共查询到20条相似文献,搜索用时 15 毫秒
1.
S. Norvidas 《Lithuanian Mathematical Journal》2008,48(1):92-99
Let
be a complex unital Banach algebra. An element a ∈
is said to be Hermitian if ‖exp(ita)‖ = 1 for all t ∈ ℝ. In the case of the algebra of bounded linear operators in a Hilbert space, this Hermitian property agrees with the ordinary
self-adjointness. If a ∈
is Hermitian, then ‖a‖ = |a|, where |a| denotes the spectral radius of a. A function F: ℝ → ℂ is called a universal symbol if ‖F(a)‖ = | F(a)| for every
and all Hermitian a ∈
. We characterize universal symbols in terms of positive-definite functions. 相似文献
2.
S. Norvidas 《Lithuanian Mathematical Journal》2009,49(2):185-189
For a compact set K in ℝ
n
, let B
2
K
be the set of all functions f ∈ L
2(ℝ2) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B
2
K
by finite exponential sums
in the space , as τ → ∞. 相似文献
3.
Consider the system with perturbation g
k
∈ ℝ
n
and output z
k
= Cx
k
. Here, A
k
,A
k
(s) ∈ ℝ
n × n
, B
k
(1) ∈ ℝ
n × p
, B
k
(2) ∈ ℝ
n × m
, C ∈ ℝ
p × n
. We construct a special Lyapunov-Krasovskii functional in order to synthesize controls u
k
(1) and u
k
(2) for which the following properties are satisfied:
$
z_{k + 1} = qz_k ,0 < q < 1(outputinvariance)
$
z_{k + 1} = qz_k ,0 < q < 1(outputinvariance)
相似文献
4.
I. V. Filimonova 《Journal of Mathematical Sciences》2007,143(4):3415-3428
One considers a semilinear parabolic equation u
t
= Lu − a(x)f(u) or an elliptic equation u
tt
+ Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ+ with the nonlinear boundary condition
, where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝn; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems
for t → ∞.
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007. 相似文献
5.
Miran Černe 《Arkiv f?r Matematik》2002,40(1):27-45
LetX(-ϱB
m
×C
n
be a compact set over the unit sphere ϱB
m
such that for eachz∈ϱB
m
the fiberX
z
={ω∈C
n
;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain inC
n
. The polynomial hull
ofX is described in terms of the Perron-Bremermann function for the homogeneous defining function ofX. Moreover, for each point (z
0,w
0)∈Int
there exists a smooth up to the boundary analytic discF:Δ→B
m
×C
n
with the boundary inX such thatF(0)=(z
0,w
0).
This work was supported in part by a grant from the Ministry of Science of the Republic of Slovenia. 相似文献
6.
A. L. Gladkov 《Mathematical Notes》1996,60(3):264-268
We study the Cauchy problem in the layer Π
T
=ℝ
n
×[0,T] for the equationu
t
=cGΔu
t
+Δϕ(u), wherec is a positive constant and the functionϕ(p) belongs toC
1(ℝ+) and has a nonnegative monotone non-decreasing derivative. The unique solvability of this Cauchy problem is established for
the class of nonnegative functionsu(x,t) ∈C
x,t
2,1
(Π
T
) with the properties:
,
.
Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 356–362, September, 1996.
This research was partially supported by the International Science Foundation under grant No. MX6000. 相似文献
7.
G. I. Laptev 《Journal of Mathematical Sciences》2008,150(5):2384-2394
This paper deals with conditions for the existence of solutions of the equations
8.
Given two Banach spaces E,F, let B(E,F) be the set of all bounded linear operators from E into F, Σ
r
the set of all operators of finite rank r in B(E,F), and Σ
r
# the number of path connected components of Σ
r
. It is known that Σ
r
is a smooth Banach submanifold in B(E,F) with given expression of its tangent space at each A ∈ Σ
r
. In this paper,the equality Σ
r
# = 1 is proved. Consequently, the following theorem is obtained: for any nonnegative integer r, Σ
r
is a smooth and path connected Banach submanifold in B(E,F) with the tangent space T
A
Σ
r
= {B ∈ B(E,F): BN(A) ⊂ R(A)} at each A ∈ Σ
r
if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property
of Σ
r
the method presented in this paper is also interesting. As an application of this result, it is proved that if E = ℝ
n
and F = ℝ
m
, then Σ
r
is a smooth and path connected submanifold of B(ℝ
n
, ℝ
m
) and its dimension is dimΣ
r
= (m+n)r−r
2 for each r, 0 <- r < min {n,m}.
