共查询到20条相似文献,搜索用时 31 毫秒
1.
Dr. Volker Vogelsang 《manuscripta mathematica》1975,14(4):379-401
We discuss the spectrum of a symmetric elliptic differential operator A with domain \(\mathop {H^m }\limits^o (\Omega ) \cap H^{2m} (\Omega )\) in regions Ω with unbounded boundary \(\dot \Omega \) , where are \(\bar \Omega \) uniformely of class C2m and on \(\dot \Omega \) the normal condition x·ν(x)≦μ for sufficient small positiveμ. We prove the A-priori-estimate \(\parallel u\parallel _{m,\Omega } \leqq c\parallel (l + r) (A - k)u\parallel _{o,\Omega } \) and show for all k>k, k≧0 suitable, there are no eigenvalues of A and by characterizing weighted Sobolev spaces with negative norm the existence of solutions \((l + r)_2 ^{ - 1} u \in \mathop H\limits^0{^m} (\Omega ) \cap H^{2m} (\Omega )\) of the equation (A?k)u=f, (1+r)f∈L2(Ω). 相似文献
2.
Letf be a continuous function on the circle ¦z¦=1. We present a theory of the (untruncated) “Carathéodory-Fejér (CF) table” of best supremumnorm approximants tof in the classes \(\tilde R_{mn} \) of functions $${{\tilde r(z) = \sum\limits_{k = - \infty }^m {a_k z^k } } \mathord{\left/ {\vphantom {{\tilde r(z) = \sum\limits_{k = - \infty }^m {a_k z^k } } {\sum\limits_{k = 0}^n {b_k } z^k ,}}} \right. \kern-\nulldelimiterspace} {\sum\limits_{k = 0}^n {b_k } z^k ,}}$$ , where the series converges in 1< ¦z¦ <∞. (The casem=n is also associated with the names Adamjan, Arov, and Krein.) Our central result is an equioscillation-type characterization: \(\tilde r \in \tilde R_{mn} \) is the unique CF approximant \(\tilde r^* \) tof if and only if \(f - \tilde r\) has constant modulus and winding numberω≥ m+ n+1?δ on ¦z¦=1, whereδ is the “defect” of \(\tilde r\) . If the Fourier series off converges absolutely, then \(\tilde r^* \) is continuous on ¦z¦=1, andω can be defined in the usual way. For general continuousf, \(\tilde r^* \) may be discontinuous, andω is defined by a radial limit. The characterization theorem implies that the CF table breaks into square blocks of repeated entries, just as in Chebyshev, Padé, and formal Chebyshev-Padé approximation. We state a generalization of these results for weighted CF approximation on a Jordan region, and also show that the CF operator \(K:f \mapsto \tilde r^* \) is continuous atf if and only if (m, n) lies in the upper-right or lower-left corner of its square block. 相似文献
3.
Dr. Wolfgang Alt 《manuscripta mathematica》1974,13(4):381-414
Using the local Kerzman kernel we prove regularity of solutions of \(\bar \partial \) u=f, where f is a \(\bar \partial \) -closed (0,1)-form in a strongly pseudoconvex domain G in ?N. If f is in Hm,∞, then the solution is in \(\tilde C^{m,\mu } \) forμ<1, that is, the m-th derivatives are in Co,μ/2 and in addition areμ-Hölder continuous on curves “parallel” to the holomorphic tangent bundle \(\tilde T\) ?G. If f is in Cm,α with o<α<1, then the solution is in \(\tilde C^{m,1 + \mu } \) forμ<α, that is, the m-th derivatives are in Co,(1+μ/2, and they have first derivatives “parallel” to \(\tilde T\) ?G lying in \(\tilde C^{o,\mu } \) . We derive the same results for the global solution constructed by Grauert and Lieb, and similar estimates on complex manifolds. 相似文献
4.
After recalling the basic properties of para-coKählerian manifolds \(\tilde M\) with concircular structure vector field ξ, the infinitesimal auto morphismsX of the structure 1-form \(\tilde \eta \) are considered. One of the results is that the Lie derivative of all powers of the structure 2-form \(\tilde \Omega ,\) i.e. \(\mathcal{L}x\tilde \Omega ^p ;p = 1,...,m,\) is exterior recurrent. Further two types of horizontal distributionsD n which are normal to ξ. IfD t (resp.D n ) is involutive, the corresponding leafM t (resp.M n ) is a minimal submanifold of \(\tilde M\) . FurtherM n is a symplectic submanifold and ξ is an umbilical normal section ofM n . Finally proper immersion \(M \to \tilde M\) are discussed, whereM is aCR-sub-manifold whose horizontal distribution isD t . It is shown that the vertical distribution is involutive, and the restriction of ξ toM is an symptotic direction. Some interesting special cases are treated. 相似文献
5.
