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1.
Consider the probability space W={−1, 1}
n
with the uniform (=product) measure. Let f: W → R be a function. Let f=Σ f
IX I be its unique expression as a multilinear polynomial where X
I=Π
i∈I
x
i. For 1≤ m≤ n let
=Σ |I|=m
f
IX I. Let T
ɛ
( f)=Σ f
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 any q≥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
4.
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
5.
Suppose that G is a finitely connected domain with rectifiable boundary γ, ∞ε G, the domains D
1,..., D
s
are the complements of G, the subsets F
j
⊂D
j
are infinite and compact, n
j
≥1, j=1,..., s, are integers, λ 0 is a complex-valued measure on γ, and
We consider the extremum problem where μ
j
, j=1,..., s, are complex-valued measures on F
j
and are Golubev sums. We prove that β=Δ, where
We also establish several other relations between these and other extremal variables.
Translated from Matematicheskie Zametki, Vol. 65, No. 5, pp. 738–745, May, 1999. 相似文献
6.
A theorem from the classical complex analysis proved by Davydov in 1949 is extended to the theory of solution of a special
case of the Beltrami equation in the z-complex plane (i.e., null solutions of the differential operator ).
It is proved that if γ is a rectifiable Jordan closed curve and f is a continuous complex-valued function on γ such that the integral
converges uniformly on γ as r → 0, where n(ζ) is the unit vector of outer normal on γ at a point ζ and ds is the differential of arc length, then the β-Cauchy-type integral
admits a continuous extension to γ and a version of the Sokhotski–Plemelj formulas holds.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 11, pp. 1443–1448, November, 2008. 相似文献
7.
The system
, where Λ={λ
n
} is the set of zeros (of multiplicities m
n
) of the Fourier transform of a singular Cantor-Lebesgue measure, is examined. We prove that e(Λ) is complete and minimal in L
p
(−a, a) with p≥1, and that | L(x+iy)| 2 does not satisfy the Muckenhoupt condition on any horizontal line Im z=y≠0 in the complex plane. This implies that e(Λ) does not have the property of convergence extension.
Translated from Matematicheskie Zametki, Vol. 64, No. 5, pp. 728–733, November, 1998. 相似文献
8.
Let Ω be an open bounded domain in ℝ N( N ≥ 3) and
. We are concerned with two kinds of critical elliptic problems. The first one is where 0 ∈ Ω,
, 2 < m < 2* and λ > 0. By using the fountain theorem and concentration estimates, if N ≥ 7 and θ > 0, we establish the existence of infinitely many solutions for the following regularization of (*) with small number ϵ > 0 Then if θ > 0 is suitably small, we obtain many solutions for problem (*) by taking the process of approximation.
The second problem is where q ∈ (0, 1), t > 0. By using similar methods as in (*), we prove that if N ≥ 7,
and t > 0, there exist infinitely many solutions with positive energy. In particular, we give a positive answer to one open problem
proposed by Ambrosetti, Brezis and Cerami [1]. 相似文献
9.
We study the solvability of the integral equation , where f∈ L
1
loc(ℝ) is the unknown function and g, T
1, and T
2 are given functions satisfying the conditions .
Most attention is paid to the nontrivial solvability of the homogeneous equation .
Translated from Matematicheskie Zametki, Vol. 62, No. 3, pp. 323–331, September, 1997.
Translated by M. A. Shishkova 相似文献
10.
For two complex Banach spaces X and Y,
( B
X; Y) will denote the space of bounded and continuous functions from B
X
to Y that are holomorphic on the open unit ball. The numerical radius of an element h in
( B
X; X) is the supremum of the set . We prove that every complex Banach space X with the Radon-Nikodym property satisfies that the subset of numerical radius attaining functions in
( B
X; X) is dense in
( B
X; X). We also show the denseness of the numerical radius attaining elements of
in the whole space, where
is the subset of functions in
which are uniformly continuous on the unit ball. For C( K) we prove a denseness result for the subset of the functions in
( B
C(K); C( K)) which are weakly uniformly continuous on the closed unit ball. For a certain sequence space X, there is a 2-homogenous polynomial P from X to X such that for every R > e, P cannot be approximated by bounded and numerical radius attaining holomorphic functions defined on RB
X
. If Y satisfies some isometric conditions and X is such that the subset of norm attaining functions of
( B
X; ℂ) is dense in
( B
X; ℂ), then the subset of norm attaining functions in
( B
X; Y) is dense in the whole space.
