共查询到20条相似文献,搜索用时 31 毫秒
1.
Let Δ3 be the set of functions three times continuously differentiable on [−1, 1] and such that f″′(x) ≥ 0, x ∈ [−1, 1]. We prove that, for any n ∈ ℕ and r ≥ 5, there exists a function f ∈ C
r
[−1, 1] ⋂ Δ3 [−1, 1] such that ∥f
(r)∥
C[−1, 1] ≤ 1 and, for an arbitrary algebraic polynomial P ∈ Δ3 [−1, 1], there exists x such that
| f(x) - P(x) | 3 C?n \uprhonr(x), \left| {f(x) - P(x)} \right| \geq C\sqrt n {{\uprho}}_n^r(x), 相似文献
2.
Z. Ditzian 《Israel Journal of Mathematics》1985,52(4):341-354
Equivalences between the condition |P
n
(k)
(x)|≦K(n
−1√1−x
2+1/n
2)
k
n
-a, whereP
n(x) is the bestn-th degree polynomial approximation tof(x), and the Peetre interpolation space betweenC[−1,1] and the space (1−x
2)
k
f
(2k)(x)∈C[−1,1] is established. A similar result is shown forE
n(f)=
‖f−P
n‖
C[−1,1]. Rates other thann
-a are also discussed.
Supported by NSERC grant A4816 of Canada. 相似文献
3.
Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ2 the set of convex functions f ∈ ℂ[−,1]. Also, let E
n
(f) and E
n
(2) (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ2 by algebraic polynomials of degree < n, respectively. Clearly, En (f) ≦ E
n
(2) (f), and Lorentz and Zeller proved that the inverse inequality E
n
(2) (f) ≦ cE
n
(f) is invalid even with the constant c = c(f) which depends on the function f ∈ Δ2.
In this paper we prove, for every α > 0 and function f ∈ Δ2, that
4.
Raffaele Mosca 《Graphs and Combinatorics》2001,17(3):517-528
Let G be a graph with n vertices, and denote as γ(G) (as θ(G)) the cardinality of a minimum edge cover (of a minimum clique cover) of G. Let E (let C) be the edge-vertex (the clique-vertex) incidence matrix of G; write then P(E)={x∈ℜ
n
:Ex≤1,x≥0}, P(C)={x∈ℜ
n
:Cx≤1,x≥0}, α
E
(G)=max{1
T
x subject to x∈P(E)}, and α
C
(G)= max{1
T
x subject to x∈P(C)}. In this paper we prove that if α
E
(G)=α
C
(G), then γ(G)=θ(G).
Received: May 20, 1998?Final version received: April 12, 1999 相似文献
5.
Bao Yongguang 《分析论及其应用》1995,11(4):15-23
Let ξn −1 < ξn −2 < ξn − 2 < ... < ξ1 be the zeros of the the (n−1)-th Legendre polynomial Pn−1(x) and −1=xn<xn−1<...<x1=1, the zeros of the polynomial
. By the theory of the inverse Pal-Type interpolation, for a function f(x)∈C
[−1,1]
1
, there exists a unique polynomial Rn(x) of degree 2n−2 (if n is even) satisfying conditions Rn(f, ξk) = f (εk) (1 ⩽ k ⩽ n −1); R1
n(f,xk)=f1(xk)(1≤k≤n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation
polynomial {Rn(f, x)} (n is even) and the main result of this paper is that if f∈C
[1,1]
r
, r≥2, n≥r+2, and n is even then |R1
n(f,x)−f1(x)|=0(1)|Wn(x)|h(x)·n3−r·E2n−r−3(f(r)) holds uniformly for all x∈[−1,1], where
. 相似文献
6.