Prof. T. Thrivikraman 《Monatshefte für Mathematik》1975,79(2):151-155
In this paper, we obtain analogues, in the situation of \(\mathfrak{E}\) -extensions, of Magill's theorem on lattices of compactifications. We define an epireflective subcategory of the categoryT 2 of all Hausdorff spaces to be admissive (respectively finitely admissive) if for any \(\mathfrak{E}\) -regular spaceX, every Hausdorff quotient of \(\beta _\mathfrak{E} X\) which is Urysohn on \(\beta _\mathfrak{E} X - X\) (respectively which is finitary on \(\beta _\mathfrak{E} X - X\) ) and which is identity onX, has \(\mathfrak{E}\) . We notice that there are many proper epireflective subcategories ofT 2 containing all compact spaces and which are admissive; there are many such which are not admissive but finitely admissive. We prove that when \(\mathfrak{E}\) is a finitely admissive epireflective subcategory ofT 2, then the lattices of finitary \(\mathfrak{E}\) -extensions of two spacesX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic. Further if \(\mathfrak{E}\) is admissive, then the lattices of Urysohn \(\mathfrak{E}\) -extensions ofX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic. 相似文献
6.
Let \(\tilde W_p^r : = \left\{ {f\left| {f \in C^{r - 1} } \right.} \right.\left[ {0,2\pi } \right],f^{(i)} (0) = f^{(i)} (2\pi ),i = 0, \ldots ,r - 1,f^{(r - 1)}\) , abs. cont. on [0, 2π] andf (r)∈L p[0, 2π]}, and set \(\tilde B_p^r : = \left\{ {f\left| {f \in \tilde W_p^r ,} \right.\left\| {f^{(r)} } \right\|_p \leqslant 1} \right\}\) . We find the exact Kolmogrov, Gel'fand, and linearn-widths of \(\tilde B_p^r\) inL p forn even and allp∈(1, ∞). The strong asymptotic estimates forn-widths of \(\tilde B_p^r\) inL p are also obtained. 相似文献
7.
郑祖康 《应用数学学报(英文版)》1987,3(2):180-192
Let{Y_t,t=1,2,…} be independent random variables with continuous distribution functionsF_i(y).For any y,dencte s=F_t(y)=1/t sum from i=1 to t F_i(y).The empirical process is defind by t~(-1/2)R(s,t) whereR(s,t)=t(1/t sum from i=1 to t I_((?)_t(Y_i)≤s)-s)=sum from i=1 to t I_(?)-ts=sum from i=1 to t I_(?)-(?)_t(y)=sum from i=1 to t I_(Y_(?)≤y)-sum from i=1 to t F_i(y).The purpose of this paper is to investigate the asymptotic properties of the empirical processR(s,t).We shall prove that for some integer sequence {t_k},there is a (?)-process (?)(s,t) such that(?)|R(s,t_k)-(?)(s,t_k)|=O(t_k~(1/2)(log t_k)~(-1/4)(log log t_k)~(1/2))a.s.where (?)(s,t) is a two-parameter Gaussian process defined in §1. 相似文献
8.
Hans Triebel 《Analysis Mathematica》1984,10(1):53-77
В статье рассматрива ются анизотропные пр остранства Бесова \(B_p^{\bar s} \) и Соболева \(W_p^{\bar s} \) н а плоскости и на единич ном круге, где 1<р<∞ и \(1< p< \infty \) И \(\bar s = (s_1 ,s_2 )\) . Основная цель состои т в доказательстве анизотропных нераве нств Харди и в изучени и соответствующих про странств \(\dot B_p^{\bar s} \) и \(\dot W_p^{\bar s} \) типа Бесова—Соболе ва. Эти результаты буд ут использованы во втор ой работе для точного описания следов упом янутых пространств н а плоских кривых. 相似文献
9.