The first author was supported in part by D.G.E.S. Project BFM2003-01681.
The second author’s work was performed during a visit to the Departamento de Análisis Matem’atico of Universidad de Granada,
with a grant supported by the Korea Research Foundation under grant (KRF-2002-070-C00006). 相似文献
11.
We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space W N,2([0, ∞); e −x) and the Sobolev-Legendre space W N,2([−1, 1]) with respect to the Sobolev-Laguerre inner product and with respect to the Sobolev-Legendre inner product respectively, where a 0 = 1, a k ≥0, 1 ≤ k ≤ N −1, γ > 0, and N ≥1 is an integer. 相似文献
13.
Let
(z ∈ ℝ). Further let λ denote a large real parameter. We show that for arbitrary real numbers k and α with k>=2.7013 and 0<α≦1, 相似文献
14.
The Nikolskii type inequality for cardinal splines is proved, which is exact in the sense of order, where ∈ ℒ
m,h
, and ℒ
m,k
is the space of cardinal splines with nodes
Project supported by the National Natural Science Foundation of China (Grant No. 19671012), and Doctoral Programme Foundation
of Institution of Higher Education. 相似文献
15.
Let X, Y be vector spaces. It is shown that if a mapping f ： X → Y satisfies f（（x＋y）/2＋z）＋f（（x-y）/2＋z=f（x）＋2f（z）,（0.1） f（（x＋y）/2＋z）-f（（x-y）/2＋z）f（y）,（0.2） or 2f（（x＋y）/2＋x）=f（x）＋f（y）＋2f（z）（0.3）for all x, y, z ∈ X, then the mapping f ： X →Y is Cauchy additive.
Furthermore, we prove the Cauchy-Rassias stability of the functional equations （0.1）, （0.2） and （0.3） in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras. 相似文献
16.
A trace formula expressing the mean values of the form (k=2,3,...) via certain arithmetic means on the group Г 0(N 1) is proved. Here the sum is taken over a normalized orthogonal basis in the space of holomorphic cusp forms of weight 2k
with respect to Г 0(N 1). By H
f
(x)
(s) we denote the Hecke series of the form f, twisted with the primitive character χ ( mod N 2), and λ f(d), (d, N 1N 2)=1, are the eigenvalues of the Hecke operators . The trace formula is used for obtaining the estimate for the newform f for all ε>0, l=0,1,2,.... This improves the known result (Duke-Friedlander-Iwaniec, 1993) with upper bound
(1+|t|) 2N
2
1/2−1/22+ε
on the right-hand side. As a corollary, we obtain the estimate for the Fourier coefficients of holomorphic cusp forms of weight k+1/2, which improves Iwaniec' result (1987) with exponent
1/4–1/28+ε. Bibliography: 25 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 14–36. 相似文献
17.
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
18.
Suppose that a lower triangular matrix μ:[μ
m
(n)
] defines a conservative summation method for series, i.e., and the sequence (ρ
m
, m ∈ ℤ 0), is bounded away from zero. Then the trigonometric series
is the Fourier series of a function f ∈ L
p
(
), where p ε ]1; ∞[, if and only if the sequence of p-norms of its μ-means is bounded: In the case of the Fejér method, we have the test due to W. and G. Young (1913). In the case of the Fourier method, we obtain
the converse of the Riesz theorem (1927).
Translated from Matematicheskie Zametki, Vol. 62, No. 5, pp. 677–686, November, 1997.
Translated by N. K. Kulman 相似文献
19.
The following regularity of weak solutions of a class of elliptic equations of the form are investigated. 相似文献
20.
Let S k(Γ) be the space of holomorphic Γ-cusp forms f(z) of even weight k ≥ 12 for Γ = SL(2, ℤ), and let S k(Γ) + be the set of all Hecke eigenforms from this space with the first Fourier coefficient a f(1) = 1. For f ∈ S k(Γ) +, consider the Hecke L-function L(s, f). Let It is proved that for large K, where ε > 0 is arbitrary. For f ∈ S k(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
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