 - x^2}}\) . For the proof we first find out the polynomials of the best L ¹[?1, 1] approximation of some truncated functions.     

Local properties of functions and approximation by trigonometric polynomials

Suppose Φ_{p, E} (p>0 an integer, E ?[0, 2π]) is a family of positive nondecreasing functions? _x(t) (t>0, x∈ E) such that? _x(nt)≤n^P ? _x(t) (n=0,1,...), t_n is a trigonometric polynomial of order at most n, and Δ _h ^l (f, x) (l>0 an integer) is the finite difference of orderl with step h of the functionf.THEOREM. Supposef (x) is a function which is measurable, finite almost everywhere on [0, 2π], and integrable in some neighborhood of each point xε E,? _X εΦ_p,E and $$\overline {\mathop {\lim }\limits_{\delta \to \infty } } |(2\delta )^{ - 1} \smallint _{ - \delta }^\delta \Delta _u^l (f,x)du|\varphi _x^{ - 1} (\delta ) \leqslant C(x)< \infty (x \in E).$$ . Then there exists a sequence {t _n } _n=1 ^∞ which converges tof (x) almost everywhere, such that for x ε E $$\overline {\mathop {\lim }\limits_{n \to \infty } } |f(x) - l_n (x)|\varphi _x^{ - 1} (l/n) \leqslant AC(x),$$ where A depends on p andl.     

A system of functions

A system of functions $$f_k (x) = \sum\nolimits_{i = 1}^r a _i \varphi _\iota (x)^k + b_i \overline {\varphi _\iota } (x)^k , k = 1,2,...$$ is considered on the interval [0,l]. Under certain conditions on the? _i(x), it is proved that the system 1 ∪ {f_k(x)} _k=1 ^∞ is complete in the space L_p(0,l). In the case r=1 it is proved, under certain additional assumptions, that the system {f_k(x)} _k=0 ^∞ is minimal.     

On the mean value of certain functions connected with the convergence of orthogonal series

В статье рассматрива ются множестваQ ⁿ , 1≦п<∞, ортонормированных с истемΦ={φ _i (x)} _i ⁿ =1, состоящих из функций, постоянных на интервалах \(\left( {\frac{{j - 1}}{n}, \frac{j}{n}} \right)\) , 1 ≦j ≦j ≦п. НаQ ⁿ естественно перенос ится с группы ортогон альных матриц порядкаn мера Хаара. Изучается поведение наQ ⁿ функци и $$S(\Phi ) = \mathop {\sup }\limits_{\mathop \sum \limits_{i = 1}^n y_i^2 = 1} (\int\limits_0^1 {\mathop {sup}\limits_{1 \leqq r \leqq n} } (\mathop \sum \limits_{i = 1}^n y_i \varphi (x))^2 dx)^{1/2} $$ . Доказывается, что приt > 0 иn=1,2,... $$\mu \{ \Phi \in Q^n :s(\Phi ) \geqq t\} \leqq (Ce^{ - \gamma t^2 } )^n $$ .     

Commutators of Marcinkiewicz integral with rough kernels on Sobolev spaces

In this paper, the authors give the boundedness of the commutator [b, ??_??,?? ] from the homogeneous Sobolev space $\dot L_\gamma ^p \left( {\mathbb{R}^n } \right)$ to the Lebesgue space L ^p (? ⁿ ) for 1 < p < ??, where ??_??,?? denotes the Marcinkiewicz integral with rough hypersingular kernel defined by $\mu _{\Omega ,\gamma } f\left( x \right) = \left( {\int_0^\infty {\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega \left( {x - y} \right)}} {{\left| {x - y} \right|^{n - 1} }}f\left( y \right)dy} } \right|^2 \frac{{dt}} {{t^{3 + 2\gamma } }}} } \right)^{\frac{1} {2}} ,$ , with ?? ?? L ¹(S ^n?1) for $0 < \gamma < min\left\{ {\frac{n} {2},\frac{n} {p}} \right\}$ or ?? ?? L(log⁺ L) ^?? (S ^n?1) for $\left| {1 - \frac{2} {p}} \right| < \beta < 1\left( {0 < \gamma < \frac{n} {2}} \right)$ , respectively.     

