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1.
For a trigonometric series
defined on [−π, π)
m
, where V is a certain polyhedron in R
m
, we prove that
if the coefficients a
k
satisfy the following Sidon-Telyakovskii-type conditions:
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 579–585, May, 2008. 相似文献
2.
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
3.
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
4.
I. I. Sharapudinov 《Mathematical Notes》2000,67(3):389-397
Let
N+2m
={−m, −m+1, …, −1, 0, 1, …,N−1,N, …,N−1+m}. The present paper is devoted to the approximation of discrete functions of the formf :
N+2m
→ ℝ by algebraic polynomials on the grid Ω
N
={0, 1, …,N−1}. On the basis of two systems of Chebyshev polynomials orthogonal on the sets Ω
N+m
and Ω
N
, respectively, we construct a linear operatorY
n+2m, N
=Y
n+2m, N
(f), acting in the space of discrete functions as an algebraic polynomial of degree at mostn+2m for which the following estimate holds (x ε Ω
N
):
whereE
n+m[g,l
2(Ω
N+m
)] is the best approximation of the function
by algebraic polynomials of degree at mostn+m in the spacel
2 (Ω
N+m
) and the function Θ
N, α
(x) depends only on the weighted estimate for the Chebyshev polynomialsτ
k
α,α
(x, N).
Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 460–470, March, 2000. 相似文献
(1) |
(1) |
5.
We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space WN,2([0, ∞); e−x) and the Sobolev-Legendre space WN,2([−1, 1]) with respect to the Sobolev-Laguerre inner product
and with respect to the Sobolev-Legendre inner product
respectively, where a0 = 1, ak ≥0, 1 ≤k ≤N −1, γ > 0, and N ≥1 is an integer. 相似文献
6.
Yuexu Zhao 《Bulletin of the Brazilian Mathematical Society》2006,37(3):377-391
Let X1, X2, ... be i.i.d. random variables with EX1 = 0 and positive, finite variance σ2, and set Sn = X1 + ... + Xn. For any α > −1, β > −1/2 and for κn(ε) a function of ε and n such that κn(ε) log log n → λ as n ↑ ∞ and
, we prove that
*Supported by the Natural Science Foundation of Department of Education of Zhejiang Province (Grant No. 20060237 and 20050494). 相似文献
7.
Let X
1, X
2, ... be i.i.d. random variables. The sample range is R
n
= max {X
i
, 1 ≤ i ≤ n} − min {X
i
, 1 ≤ i ≤ n}. If for a non-degenerate distribution G and some sequences (α
k
), (β
k
) then we have
and
almost surely for any continuity point x of G and for any bounded Lipschitz function f: R → R.
相似文献
8.
E. G. Goluzina 《Journal of Mathematical Sciences》1996,79(5):1304-1307
We study the structural properties of the class Mk,λ,b(k≥2, 0≤λ≤1, b∈ℂ\{0}) of functions f(z)=z+ ... which are regular in |z|<1 and satisfy the conditions f(z)f′(z)z−1≠0 and
, where J(z)=λ(1+b−1zf″(z)/f′(z)+(1−λ)(b−1zf′(z)/f(z)+1−b−1). The value regions of some functionals on this class are found. The case λ=1 was considered in our previous paper. Bibliography:
4 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 55–60.
Translated by O. A. Ivanov. 相似文献
9.
S. BERHANU F. CUCCU G. PORRU 《数学学报(英文版)》2007,23(3):479-486
For γ≥1 we consider the solution u=u(x) of the Dirichlet boundary value problem Δu + u^-γ=0 in Ω, u=0 on δΩ. For γ= 1 we find the estimate
u(x)=p(δ(x))[1+A(x)(log 1/δ(x)^-6],
where p(r) ≈ r r√2 log(1/r) near r = 0,δ(x) denotes the distance from x to δΩ, 0 〈ε 〈 1/2, and A(x) is a bounded function. For 1 〈 γ 〈 3 we find
u(x)=(γ+1/√2(γ-1)δ(x))^2/γ+[1+A(x)(δ(x))2γ-1/γ+1]
For γ3= we prove that
u(x)=(2δ(x))^1/2[1+A(x)δ(x)log 1/δ(x)] 相似文献
10.
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
11.