M. Laczkovich 《Analysis Mathematica》1977,3(3):199-206
qVЕРхНИИ пРЕДЕл пОслЕД ОВАтЕльНОстИ МНОжЕс тВA n ОпРЕДЕльЕтсь сООтНО шЕНИЕМ \(\mathop {\lim sup}\limits_{n \to \infty } A_n = \mathop \cap \limits_{k = 1}^\infty \mathop \cup \limits_{n = k}^\infty A_n . B\) стАтьЕ РАссМАтРИВА Етсь слЕДУУЩИИ ВОпРО с: ЧтО МОжНО скАжАть О ВЕРхНИх пРЕДЕлАх \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) , еслИ ИжВЕстНО, ЧтО пРЕсЕЧЕНИь \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) «МАлы» Дль кАж-ДОИ пОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) ? ДОкАжыВАЕтсь, Ч тО
- ЕслИ \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) — кОНЕЧНОЕ МНО жЕстВО Дль кАжДОИ пОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) , тО НАИДЕтсь тАкАь пОДпО слЕДОВАтЕльНОсть, Дл ь кОтОРОИ МНОжЕстВО \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) сЧЕтНО;
- ЕслИ \(2^{\aleph _0 } = \aleph _1\) , тО сУЩЕстВУЕ т тАкАь пОслЕДОВАтЕл ьНОсть (An), ЧтО \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) — сЧЕтНОЕ МНОжЕстВО Дль лУБОИ п ОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) , НО \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) ИМЕЕт МОЩ-НОсть кОНтИНУУМА;
- ЕслИA n — БОРЕлЕ ВскИЕ МНОжЕстВА В НЕкОтОРО М пОлНОМ сЕпАРАБЕльНО М МЕтРИЧЕскОМ пРОстРАНстВЕ, И \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) — сЧЕт НОЕ МНОжЕстВО Дль кАж ДОИ пОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) , тО сУЩЕстВУЕт тАкАь п ОДпОслЕДОВАтЕльНОсть, ЧтО \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) — сЧЕтНОЕ МНОжЕстВО. кРОМЕ тОгО, ДОкАжАНО, Ч тО В слУЧАьх А) И В) В пОслЕДОВАтЕльНОстИ (A n ) сУЩЕстВУЕт схОДьЩА ьсь пОДпОслЕДОВАтЕльНО сть.
10.
Thomas Peternell 《manuscripta mathematica》1983,42(2-3):259-263
Let X be a complex space and A?X a compact subspace. Let \(\tilde X\) be the blowing up of A in X and \(\tilde A\) ? \(\tilde X\) the resulting hyper-surface. Then the normal bundle of \(\tilde A\) in { \(\tilde X\) is weakly negative iff the normal bundle of the k-th infinitesimal neighborhood of A in X is weakly negative for all k?0. This corrects a theorem in [5]. 相似文献
11.
12.
Rolf Trautner 《Analysis Mathematica》1988,14(2):111-122
Рассматриваются слу чайная величина \(\mathfrak{X} = (X_n (\omega ))\) , удовлетворяющая усл овиюE(X n 4 )≦M, и соответствующ ий случайный степенн ой ряд \(f_x (z;\omega ) = \mathop \sum \limits_{n = 0}^\infty a_n X_n (\omega )z^n\) . Устанавливаются тео ремы непродолжимост и почти наверное:
- дляf x при условиях с лабой мультипликати вности на \(\mathfrak{X}\) ,
- для \(f_{\tilde x}\) , где \(\mathop \mathfrak{X}\limits^ \sim = (\mathop X\limits^ \sim _n )\) есть подп оследовательность в \(\mathfrak{X}\) ,
- для по крайней мере од ного из рядовf x′ илиf x″ , где \(\mathfrak{X}'\) и \(\mathfrak{X}''\) — некоторые п ерестановки \(\mathfrak{X}\) , выбираемые универс ально, т. е. независимо от коэффициентовa n .
13.
14.
D. Leviatan 《Israel Journal of Mathematics》1970,8(2):138-146
The purpose of this paper is to prove that for a large set of absolute Hausdorff and quasi-Hausdorff methods the condition $$\sum\limits_{k = 1}^\infty {\left| {\lambda _n a_n - \lambda _{n - 1} a_{n - 1} } \right|< } \infty $$ is a Tauberian condition, i.e., its fulfillment together with the absolute summability of \(\sum\limits_{n = 0}^\infty {a_n } \) tos implies that \(\sum\limits_{n = 0}^\infty {\left| {a_n } \right|}< \infty \) and \(\sum\limits_{n = 0}^\infty {a_n } = s.\) a n =s. 相似文献
15.
N. D. Nagaev 《Mathematical Notes》1976,20(1):581-588
For fixed ?>0, the following inequality holds: $$\left| {\frac{u}{\upsilon } - \wp } \right| > Cexp\left( { - \left( {lnH} \right)^{2 + \varepsilon } } \right)$$ for all numbers β belonging to a field K of finite degree over Q. The constant C>0 does not depend on β. H is the height of β. \(\wp \) (u) and \(\wp \) (v) are algebraic numbers, and u/v is a transcendental number. \(\wp \) (z) is the Weierstrass function with complex multiplication and algebraic invariants. The proof is ineffective. 相似文献
16.
