共查询到20条相似文献,搜索用时 40 毫秒
1.
V. V. Lebedev 《Functional Analysis and Its Applications》2012,46(2):121-132
We consider the space
A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle
\mathbbT\mathbb{T} such that the sequence of Fourier coefficients
[^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l
1(ℤ). The norm on
A(\mathbbT)A(\mathbb{T}) is defined by
|| f ||A(\mathbbT) = || [^(f)] ||l1 (\mathbbZ)\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if
f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that
|| einf ||A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that
|| einf ||A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/
{\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right.
\kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/
{\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right.
\kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear. 相似文献
2.
S. E. Pastukhova 《Journal of Mathematical Sciences》2012,181(5):668-700
We consider the operator exponential e
−tA
, t > 0, where A is a selfadjoint positive definite operator corresponding to the diffusion equation in
\mathbbRn {\mathbb{R}^n} with measurable 1-periodic coefficients, and approximate it in the operator norm
|| · ||L2( \mathbbRn ) ? L2( \mathbbRn ) {\left\| {\; \cdot \;} \right\|_{{{L^2}\left( {{\mathbb{R}^n}} \right) \to {L^2}\left( {{\mathbb{R}^n}} \right)}}} with order
O( t - \fracm2 ) O\left( {{t^{{ - \frac{m}{2}}}}} \right) as t → ∞, where m is an arbitrary natural number. To construct approximations we use the homogenized parabolic equation with constant
coefficients, the order of which depends on m and is greater than 2 if m > 2. We also use a collection of 1-periodic functions N
α
(x),
x ? \mathbbRn x \in {\mathbb{R}^n} , with multi-indices α of length
| a| \leqslant m \left| \alpha \right| \leqslant m , that are solutions to certain elliptic problems on the periodicity cell. These results are used to homogenize the diffusion
equation with ε-periodic coefficients, where ε is a small parameter. In particular, under minimal regularity conditions, we construct approximations of order O(ε
m
) in the L
2-norm as ε → 0. Bibliography: 14 titles. 相似文献
3.
Given an isotropic random vector X with log-concave density in Euclidean space
\mathbbRn{\mathbb{R}^n} , we study the concentration properties of |X| on all scales, both above and below its expectation. We show in particular that
l \mathbbP( | |X| - ?n | 3 t?n ) £ C exp ( -cn1/2 min(t3, t) ) "t 3 0, \begin{array}{l} \mathbb{P}\left ( \left | |X| - \sqrt{n} \right | \geq t\sqrt{n} \right ) \leq C \, {\rm exp} \left ( -cn^{1/2} {\rm min}(t^{3}, t) \right) \; \forall t \geq 0, \end{array} 相似文献
4.
E. A. Sevost’yanov 《Ukrainian Mathematical Journal》2011,63(1):84-97
For open discrete mappings
f:D\{ b } ? \mathbbR3 f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain
D ì \mathbbR3 D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point
b ? \mathbbR3 b \in {\mathbb{R}^3} , we prove the following statement: Let a point y
0 belong to
[`(\mathbbR3)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K
I
(x, f) and outer dilatation K
O
(x, f) of the mapping f at the point x satisfy certain conditions. Let B
f
denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y
0, the set V ∩ f(B
f
) cannot be contained in a set A such that g(A) = I, where
I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and
g:U ? \mathbbRn g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain
U ì \mathbbRn U \subset {\mathbb{R}^n} such that A ⊂ U. 相似文献
5.
Let
C( \mathbbRm ) C\left( {{\mathbb{R}^m}} \right) be the space of bounded and continuous functions
x:\mathbbRm ? \mathbbR x:{\mathbb{R}^m} \to \mathbb{R} equipped with the norm
|