Aleksandar Ivić 《Central European Journal of Mathematics》2004,2(4):494-508
Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of
. If
with
, then we obtain
. We also show how our method of proof yields the bound
, where T
1/5+ε≤G≪T, T<t
1<...<t
R
≤2T, t
r
+1−t
r
≥5G (r=1, ..., R−1). 相似文献
12.
For two collections of nonnegative and suitably normalized weights W = (Wj) and V = (Vn,k), a probability distribution on the set of partitions of the set {1, …, n} is defined by assigning to a generic partition
{Aj, j ≤ k} the probability Vn,k
, where |Aj| is the number of elements of Aj. We impose constraints on the weights by assuming that the resulting random partitions Π n of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers.
This implies that the weights W must be of a very special form depending on a single parameter α ∈ [− ∞, 1]. The case α =
1 is trivial, and for each value of α ≠ = 1 the set of possible V-weights is an infinite-dimensional simplex. We identify
the extreme points of the simplex by solving the boundary problem for a generalized Stirling triangle. In particular, we show
that the boundary is discrete for − ∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members
of the Ewens-Pitman family of random partitions indexed by (α,θ), while for 0 < α < 1 the extremes are obtained by conditioning
an (α,θ)-partition on the asymptotics of the number of blocks of Πn as n tends to infinity. Bibliography: 29 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 83–102. 相似文献
13.
E. G. Goluzina 《Journal of Mathematical Sciences》1997,83(6):745-749
Let Mk,λ(0≤λ≤1, k≥2) be the class of functions f(z)=1/z+ao+a1z+... that are regular and locally univalent for 0<⩛z⩛<1 and satisfy the condition
where Jλ(z)=λ(1+zf″(z)/f'(z))+(1-λ)zf'(z)/f(z). In the class Mk,λ we consider sorne coefficient problems and problems concerning distortion theorems.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 91–96.
Translated by N. Yu. Netsvetaev. 相似文献
14.
Let {Xk} be a sequence of i.i.d. random variables with d.f. F(x). In the first part of the paper the weak convergence of the d.f.'s
Fn(x) of sums
is studied, where 0<α≤2, ank>0, 1≤k≤mn, and, as n→∞, bothmax
1≤k≤mna
nk→0 and
. It is shown that such convergence, with suitably chosen An's and necessarily stable limit laws, holds for all such arrays {αnk} provided it holds for the special case αnk=1/n, 1≤k≤n. Necessary and sufficient conditions for such convergence are classical. Conditions are given for the convergence
of the moments of the sequence {Fn(x)}, as well as for its convergence in mean. The second part of the paper deals with the almost sure convergence of sums
, where an≠0, bn>0, andmax
1≤k≤n ak/bn→0. The strong law is said to hold if there are constants An for which Sn→0 almost surely. Let N(0)=0 and N(x) equal the number of n≥1 for which bn/|an|<x if x>0. The main result is as follows. If the strong law holds,EN (|X1|)<∞. If
for some 0<p≤2, then the strong law holds with
if 1≤p≤2 and An=0 if 0<p<1. This extends the results of Heyde and of Jamison, Orey, and Pruitt. The strong law is shown to hold under various
conditions imposed on F(x), the coefficients an and bn, and the function N(x).
Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993. 相似文献
15.
Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables 总被引:1,自引:0,他引:1
Ye JIANG Li Xin ZHANG 《数学学报(英文版)》2006,22(3):781-792
Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1+X2+…+Xn,Mn=max k≤n│Sk│,n≥1.Let an=O(1/loglogn).In this paper,we prove that,for b〉-1,lim ε→0 →^2(b+1)∑n=1^∞ (loglogn)^b/nlogn n^1/2 E{Mn-σ(ε+an)√2nloglogn}+σ2^-b/(b+1)(2b+3)E│N│^2b+3∑k=0^∞ (-1)k/(2k+1)^2b+3 holds if and only if EX=0 and EX^2=σ^2〈∞. 相似文献
16.
We prove a general theorem on the zeros of a class of generalised Dirichlet series. We quote the following results as samples.
Theorem A.Let 0<θ<1/2and let {a
n
}be a sequence of complex numbers satisfying the inequality
for N = 1,2,3,…,also for n = 1,2,3,…let α
n
be real and |αn| ≤ C(θ)where C(θ) > 0is a certain (small)constant depending only on θ. Then the number of zeros of the function
in the rectangle (1/2-δ⩽σ⩽1/2+δ,T⩽t⩽2T) (where 0<δ<1/2)is ≥C(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided T ≥T
0(θ,δ)a large positive constant.