Supported by the National Science Foundation of China (Grant No.10671049 and 10771101). 相似文献
9.
Let ℝℝ denote the set of real valued functions defined on the real line. A map D: ℝℝ → ℝℝ is said to be a difference operator if there are real numbers a
i, b
i (i = 1, …, n) such that (Dƒ)(x) = ∑
i=1
n
a
i
ƒ(x + b
i) for every ƒ ∈ ℝℝand x ∈ ℝ. By a system of difference equations we mean a set of equations S = {D
i
ƒ = g
i: i ∈ I}, where I is an arbitrary set of indices, D
i is a difference operator and g
i is a given function for every i ∈ I, and ƒ is the unknown function. One can prove that a system S is solvable if and only if every finite subsystem of S is solvable. However, if we look for solutions belonging to a given class of functions then the analogous statement is no
longer true. For example, there exists a system S such that every finite subsystem of S has a solution which is a trigonometric polynomial, but S has no such solution; moreover, S has no measurable solutions.
This phenomenon motivates the following definition. Let
be a class of functions. The solvability cardinal sc(
) of
is the smallest cardinal number κ such that whenever S is a system of difference equations and each subsystem of S of cardinality less than κ has a solution in
, then S itself has a solution in
. In this paper we determine the solvability cardinals of most function classes that occur in analysis. As it turns out, the
behaviour of sc(
) is rather erratic. For example, sc(polynomials) = 3 but sc(trigonometric polynomials) = ω
1, sc({ƒ: ƒ is continuous}) = ω
1 but sc({f : f is Darboux}) = (2
ω
)+, and sc(ℝℝ) = ω. We consistently determine the solvability cardinals of the classes of Borel, Lebesgue and Baire measurable functions, and
give some partial answers for the Baire class 1 and Baire class α functions.
Partially supported by Hungarian Scientific Foundation grants no. 49786,37758,F 43620 and 61600.
Partially supported by Hungarian Scientific Foundation grant no. 49786. 相似文献
10.
Guo-en HU~ Da-chun YANG~ 《中国科学A辑(英文版)》2007,50(11):1621-1641
Letμbe a nonnegative Radon measure on R~d which only satisfiesμ(B(x,r))≤C_0r~n for all x∈R~d,r>0,and some fixed constants C_0>0 and n∈(0,d].In this paper,some weighted weak type estimates with A_(p,(log L)~σ)~ρ(μ) weights are established for the commutators generated by Calder■n-Zygmund singular integral operators with RBMO(μ) functions. 相似文献
11.
O. V. Matveev 《Mathematical Notes》1997,62(3):339-349
Supposem, n ∈ℕ,m≡n (mod 2),K(x)=|x|
m
form odd,K(x)=|x|
m
In |x| form even (x∈ℝ
n
),P is the set of real polynomials inn variables of total degree ≤m/2, andx
1,...,x
N
∈ℝ
n
. We construct a function of the form
12.
Djemaïa Bensikaddour Sadek Gala Amina Lahmar-Benbernou 《Periodica Mathematica Hungarica》2008,57(1):1-22
The most important result stated in this paper is to show that the solutions of the Poisson equation −Δu = f, where f ∈ (Ḣ1(ℝ
d
) → (Ḣ−1(ℝ
d
)) is a complex-valued distribution on ℝ
d
, satisfy the regularity property D
k
u ∈ (Ḣ1 → Ḣ−1) for all k, |k| = 2. The regularity of this equation is well studied by Maz’ya and Verbitsky [12] in the case where f belongs to the class of positive Borel measures.
相似文献
13.
