共查询到20条相似文献，搜索用时 46 毫秒
1.
J. S. Hwang 《数学学报(英文版)》1998,14(1):57-66
Letf(X) be an additive form defined by
wherea
i
≠0 is integer,i=1,2…,s. In 1979, Schmidt proved that if ∈>0 then there is a large constantC(k,∈) such that fors>C(k,∈) the equationf(X)=0 has a nontrivial, integer solution in σ1, σ2, …, σ3,x
1,x
2, …,x
3 satisfying
Schmidt did not estimate this constantC(k,∈) since it would be extremely large. In this paper, we prove the following result 相似文献
2.
Let f(x, y) be a periodic function defined on the region D
with period 2π for each variable. If f(x, y) ∈ C
p (D), i.e., f(x, y) has continuous partial derivatives of order p on D, then we denote by ω
α,β(ρ) the modulus of continuity of the function
and write
For p = 0, we write simply C(D) and ω(ρ) instead of C
0(D) and ω
0(ρ).
Let T(x,y) be a trigonometrical polynomial written in the complex form
We consider R = max(m
2 + n
2)1/2 as the degree of T(x, y), and write T
R(x, y) for the trigonometrical polynomial of degree ⩾ R.
Our main purpose is to find the trigonometrical polynomial T
R(x, y) for a given f(x, y) of a certain class of functions such that
attains the same order of accuracy as the best approximation of f(x, y).
Let the Fourier series of f(x, y) ∈ C(D) be
and let
Our results are as follows
Theorem 1 Let f(x, y) ∈ C
p(D (p = 0, 1) and
Then
holds uniformly on D.
If we consider the circular mean of the Riesz sum S
R
δ
(x, y) ≡ S
R
δ
(x, y; f):
then we have the following
Theorem 2 If f(x, y) ∈ C
p (D) and ω
p(ρ) = O(ρ
α (0 < α ⩾ 1; p = 0, 1), then
holds uniformly on D, where λ
0
is a positive root of the Bessel function J
0(x)
It should be noted that either
or
implies that f(x, y) ≡ const.
Now we consider the following trigonometrical polynomial
Then we have
Theorem 3 If f(x, y) ∈ C
p(D), then uniformly on D,
Theorems 1 and 2 include the results of Chandrasekharan and Minakshisundarm, and Theorem 3 is a generalization of a theorem
of Zygmund, which can be extended to the multiple case as follows
Theorem 3′ Let f(x
1, ..., x
n) ≡ f(P) ∈ C
p
and let
where
and
being the Fourier coefficients of f(P). Then
holds uniformly.
__________
Translated from Acta Scientiarum Naturalium Universitatis Pekinensis, 1956, (4): 411–428 by PENG Lizhong. 相似文献
3.
The existence of the singular integral ∫K(x, y)f(y)dy associated to a Calderón-Zygmund kernel where the integral is understood
in the principal value sense TF(x)=limε→0+∫|x−y|>εK(x, y)f(y)dy has been well studied. In this paper we study the existence of the above integral in the Cesàro-α sense. More
precisely, we study the existence of
for −1<α<0 in the setting of weighted spaces. 相似文献
4.
Positive Solutions for Semipositone
<Emphasis Type="Italic">m</Emphasis>-point Boundary-value
Problems 总被引：7，自引：0，他引：7
Abstract
Let ξ
i
∈ (0, 1) with 0 <
ξ1 < ξ2 <
··· < ξ
m−2 < 1,
a
i
, b
i
∈ [0,∞) with
and
. We consider the
m-point boundary-value
problem
where f(x, y) ≥ −M, and M is a positive constant. We show the
existence and multiplicity of positive solutions by applying the
fixed point theorem in cones.
*Supported by the NSFC (10271095).
GG-110-10736-1003, NWNU-KJCXGC-212 and the Foundation of Major
Project of Science and Technology of Chinese Education
Ministry 相似文献
5.