Theorem B.In the above theorem we can relax the condition on a
n
to
and |aN| ≤ (1/2-θ)-1.Then the lower bound for the number of zeros in (σ⩾1/3−δ,T⩽t⩽2T)is > C(θ,δ) Tlog T(log logT)-1.The upper bound for the number of zeros in σ⩾1/3+δ,T⩽t⩽2T) isO(T)provided
for every ε > 0.
Dedicated to the memory of Professor K G Ramanathan 相似文献
17.
Equations with two time scales (refinement equations or dilation equations) are central to wavelet theory. Several applications
also include an inhomogeneous forcing term F(t). We develop here a part of the existence theory for the inhomogeneous refinement
equation
where a (k) is a finite sequence and F is a compactly supported distribution on ℝ.
The existence of compactly supported distributional solutions to an inhomogeneous refinement equation is characterized in
terms of conditions on the pair (a, F).
To have Lp solutions from F ∈ Lp(ℝ), we construct by the cascade algorithm a sequence of functions φ0 ∈ Lp(ℝ) from a compactly supported initial function ℝ as
A necessary and sufficient condition for the sequence {φn} to converge in Lp(ℝ)(1 ≤ p ≤ ∞) is given by the p-norm joint spectral radius of two matrices derived from the mask a. A convexity property
of the p-norm joint spectral radius (1 ≤ p ≤ ∞) is presented.
Finally, the general theory is applied to some examples and multiple refinable functions.
Acknowledgements and Notes. Research supported in part by Research Grants Council and City University of Hong Kong under Grants #9040281, 9030562, 7000741. 相似文献
18.
A. G. Shukhov 《Mathematical Notes》1999,65(4):510-515
Let {
} be a sequence of finitely presented groups with generating setA={a1, …, am}, and letRk be the symmetrized set of words over the alphabetA∪A−1 obtained from the defining words and their inverses by all cyclic shifts. We shall assume that the words inRk are cyclically irreducible, and their lengths tend to ∞ ask increases. In the paper, it is proved that ifRk satisfies the small cancellation conditionC'(1/6) and the number of relators increases not very rapidly with increasingk, then the growth rate ψ(Gk) tends to 2m−1 ask→∞.
Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 611–617, April, 1999. 相似文献
19.
J. C. Gupta 《Proceedings Mathematical Sciences》2000,110(4):415-430
Let G
n,k
be the set of all partial completely monotone multisequences of ordern and degreek, i.e., multisequencesc
n(β1,β2,…, β
k
), β1,β2,…, βk
= 0,1,2,…, β1+β2 + … +β
k
≤n,c
n(0,0,…, 0) = 1 and
whenever β0 ≤n - (β1 + β2 + … + β
k
) where Δc
n(β1,β2,…, β
k
) =c
n(β1 + 1, β2,…, β
k
)+c
n(β1,β2+1,…, β
k
)+…+c
n (β1,β2,…, β
k
+1) -c
n(β1,β2,…, β
k
). Further, let Π
n,k
be the set of all symmetric probabilities on {0,1,2,…,k}
n
. We establish a one-to-one correspondence between the sets G
n,k
and Π
n,k
and use it to formulate and answer interesting questions about both. Assigning to G
n,k
the uniform probability measure, we show that, asn→∞, any fixed section {it{cn}(β1,β2,…, β
k
), 1 ≤ Σβ
i
≤m}, properly centered and normalized, is asymptotically multivariate normal. That is,
converges weakly to MVN[0, Σ
m
]; the centering constantsc
0(β1, β2,…, β
k
) and the asymptotic covariances depend on the moments of the Dirichlet (1, 1,…, 1; 1) distribution on the standard simplex
inR
k. 相似文献
20.
Francesco Leonetti 《Annali dell'Universita di Ferrara》1985,31(1):169-184
Riassunto In questo lavoro si prova la regolarità h?lderiana delle derivate, fino all'ordinek, dei minimi locali
dei funzionali
sotto opportune ipotesi suA
ij
αβ
e sug.
Summary In this paper we prove h?lder-continuity of the derivates, up to orderk, of local minima of functionals under suitable hypotheses forA ij αβ andg.相似文献