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1.
In the paper, the equation
is considered in the scale of the weighted spaces H
β
s
(ℝ
n
) (q > 1, a
kα
∈ ℂ). We prove that if the expression
does not vanish on the set {ξ ∈ ℝ
n
∖ 0, |z| ≤ q
β−s+n
/2−2m}, then this equation has a unique solution u ∈ H
β
s+2m
(ℝ
n
) for every function f ∈ H
β
s
(ℝ
n
) provided that β, s ≠ ∈ ℝ, β − s ≠ n/2 + p, and β − s − 2m ≠ − n/2 − p (p = 0, 1, ...).
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 37–55, 2007. 相似文献
2.
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
3.
Piotr Niemiec 《Rendiconti del Circolo Matematico di Palermo》2008,57(3):391-399
The aim of the paper is to prove that every f ∈ L
1([0,1]) is of the form f = , where j
n,k
is the characteristic function of the interval [k- 1 / 2
n
, k / 2
n
) and Σ
n=0∞Σ
k=12n
|a
n,k
| is arbitrarily close to ||f|| (Theorem 2). It is also shown that if μ is any probabilistic Borel measure on [0,1], then for any ɛ > 0 there exists a sequence (b
n,k
)
n≧0
k=1,...,2n
of real numbers such that and for each Lipschitz function g: [0,1] → ℝ (Theorem 3).
相似文献
4.
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
considered in the whole space ℝn, n ≥ 2. The functions A
i
(x, u, ξ), i = 1,…, n, A
0(x, u), and f(x) can arbitrarily grow as |x| → ∞. These functions satisfy generalized conditions of the monotone operator theory in the arguments u ∈ ℝ and ξ ∈ ℝn. We prove the existence theorem for a solution u ∈ W
loc
1,p
(ℝn) under the condition p > n.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 133–147, 2006. 相似文献
5.
Two Inequalities for Convex Functions 总被引:1,自引:0,他引:1
Let a 0 < a 1 < ··· < a n be positive integers with sums $ {\sum\nolimits_{i = 0}^n {\varepsilon _{i} a_{i} {\left( {\varepsilon _{i} = 0,1} \right)}} } Let a
0 < a
1 < ··· < a
n
be positive integers with sums
distinct.
P. Erd?s conjectured that
The best known result along this line is that
of Chen: Let f be any given convex decreasing function on [A, B] with α
0, α
1, ... , α
n
, β
0, β
1, ... , β
n
being real numbers in [A, B] with α
0 ≤ α
1 ≤ ··· ≤ α
n
,
Then
In this paper, we obtain two generalizations of the above result; each is of
special interest in itself. We prove:
Theorem 1
Let f and g be two given non-negative convex decreasing functions on [A, B], and α
0, α
1, ... ,
α
n
, β
0, β
1, ... , β
n
, α'
0, α'
1, ... , α'
n
, β'
0
, β'
1
, ... , β'
n
be real numbers in [A, B] with
α
0 ≤ α
1 ≤ ··· ≤
α
n
,
α'
0 ≤ α'
1 ≤ ··· ≤ α'
n
,
Then
Theorem 2
Let f be any given convex decreasing function on [A, B] with
k
0, k
1, ... , k
n
being nonnegative
real numbers and
α
0, α
1, ... , α
n
, β
0, β
1, ... , β
n
being real numbers in [A, B] with
α
0 ≤ α
1 ≤
··· ≤ α
n
,
Then
相似文献
6.
Cao Jiading 《分析论及其应用》1989,5(2):99-109
Let an≥0 and F(u)∈C [0,1], Sikkema constructed polynomials:
, ifα
n
≡0, then Bn (0, F, x) are Bernstein polynomials.
Let
, we constructe new polynomials in this paper:
Q
n
(k)
(α
n
,f(t))=d
k
/dx
k
B
n+k
(α
n
,F
k
(u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα
n
≡0, k=1, then Qn
(1) (0, f(t), x) are Kantorovič polynomials Pn(f). Ifα
n
=0, k=2, then Qn
(2), (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is:
Theorem 2. Let 1≤p≤∞, in order that for every f∈LP [0, 1],
, it is sufficient and necessary that
,
§ 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define[1–2],[8–10]:
.
As usual, for the space Lp [a,b](1≤p<∞), we have
and L[a, b]=l1[a, b].
Letα
n
⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials
[3] [4].
The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports. 相似文献
7.
A. V. Harutyunyan W. Lusky 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(3):128-135
Let U
n
be the unit polydisk in C
n
and S be the space of functions of regular variation. Let 1 ≤ p < ∞, ω = (ω
1, ..., ω
n
), ω
j
∈ S(1 ≤ j ≤ n) and f ∈ H(U
n
). The function f is said to be in holomorphic Besov space B
p
(ω) if
$
\left\| f \right\|_{B_p (\omega )}^p = \int_{U^n } {\left| {Df(z)} \right|^p \prod\limits_{j = 1}^n {\frac{{\omega _j (1 - |z_j |)}}
{{(1 - |z_j |^{2 - p} )}}} dm_{2n} (z) < + \infty }
$
\left\| f \right\|_{B_p (\omega )}^p = \int_{U^n } {\left| {Df(z)} \right|^p \prod\limits_{j = 1}^n {\frac{{\omega _j (1 - |z_j |)}}
{{(1 - |z_j |^{2 - p} )}}} dm_{2n} (z) < + \infty }
相似文献
8.
The asymptotic estimate for the expected number of real zeros of a random algebraic polynomial
is known. The identical random coefficients aj(ω) are normally distributed defined on a probability space
, ω ∈Ω. The estimate for the expected number of zeros of the derivative of the above polynomial with respect to x is also known, which gives the expected number of maxima and minima of Qn(x, ω). In this paper we provide the asymptotic value for the expected number of zeros of the integration of Qn(x,ω) with respect to x. We give the geometric interpretation of our results and discuss the difficulties which arise when we consider a similar
problem for the case of
. 相似文献
9.
Ilham A. Aliev 《Integral Equations and Operator Theory》2009,65(2):151-167
We introduce new potential type operators Jab = (E+(-D)b/2)-a/bJ^{\alpha}_{\beta} = (E+(-\Delta)^{\beta/2})^{-\alpha/\beta}, (α > 0, β > 0), and bi-parametric scale of function spaces
Hab, p(\mathbbRn)H^{\alpha}_{\beta , p}({\mathbb{R}}^n) associated with Jαβ. These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization
of the spaces
Hab, p(\mathbbRn)H^{\alpha}_{\beta, p}({\mathbb{R}}^n) is given with the aid of a special wavelet–like transform associated with a β-semigroup, which generalizes the well-known
Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1). 相似文献
10.
Let f∈C
[−1,1]
″
(r≥1) and Rn(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions Rn(f,α,β,xk)=f(xk),Rn
′(f,α,β,xk)=f′(xk)(k=1,2,…,n), where {xk} are the roots of n-th Jacobi polynomial Pn(α,β,x),α,β>−1 and {x
k
″
} are the roots of (1−x2)Pn″(α,β,x). In this paper, we prove that
holds uniformly on [0,1].
In Memory of Professor M. T. Cheng
Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang. 相似文献
11.
In this paper we study the problem of explicitly constructing a dimension expander raised by [3]: Let
\mathbbFn \mathbb{F}^n be the n dimensional linear space over the field
\mathbbF\mathbb{F}. Find a small (ideally constant) set of linear transformations from
\mathbbFn \mathbb{F}^n to itself {A
i
}
i∈I
such that for every linear subspace V ⊂
\mathbbFn \mathbb{F}^n of dimension dim(V)<n/2 we have
|