LetW be an algebraically closed filed of characteristic zero, letK be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value, and letA(K) (resp. ℳ(K)) be the set of entire (resp. meromorphic) functions inK. For everyn≥7, we show that the setS
n(b) of zeros of the polynomialx
n−b (b≠0) is such that, iff, g ∈W[x] or iff, g ∈A(K), satisfyf
−1(S
n(b))=g
−1(S
n(b)), thenf
n=g
n. For everyn≥14, we show thatS
n(b) is such that iff, g ∈W({tx}) or iff, g ∈ ℳ(K) satisfyf
−1(S
n(b))=g
−1(S
n(b)), then eitherf
n=g
n, orfg is a constant. Analogous properties are true for complex entire and meromorphic functions withn≥8 andn≥15, respectively.
For everyn≥9, we show that the setY
n(c) of zeros of the polynomial
, (withc≠0 and 1) is an ursim ofn points forW[x], and forA(K). For everyn≥16, we show thatY
n(c) is an ursim ofn points forW(x), and for ℳ(K). We follow a method based on thep-adic Nevanlinna Theory and use certain improvement of a lemma obtained by Frank and Reinders. 相似文献
7.
Vincenzo De Filippis 《Israel Journal of Mathematics》2009,171(1):325-348
Let R be a prime ring with extended centroid C, δ a nonzero generalized derivation of R, f(x
1, ..., x
n
) a nonzero multilinear polynomial over C, I a nonzero right ideal of R and k ≥ a fixed integer.
If [δ(f(r
1, ..., r
n
)), f(r
1, ..., r
n
)]
k
= 0, for all r
1, ..., r
n
∈ I, then either δ(x) = ax, with (a-γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds
(1) if char(R) = 0 then f(x
1, ..., x
n
) is central valued in eRCe
(2) if char(R) = p > 0 then is central valued in eRCe, for a suitable s ≥ 0, unless when char(R) = 2 and eRCe satisfies the standard identity s
4
(3) δ(x) = ax−xb, where (a+b+α)e = 0, for α ∈ C, and f(x
1, ..., x
n
)2 is central valued in eRCe. 相似文献
8.
An Application of a Mountain Pass Theorem 总被引:3,自引:0,他引:3
We are concerned with the following Dirichlet problem:
−Δu(x) = f(x, u), x∈Ω, u∈H
1
0(Ω), (P)
where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L
∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR)
is no longer true, where F(x, s) = ∫
s
0
f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming
(AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable
conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.
Received June 24, 1998, Accepted January 14, 2000. 相似文献
9.
Basudeb Dhara 《Rendiconti del Circolo Matematico di Palermo》2008,57(3):401-410
Let R be a prime ring of char R ≠ = 2 with center Z(R) and with extended centroid C, d a nonzero derivation of R and f(x
1, ..., x
n
) a nonzero multilinear polynomial over C. Suppose that x
s
d(x)x
t
∈ Z(R) for all x ∈ {d(f(x
1, ..., x
n
))|x
1, ..., x
n
∈ ρ}, where ρ is a nonzero right ideal of R and s ≥ 0, t ≥ 0 are fixed integers. If d(ρ)ρ ≠ = 0, then ρ
C = eRC for some idempotent e in the socle of RC and f(x
1, ..., x
n
)
N
is central-valued in eRCe, where N = s + t + 1.
相似文献
10.
Let f∈C3[a,b] and L be a linear differential operator such that L(f)≥0. Then there exists a sequence Qn, n≥1, of polynomial splines with equally spaced knots, such that Q(r), approximates f(r), 0≤r≤s, simultaneously in the uniform norm. This approximation is given through inequalities with rates, involving a measure
of smoothness to f(s); so that L (Qn)≥0. The encountered cases are the continuous, periodic and discrete. 相似文献
11.
