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1.
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
2.
L. G. Arabadzhyan 《Mathematical Notes》1997,62(3):271-277
We study the solvability of the integral equation
, wheref∈L
1
loc(ℝ) is the unknown function andg,T
1, andT
2 are given functions satisfying the conditions
.
Most attention is paid to the nontrivial solvability of the homogeneous equation
.
Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 323–331, September, 1997.
Translated by M. A. Shishkova 相似文献
3.
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
4.
O. B. Skaskiv 《Mathematical Notes》1999,66(2):223-232
For an entire Dirichlet series
, sufficient conditions on the exponents
are established such that the following relations hold outside a set of finite measure asx→+∞:
, where ψ(x) is a function increasing to +∞ and such thatx≤ψ(x)≤e
x
(x≥0).
Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 282–292, August, 1999 相似文献
5.
A. V. Filinovskii 《Mathematical Notes》1997,61(5):635-643
The following boundary value problem is studied:
here the surface Г satisfies the condition(
, where
and ν is the outward (with respect to Ω) normal to Γ.
Translated fromMatematischskie Zametki, Vol. 61, No. 5, pp. 759–768, May, 1997.
Translated by N. K. Kulman 相似文献
6.
LI Xiong 《中国科学A辑(英文版)》2001,44(2):137-144
We are concerned with the boundedness of all the solutions for second order differential equation
, wheref(x) andg(x) are odd, e( t) is odd and 1-periodic, andg(x) satisfies
相似文献
7.
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
8.
F. N. Garif’yanov 《Mathematical Notes》2000,67(5):572-576
The lacunary homogeneous moment problem
in the class of entire functions of exponential type is studied.
Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 674–679, May, 2000. 相似文献
9.
O. V. Viskov 《Mathematical Notes》2000,68(3):289-294
Let us choose a positive integern and denoteF(x, y)=
, wheref(·) andg(·) are arbitrary sufficiently smooth functions. Three different proofs of the validity of the relation
are given. We also establish discrete and noncommutative analogs of this identity.
Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 332–338, September, 2000. 相似文献
10.
M. B. Khripunova 《Mathematical Notes》1998,64(3):394-400
It is proved that if ƒ(n) is a multiplicative function taking a valueζ on the set of primes such thatζ
3 = 1,ζ ≠ 1 andƒ
3(p
r)=1 forr≥2, then there exists aθ ∈ (0, 1), for which
, where
.
Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 457–464, September, 1998.
The author wishes to thank Professor N. M. Timofeev for useful discussions.
This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00502. 相似文献
11.
§ 1 IntroductionIn this note we are concerned with the asymptotically periodic second order equation-u″+α( x) u =β( x) uq +γ( x) up, x∈ R,( 1 )where1 相似文献
12.
I. M. Guseinov 《Mathematical Notes》1997,62(2):172-180
We prove the existence of a transformation operator that takes the solution of the equationy″=λ2n
y to the solution of the equation
with a condition at infinity. Some properties of the kernel of this operator are studied.
Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 206–215, August, 1997.
Translated by M. A. Shishkova 相似文献
13.
V. N. Denisov 《Journal of Mathematical Sciences》2008,150(5):2344-2357
In the paper, we study the sufficient conditions for the lower-order coefficient of the parabolic equation
under which its solution satisfying the initial condition
stabilizes to zero, i.e., there exists the limit
uniform in x from every compact set K in ℝN for any function u
0(x) belonging to a certain uniqueness class of the problem considered and growing not rapidly than
with a > 0 and b < 0 at infinity.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 79–97, 2006. 相似文献
14.
We are going to discuss special cases of a conditional functional inequality
whereX is a real inner product space. In particular, we will give conditions which force the representationf(x)=c‖x‖2+a(x) for x ∈X, where c ∈ R anda:x→ℝ is an additive functional. 相似文献
15.
A. I. Zvyagintsev 《Mathematical Notes》1997,62(5):596-606
For functions satisfying the boundary conditions
, the following inequality with sharp constants in additive form is proved:
wheren≥2, 0≤1≤n−2,−1≤m≤1, m+1≤n−3, and1≤p,q,r≤∞.
Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 712–724, November, 1997.
Translated by N. K. Kulman 相似文献
16.
We provea priori inequalities for non-subelliptic quasilinear equations related to the Monge-Ampère equation in two dimensions, for example,
equations of the type
. 相似文献
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17.
CHEN Qionglei & ZHANG Zhifei Department of Mathematics Zhejiang University Hangzhou China Academy of Mathematics System Sciences Chinese Academy of Sciences Beijing China 《中国科学A辑(英文版)》2004,47(6):842-853
In this paper we give the (Lα p, Lp) boundedness of the maximal operator of a class of super singular integrals defined bywhich improves and extends the known result. Moreover, by applying an off-Diagonal T1 Theorem, we also obtain the (Lp, Lq) boundedness of the commutator defined by 相似文献
18.
A. T. Il'ichev 《Mathematical Notes》1992,52(1):662-668
Existence is proved for a family of soliton-like solutions for the nonlinear evolution equation ut–+uux+uxxx-uxxxxx=O. The problem is reduced to investigating the fixed points of the operator
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相似文献
19.
20.
S. Ya. Khavinson 《Mathematical Notes》1999,65(5):620-626
Suppose thatG is a finitely connected domain with rectifiable boundary γ, ∞εG, the domainsD
1,...,D
s
are the complements ofG, the subsetsF
j
⊂D
j
are infinite and compact,n
j
≥1,j=1,...,s, are integers, λ0 is a complex-valued measure on γ, and
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