O. M. Fomenko 《Journal of Mathematical Sciences》2006,133(6):1733-1748
Let Sk(Γ) be the space of holomorphic Γ-cusp forms f(z) of even weight k ≥ 12 for Γ = SL(2, ℤ), and let Sk(Γ)+ be the set of all Hecke eigenforms from this space with the first Fourier coefficient af(1) = 1. For f ∈ Sk(Γ)+, consider the Hecke L-function L(s, f). Let
It is proved that for large K,
where ε > 0 is arbitrary. For f ∈ Sk(Γ)+, let L(s, sym
2 f) denote the symmetric square L-function. It is proved that as k → ∞ the frequence
converges to a distribution function G(x) at every point of continuity of the latter, and for the corresponding characteristic
function an explicit expression is obtained. Bibliography: 17 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 221–246. 相似文献
6.
This paper deals with the existence of weak solutions in W
01(Ω) to a class of elliptic problems of the form
in a bounded domain Ω of ℝ
N
. Here a satisfies
for all ξ∈ℝ
N
, a.e. x∈Ω,
, h
1∈L
loc
1(Ω), h
1(x)≧1 for a.e. x in Ω; λ
1 is the first eigenvalue for −Δ
p
on Ω with zero Dirichlet boundary condition and g, h satisfy some suitable conditions.
相似文献
7.
Let A
0, ... , A
n−1 be operators on a separable complex Hilbert space , and let α0,..., α
n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,
for 2 ≤ p < ∞, and the reverse inequality holds for 0 < p ≤ 2. Moreover, we prove that if ω0,..., ω
n−1 are the n roots of unity with ω
j
= e
2πij/n
, 0 ≤ j ≤ n − 1, then for every unitarily invariant norm,
for 2 ≤ p < ∞, and the reverse inequalities hold for 0 < p ≤ 2. These inequalities, which involve n-tuples of operators, lead to natural generalizations and refinements of some of the classical Clarkson inequalities in the
Schatten p-norms. Extensions of these inequalities to certain convex and concave functions, including the power functions, are olso
optained.
相似文献
8.
O. L. Vinogradov 《Journal of Mathematical Sciences》1999,97(4):4233-4237
Let C be the space of 2π-periodic continuous real-valued functions, let
be first- and second-order moduli of continuity of a function f∈C with step h≥0. Denote by Lip1 = {f ∈ C: ω1(f,h) = O(h)} the Lipschitz class and by Z1 = {f ∈ C: ω2(f,h) = O(h)} the Zygmund class. The class of functions W⊂C is said to be described in terms of the kth modulus of continuity
if for any functions f1, f2∈C such that ωk(f2) from f1∈W it follows that f2∈W. As is shown, the class Z1 is not described in terms of the first-order modulus of continuity, whereas the class Lip is not described in terms of the
second-order modulus of continuity. Bibliography: 3 titles.
Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 83–89. 相似文献
9.
Hung Viet Le 《Integral Equations and Operator Theory》2000,37(1):64-71
In this paper, we establish the boundedness of the following maximal operator
onL
p
(R
n
) for allp>1, n≥2, where Γ(y)≡Γ(|y|) is a real, measurable, and radial function defined onR
n−1
. 相似文献
10.
S. V. Astashkin 《Mathematical Notes》1999,65(4):407-417
In this paper it is proved that from any uniformly bounded orthonormal system {f
n}
n=1
∞
of random variables defined on the probability space (Ω, ε, P), one can extract a subsystem {fni}
i
Emphasis>=1/∞
majorized in distribution by the Rademacher system on [0, 1]. This means that {
}, whereC>0 is independent of m∈N, ai∈N (i=1,…,m) andz>0.
Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 483–495, April, 1999. 相似文献
11.
LiJunjie BianBaojun 《高校应用数学学报(英文版)》2000,15(3):273-280
The following regularity of weak solutions of a class of elliptic equations of the form are investigated. 相似文献
12.