Assume thatf is an integer transcendental solution of the differential equationP
n
(z, f, f′)=P
n−1(z, f, f′, ... f
(p)), whereP
n
andP
n−1 are polynomials in all variables, the degree ofP
n
with respect tof andf′ is equal ton, and the degree ofP
n−1 with respect tof, f′, ... f
(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E
*, whereE
* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z
1/2 (β+o(1)), β∈C, holds forz=re
iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re
iν)‖=o(r
1/2),r→+∞,r>0,
, where Δ is a certain set on the semiaxisr>0 with mes Δ<∞.
“L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77,
January, 1999. 相似文献
12.
Suppose that(T
t
)t>0 is aC
0 semi-group of contractions on a Banach spaceX, such that there exists a vectorx∈X, ‖x‖=1 verifyingJ
−1(Jx)={x}, whereJ is the duality mapping fromX toP(X
*). If |<T
t
x,f>|→1, whent→+∞ for somef∈X
*, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onx∈X is the best possible.
If the hypothesisJ
−1(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ
−1(Jx) is weakly compact, then if |<T
t
x, f>|→1, whent→+∞ for somef∈X
*, ‖f‖≤1, there existsy∈J
−1(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ1 orL
1. 相似文献
13.
An (n, d, k)-mapping f is a mapping from binary vectors of length n to permutations of length n + k such that for all x, y
{0,1}n, dH (f(x), f(y)) ≥ dH (x, y) + d, if dH (x, y) ≤ (n + k) − d and dH (f(x), f(y)) = n + k, if dH (x, y) > (n + k) − d. In this paper, we construct an (n,3,2)-mapping for any positive integer n ≥ 6. An (n, r)-permutation array is a permutation array of length n and any two permutations of which have Hamming distance at least r. Let P(n, r) denote the maximum size of an (n, r)-permutation array and A(n, r) denote the same setting for binary codes. Applying (n,3,2)-mappings to the design of permutation array, we can construct an efficient permutation array (easy to encode and decode)
with better code rate than previous results [Chang (2005). IEEE Trans inf theory 51:359–365, Chang et al. (2003). IEEE Trans
Inf Theory 49:1054–1059; Huang et al. (submitted)]. More precisely, we obtain that, for n ≥ 8, P(n, r) ≥ A(n − 2, r − 3) > A(n − 1,r − 2) = A(n, r − 1) when n is even and P(n, r) ≥ A(n − 2, r − 3) = A(n − 1, r − 2) > A(n, r − 1) when n is odd. This improves the best bound A(n − 1,r − 2) so far [Huang et al. (submitted)] for n ≥ 8.
The work was supported in part by the National Science Council of Taiwan under contract NSC-93-2213-E-009-117 相似文献
14.
V. A. Ustimenko 《Journal of Mathematical Sciences》2007,140(3):461-471
The paper is devoted to the study of a linguistic dynamical system of dimension n ≥ 2 over an arbitrary commutative ring K,
i.e., a family F of nonlinear polynomial maps f
α : K
n
→ K
n
depending on “time” α ∈ {K − 0} such that f
α
−1 = f
−αM, the relation f
α1 (x) = f
α2 (x) for some x ∈ Kn implies α1 = α2, and each map f
α has no invariant points. The neighborhood {f
α (υ)∣α ∈ K − {0}} of an element v determines the graph Γ(F) of the dynamical system on the vertex set Kn. We refer to F as a linguistic dynamical system of rank d ≥ 1 if for each string a = (α1, υ, α2), s ≤ d, where αi + αi+1 is a nonzero divisor for i = 1, υ, d − 1, the vertices υ
a = f
α1 × ⋯ × f
αs
(υ) in the graph are connected by a unique path. For each commutative ring K and each even integer n ≠= 0 mod 3, there is a family of linguistic dynamical systems Ln(K) of rank d ≥ 1/3n. Let L(n, K) be the graph of the dynamical system Ln(q). If K = Fq, the graphs L(n, Fq) form a new family of graphs of large girth. The projective limit L(K) of L(n, K), n → ∞, is well defined for each commutative
ring K; in the case of an integral domain K, the graph L(K) is a forest. If K has zero divisors, then the girth of K drops
to 4. We introduce some other families of graphs of large girth related to the dynamical systems Ln(q) in the case of even q. The dynamical systems and related graphs can be used for the development of symmetric or asymmetric
cryptographic algorithms. These graphs allow us to establish the best known upper bounds on the minimal order of regular graphs
without cycles of length 4n, with odd n ≥ 3. Bibliography: 42 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 214–234. 相似文献
15.