WU Hao & LI Weigu School of Mathematical Sciences Peking University Beijing China 《中国科学A辑(英文版)》2005,48(12):1670-1682
In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case. 相似文献
14.
Xian ZHANG 《数学学报(英文版)》2006,22(3):873-878
Suppose F is a field different from F2, the field with two elements. Let Mn(F) and Sn(F) be the space of n × n full matrices and the space of n ×n symmetric matrices over F, respectively. For any G1, G2 ∈ {Sn(F), Mn(F)}, we say that a linear map f from G1 to G2 is inverse-preserving if f(X)^-1 = f(X^-1) for every invertible X ∈ G1. Let L (G1, G2) denote the set of all inverse-preserving linear maps from G1 to G2. In this paper the sets .L(Sn(F),Mn(F)), L(Sn(F),Sn(F)), L (Mn(F),Mn(F)) and L(Mn (F), Sn (F)) are characterized. 相似文献
15.
The wave equation, ∂
tt
u=Δu, in ℝ
n+1, considered with initial data u(x,0)=f∈H
s
(ℝ
n
) and u’(x,0)=0, has a solution which we denote by . We give almost sharp conditions under which and are bounded from H
s
(ℝ
n
) to L
q
(ℝ
n
). 相似文献
16.
I. Kocsis 《Acta Mathematica Hungarica》2008,118(4):307-312
Let X ⊂ ℝ be an interval of positive length and define the set Δ = {(x, y) ∈ X × X | x ≧ y}. We give the solution of the equation
17.
Let (S)⊄L
2(S′(∔),μ)⊄(S)* be the Gel'fand triple over the white noise space (S′(∔),μ). Let (e
n
,n>-0) be the ONB ofL
2(∔) consisting of the eigenfunctions of the s.a. operator
. In this paper the Euler operator Δ
E
is defined as the sum
, where ∂
i
stands for the differential operatorD
e
i. It is shown that Δ
E
is the infinitesimal generator of the semigroup (T
t
), where (T
t
ϕ)(x)=ϕ(e
t
x) for ϕ∈(S). Similarly to the finite dimensional case, the λ-order homogeneous test functionals are characterized by the Euler equation:
Δ
Eϕ
=λϕ. Via this characterization the λ-order homogeneous Hida distributions are defined and their properties are worked out.
Supported by the National Natural Science Foundation of China. 相似文献
18.
Liguang Liu 《Frontiers of Mathematics in China》2007,2(4):599-611
Let ℐ(ℝn) be the Schwartz class on ℝn and ℐ∞(ℝn) be the collection of functions ϕ ∊ ℐ(ℝn) with additional property that
19.
Jerome A. Goldstein 《Semigroup Forum》1996,52(1):37-47
Of concern are semigroups of linear norm one operators on Hilbert space of the form (discrete case)T={T
n
/n=0,1,2,...} or (continuous case)T={T(t)/t=≥0}. Using ergodic theory and Hilbert-Schmidt operators, the Cesàro limits (asn→∞) of |〈T
n
f,f〉|2, |〈T
(n)f,f〉|2 are computed (withn∈ℤ+ orn∈ℤ+). Specializing the Hilbert space to beL
2(T,μ) (discrete case) orL
2(ℝ,μ) (continuous case) where μ is a Borel probability measure on the circle group or the line, the Cesàro limit of
(asn→±∞, with,n∈ℤ orn∈ℝ) is obtained and interpreted. Extensions toT
M
, and ℝ
M
are given. Finally, we discuss recent operator theoretic extensions from a Hilbert to a Banach space context.
Partially supported by an NSF grant 相似文献
20.
LetX andY denote two complex Banach spaces and letB(Y, X) denote the algebra of all bounded linear operators fromY toX. ForA∈B(X)
n
,B∈B(Y)
n
, the elementary operator acting onB(Y, X) is defined by
. In this paper we obtain the formulae of the spectrum and the essential spectrum of Δ(A, B) by using spectral mapping theorems. Forn=1, we prove thatS
p
(L
A
,R
B
)=σ(A)×σ(B) and
. 相似文献
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