L. V. Kritskov 《Mathematical Notes》1999,65(4):454-461
Suppose thatА is a nonnegative self-adjoint extension to {
} of the formal differential operator−Δu+q(x)u with potentialq(x) satisfying the condition {
} or the condition {
} in which the nonnegative function itχ(r) is such that {
}. For each α∈(0, 2], we establish an estimate of the generalized Fourier transforms of an arbitrary function {
} of the form {
} If, in addition, {
}, then, along with this estimate, a similar lower bound is established.
Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 542–551, April, 1999. 相似文献
13.
Linghai ZHANG 《数学年刊B辑(英文版)》2008,29(2):179-198
Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0. 相似文献
14.
Let H be an infinite-dimensional real Hilbert space equipped with the scalar product (⋅,⋅)
H
. Let us consider three linear bounded operators,
We define the functions
where a
i
∈H and α
i
∈ℝ. In this paper, we discuss the closure and the convexity of the sets Φ
H
⊂ℝ2 and F
H
⊂ℝ3 defined by
Our work can be considered as an extension of Polyak’s results concerning the finite-dimensional case. 相似文献
15.
Suppose C
r
= (r
C
r
) ∪ (r
C
r
+ 1 − r) is a self-similar set with r ∈ (0, 1/2), and Aut(C
r
) is the set of all bi-Lipschitz automorphisms on C
r
. This paper proves that there exists f* ∈ Aut(C
r
) such that
where and blip(g) = max(lip(g), lip(g
−1)).
This work was supported by National Natural Science Foundation of China (Grant Nos. 10671180, 10571140, 10571063, 10631040,
11071164) and Morningside Center of Mathematics 相似文献
16.
A. Michael Alphonse 《Journal of Fourier Analysis and Applications》2000,6(4):449-456
In this paper we prove that the maximal commutator of singular integral operator [b, T]* satisfies the inequality:
where f is any smooth function with compact support, λ>0 and C is a positive constant independent of f and λ. 相似文献
17.
V. V. Zhuk 《Journal of Mathematical Sciences》2009,157(4):592-606
Let
be the Fejér kernel, C be the space of contiuous 2π-periodic functions f with the norm
, let
be the Jackson polynomials of the function f, and let
be the Fejér sums of f. The paper presents upper bounds for certain quantities like
which are exact in order for every function f ∈ C. Special attention is paid to the constants occurring in the inequalities obtained. Bibliography: 14 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 90–114. 相似文献
18.
Copositive approximation of periodic functions 总被引：1，自引：0，他引：1
Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y
i
∈ Y:= {y
i
}
i∈ℤ such that for x ∈ [y
i
, y
i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each n ≧ N(Y) we construct a trigonometric polynomial P
n
of order ≦ n, changing its sign at the same points y
i
∈ Y as f, and
where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω
3(f, t) is the third modulus of smoothness of f and ∥ · ∥ is the max-norm.
This work was done while the first author was visiting CPT-CNRS, Luminy, France, in June 2006. 相似文献
19.
Consider the probability spaceW={−1, 1}
n
with the uniform (=product) measure. Letf: W →R be a function. Letf=Σf
IXI be its unique expression as a multilinear polynomial whereX
I=Π
i∈I
x
i. For 1≤m≤n let
=Σ|I|=m
f
IXI. LetT
ɛ
(f)=Σf
Iɛ|I|
X
I where 0<ɛ<1 is a constant. A hypercontractive inequality, proven by Bonami and independently by Beckner, states that
This inequality has been used in several papers dealing with combinatorial and probabilistic problems. It is equivalent to
the following inequality via duality: For anyq≥2
In this paper we prove a special case with a slightly weaker constant, which is sufficient for most applications. We show
where
. Our proof uses probabilistic arguments, and a generalization of Shearer’s Entropy Lemma, which is of interest in its own
right.
Supported partially by NSF Award Abstract #0071261. 相似文献
20.
Bogdan Batko Jacek Tabor 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》1999,69(1):67-73
Let G be a commutative semigroup and letL be a complete Archimedean Riesz Space. Suppose thatF: G → L satisfies for somee ∈ L
+ the inequality
Then there exists a unique additive mappingA : G → L such that
As the method of the proof we use the Johnson-Kist Representation Theorem. 相似文献