Mei-Chi Shaw 《Journal of Geometric Analysis》1991,1(2):139-163
LetM be the boundary of a strongly pseudoconvex domain in \(\mathbb{C}^n \) ,n≥4 and ω be an open subset inM such that ?ω is the intersection ofM with a flat hypersurface. We establish theL 2 existence theorems of the \(\bar \partial _b - Neumann\) problem on ω. In particular, we prove that the \(\bar \partial _b - Laplacian\) \(\square _b = \bar \partial _b \bar \partial _b^* + \bar \partial _b^* \bar \partial _b \) equipped with a pair of natural boundary conditions, the so-called \(\bar \partial _b - Neumann\) boundary conditions, has closed range when it acts on (0,q) forms, 1≤q≤n?3. Thus there exists a bounded inverse operator for \(\square _b \) , the \(\bar \partial _b - Neumann\) operatorN b, and we have the following Hodge decomposition theorem on ω for \(\bar \partial _b \bar \partial _b^* N_b \alpha + \bar \partial _b^* \bar \partial _b N_b \alpha \) , for any (0,q) form α withL 2(ω) coefficients. The proof depends on theL p regularity of the tangential Cauchy-Riemann operators \(\bar \partial _b u = \alpha \) on ω?M under the compatibility condition \(\bar \partial _b \alpha = 0\) , where α is a (p, q) form on ω, where 1≤q≤n?2. The interior regularity ofN b follows from the fact that \(\square _b \) is subelliptic in the interior of ω. The operatorN b induces natural questions on the regularity up to the boundary ?ω. Near the characteristic point of the boundary, certain compatibility conditions will be present. In fact, one can show thatN b is not a compact operator onL 2(ω). 相似文献
17.
John J. Benedetto 《Rendiconti del Circolo Matematico di Palermo》1985,34(3):407-421
We prove \(\left\| F \right\|_{2,\Omega } \leqslant c({\rm T} \Omega )\left\| f \right\|_{A{}_T} \) , whereF is the Fourier transform off,||F||2,Ω is theL 2-norm ofF on \([ - \Omega ,\Omega ],\left\| f \right\|_{A{}_T} \) is the absolutely convergent Fourier series norm for 2T-periodic functions, and $$c(T\Omega ) = (\frac{1}{\pi }\int\limits_{ - T\Omega }^{T\Omega } {\frac{{\sin ^2 \gamma }}{{\gamma ^2 }}d\gamma } )^{1/2} $$ Analogous inequalities, depending on prolate spheroidal wave functions, are more difficult to prove and their constants are less explicit. 相似文献
18.
So-Chin Chen 《Inventiones Mathematicae》1988,92(1):173-185
In this paper we show that if \(D \subseteq \mathbb{C}^n ,n \geqq 2\) , is a smooth bounded pseudoconvex circular domain with real analytic defining functionr(z) such that \(\sum\limits_{k = 1}^n {z_k \frac{{\partial r}}{{\partial z_k }}} \ne 0\) for allz near the boundary, then the solutionu to the \(\bar \partial\) -Neumann problem, $$square u = (\bar \partial \bar \partial * + \bar \partial *\bar \partial )u = f,$$ is real analytic up to the boundary, if the given formf is real analytic up to the boundary. In particular, if \(D \subseteq \mathbb{C}^n ,n \geqq 2\) , is a smooth bounded complete Reinhardt pseudoconvex domain with real analytic boundary. Then ? is analytic hypoelliptic. 相似文献
19.
20.
V. Totik 《Analysis Mathematica》1978,4(2):145-152
Пустьf(x) — интегрируемая 2π-периодическая функция, aω(f,δ) иs n(x)=sn(f, x). соответственно, модуль непрерывности иn-ая сумма Фурье этой функции. В настоящей работе, продолжающей исследования Г. Фрейда, Л. Лейндлера—E. M. Никищина, И. Сабадоша и К. И. Осколкова, доказывается следующая теорема.Если Ω(u) — выпуклая или вогнутая непрерывная функция и если (1) 1 $$\left\| {\left. {\sum\limits_{k = 1}^\infty \Omega (|S_k (x) - f(x)|)} \right\|_C } \right.$$ то 1 $$\omega (f;\delta ) = O\left( {\delta \int\limits_\delta ^1 {\frac{{\bar \Omega (v)}}{{v^2 }}dv} } \right),$$ где ¯Ω(v) —функция, обратная к Ω(и). При этом существует функция f0(х), удовлетворяющая условию (1), для которой $$\omega (f;\delta ) = c\delta \int\limits_\delta ^1 {\frac{{\bar \Omega (v)}}{{v^2 }}dv} (c > 0).$$ ЕслиΩ(u)— вогнутая функция, то интеграл \(\int\limits_\delta ^1 {\frac{{\bar \Omega (v)}}{{v^2 }}dv} \) можно заменить на \(\int\limits_{\bar \Omega (\delta )}^1 {\frac{{du}}{{\Omega (u)}}.} \) . Отсюда вытекает, что еслиΩ(u) — функция типа модуля непрерывности, то для того, чтобы (1) всегда влекло принадлежность f(x) классу Lip 1, необходимо и достаточно условие \(\int\limits_0^1 {\frac{{du}}{{\Omega (u)}}}< \infty .\) 相似文献