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1.
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
2.
I. V. Astashova 《Journal of Mathematical Sciences》2007,143(4):3198-3204
One considers the differential inequality
, where a
j
(x) are continuous functions, p* > 0, n ≥ 1, k > 1, and its special case
, where all r
j
(x) are sufficiently smooth positive functions. Uniform estimates are obtained for solutions defined in the same domain.
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 27–36, 2007. 相似文献
3.
A. A. Kon'kov 《Journal of Mathematical Sciences》2006,134(3):2073-2237
The paper considers solutions of the coercive inequalities
defined on an arbitrary (possibly, unbounded) subset ℝ
n
, where n ≥ 2, L and
are elliptic operators of the form
, and F is a certain function.
__________
Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions),
Vol. 7, Partial Differential Equations, 2004. 相似文献
4.
We study a quasilinear elliptic problem
with nonhomogeneous principal part φ. Under the hypothesis f(x,t)= o(φ(t)t) at t= 0 and ∞, the existence of multiple positive solutions is proved by using the variational arguments in the Orlicz–Sobolev
spaces.
Mathematics Subject Classification (2000) 35J20; 35J25; 35J70; 47J10; 47J30 相似文献
5.
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
6.
§ 1 IntroductionRecently,certain three-point boundary value problems for nonlinear ordinarydifferential equations have been studied by many authors[1— 6] .However,few papers havebeen published on the same problems for nonlinear functional differential equations.In thispaper,we are concerned with the following second order differential equation with anadvanced argumentu″(t) +λa(t) f(u(h(t) ) ) =0 ,t∈ (0 ,1 ) (1 .1 )with the three-point boundary conditionsu(0 ) =0 ,αu(η) =u(1 ) ,(1 .2 )… 相似文献
7.
In the rectangle D = (0,
,
is considered, where p and
are locally summable functions and may have nonintegrable singularities on . The effective conditions guaranteeing the unique solvability of this problem and the stability of its solution with respect to small perturbations of the coefficients of the equation under consideration are established. 相似文献
8.
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
9.
10.
11.
Choonkil BAAK 《数学学报(英文版)》2006,22(6):1789-1796
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
12.
WANGSHU WENSHIHLIANG YEQIXIAO 《高校应用数学学报(英文版)》1997,12(2):127-138
ＦＩＮＩＴＥＴＲＡＶＥＬＩＮＧＷＡＶＥＳＦＯＲＡＲＥＡＣＴＩＯＮ┐ＤＩＦＦＵＳＩＯＮＳＹＳＴＥＭＷＩＴＨＮ（３）ＣＯＭＰＯＮＥＮＴＳＷＡＮＧＳＨＵ,ＷＥＮＳＨＩＨＬＩＡＮＧＡＮＤＹＥＱＩＸＩＡＯＡｂｓｔｒａｃｔ．Ｉｎｔｈｉｓｐａｐｅｒ,ｔｈｅｅｘ... 相似文献
13.
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
14.
15.
16.
V. P. Burskii 《Mathematical Notes》1999,65(1):20-23
We study the Dirichlet problem for the system of elliptic equations with matrix complex-valued coefficients
We are concerned with the case in which this homogeneous system of equations can have a countable number of linearly independent
solutions.
Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 23–27, January, 1999. 相似文献
17.
It is proved that the Dirichlet problem is correct in the characteristic rectangle D
ab
= [0, a] × [0, b] for the linear hyperbolic equation
with the summable in D
ab
coefficients p
0, p
1, p
2, p
3 and q if and only if the corresponding homogeneous problem has only the trivial solution. The effective and optimal in some sense restrictions on p
0, p
1, p
2 and p
3 guaranteeing the correctness of the Dirichlet problem are established. 相似文献
18.
19.
Pigong Han 《Israel Journal of Mathematics》2008,164(1):125-152
Let Ω be an open bounded domain in ℝN(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
| (*) |
20.
Let L(x, v) be a Lagrangian which is convex and superlinear in the velocity variable v, and let H(x, p) be the associated Hamiltonian. Conditions are obtained under which every viscosity solution
of the Hamilton-Jacobi equation
is an action function in the large, i.e.,
for all
Received: 13 June 2003 相似文献