Приближение алгебра ическими многочлена ми классов функций, являющихся д робными интегралами от суммируемых функц ий

Approximation in the mean (E _n(f)₁) by algebraic polynomials of order ≦n is studied in the paper, for classesW ₁ ^r of functionsf, which can be represented as $$f(x) = \frac{1}{{\Gamma (r)}}\int\limits_{ - 1}^1 {(x - t)_ + ^{^{r - 1} } } \varphi (t)dt,$$ where??L ₁-1, 1], ∥?∥₁≧1, (x-t) ₊ ^r1 =[max(0, x-t)]^r1, Г (r) stands for Euler's gamma-function. It is proved that for all realr≧1 and positive integersn≧[r]?1 the relation sup E_n(f)₁:f?W₁ ^r=∥(S_n)_r∥_t8, is valid, where $$(s_\Lambda )_{_r } (t) = \frac{1}{{\Gamma (r)}}\int\limits_{ - 1}^1 {(x - t)_ + ^{r - 1} } $$ sgn sin (n+2) arc cosx dx.     

Generalized monotone approximation inL p Space

Letf(x) ∈L _p[0,1], 1?p? ∞. We shall say that functionf(x)∈Δ^k (integerk?1) if for anyh ∈ [0, 1/k] andx ∈ [0,1?kh], we have Δ _h ^k f(x)?0. Denote by ∏ _n the space of algebraic polynomials of degree not exceedingn and define $$E_{n,k} (f)_p : = \mathop {\inf }\limits_{\mathop {P_n \in \prod _n }\limits_{P_n^{(\lambda )} \geqslant 0} } \parallel f(x) - P_n (x)\parallel _{L_p [0,1]} .$$ We prove that for any positive integerk, iff(x) ∈ Δ ^k ∩ L _p[0, 1], 1?p?∞, then we have $$E_{n,k} (f)_p \leqslant C\omega _2 \left( {f,\frac{1}{n}} \right)_p ,$$ whereC is a constant only depending onk.     

STRONG CONVERSE INEQUALITY FOR SZASZ-DURRMEYER OPERATORS

For Szasz-Durrmeyer operators Ln (f,z), 1＜ p≤∞, we prove that, forsome m, w^2φ(f,1/√n)p ≤(≤M(││Ln,f,x) - ,f││p ││Lmn(f, x) -f││p),where φ(x)^2 =x, M ＞0,w^2φ(f,t)p is Ditzian-Totik modulus of smoothness.     

Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

Let $\mathcal{X}$ be a metric space with doubling measure and L a nonnegative self-adjoint operator in $L^{2}(\mathcal{X})$ satisfying the Davies–Gaffney estimates. Let $\varphi:\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that φ(x,?) is an Orlicz function, $\varphi(\cdot,t)\in\mathbb{A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(φ)∈(0,1], and it satisfies the uniformly reverse Hölder inequality of order 2/[2?I(φ)]. In this paper, the authors introduce a Musielak–Orlicz–Hardy space $H_{\varphi,L}(\mathcal{X})$ , by the Lusin area function associated with the heat semigroup generated by L, and a Musielak–Orlicz BMO-type space $\mathrm{BMO}_{\varphi,L}(\mathcal{X})$ , which is further proved to be the dual space of $H_{\varphi,L}(\mathcal{X})$ and hence whose φ-Carleson measure characterization is deduced. Characterizations of $H_{\varphi,L}(\mathcal{X})$ , including the atom, the molecule, and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize $H_{\varphi,L}(\mathcal{X})$ in terms of the Littlewood–Paley $g^{\ast}_{\lambda}$ -function $g^{\ast}_{\lambda,L}$ and establish a Hörmander-type spectral multiplier theorem for L on $H_{\varphi,L}(\mathcal{X})$ . Moreover, for the Musielak–Orlicz–Hardy space H _φ,L (? ⁿ ) associated with the Schrödinger operator L:=?Δ+V, where $0\le V\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ , the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom, and the molecule; finally, the authors show that the Riesz transform ?L ^?1/2 is bounded from H _φ,L (? ⁿ ) to the Musielak–Orlicz space L ^φ (? ⁿ ) when i(φ)∈(0,1], and from H _φ,L (? ⁿ ) to the Musielak–Orlicz–Hardy space H _φ (? ⁿ ) when $i(\varphi)\in(\frac{n}{n+1},1]$ , where i(φ) denotes the uniformly critical lower type index of φ.