Zbigniew Grande 《Central European Journal of Mathematics》2011,9(4):772-777
A sequence (f
n
)
n
of functions f
n
: X → ℝ almost decreases (increases) to a function f: X → ℝ if it pointwise converges to f and for each point x ∈ X there is a positive integer n(x) such that f
n+1(x) ≤ f
n
(x) (f
n+1(x) ≥ f
n
(x)) for n ≥ n(x). In this article I investigate this convergence in some families of continuous functions. 相似文献
16.
Sun Xiehua 《分析论及其应用》1993,9(4):1-8
In this paper we solve a remained problem in [2], whether the following estimate approximation for the classf∈[-1, 1]∩BV by Lagrange interpolation based on the Jacobi abscissas: L
n
(a,d)
(f,x)−f(x)=O(1/n) holds, if α≠β α,β≥−1.
The project is supported by the Natural Science Foundation of Zhejiang Province. 相似文献
17.
Boris Shekhtman 《Journal of Approximation Theory》1996,85(3):343-347
LetEbe a subspace ofC(X) and letR(E)=g/h : g, hE; h>0}. We make a simple, yet intriguing observation: if zero is a best approximation toffromE, then zero is a best approximation toffromR(E). We also prove that if {En} is dense inC(X) then for almost allf(in the sense of category)[formula]That extends the results of P. Borwein and S. Zhou who proved it for the case whenEnis the space of algebraic or trigonometric polynomials of degreen. 相似文献
18.
E. S. Dubtsov 《Journal of Mathematical Sciences》2010,166(1):23-30
Let B
n
denote the unit ball in
\mathbbC \mathbb{C}
n
, n ≥ 1. Let K \mathcal{K}
0(n) denote the class of functions defined for z ∈ B
n
as a constant plus the integral of the kernel log(1/(1 −〈z, ζ〉)) against a complex Borel measure on the sphere {ζ ∈
\mathbbC \mathbb{C}
n
,: |ζ| = 1}. Properties of holomorphic functions g such that fg ∈ K \mathcal{K}
0(n) for all f ∈ K \mathcal{K}
0(n) are studied. The extended Cesàro operators are investigated on the spaces K \mathcal{K}
0(n), n ≥ 1. Bibliography: 15 titles. 相似文献
19.
Zhou Songping 《分析论及其应用》1989,5(1):11-14
In 1980, M. Hasson raised a conjecture as follows: Let N≥1, then there exists a function f0(x)∈C
[−1,1]
2N
, for N+1≤k≤2N, such that p
n
(k)
(f0,1)→f
0
(k)
(1), n→∞, where pn(f,x) is the algebraic polynomial of best approximation of degree ≤n to f(x). In this paper, a, positive answer to this conjecture
is given. 相似文献
20.
Fu Qing GAO 《数学学报(英文版)》2007,23(8):1527-1536
Let {Xn;n≥ 1} be a sequence of independent non-negative random variables with common distribution function F having extended regularly varying tail and finite mean μ = E(X1) and let {N(t); t ≥0} be a random process taking non-negative integer values with finite mean λ(t) = E(N(t)) and independent of {Xn; n ≥1}. In this paper, asymptotic expressions of P((X1 +… +XN(t)) -λ(t)μ 〉 x) uniformly for x ∈[γb(t), ∞) are obtained, where γ〉 0 and b(t) can be taken to be a positive function with limt→∞ b(t)/λ(t) = 0. 相似文献
|