共查询到20条相似文献,搜索用时 376 毫秒
1.
V. G. Krotov 《Ukrainian Mathematical Journal》2010,62(3):441-451
We prove the following statement, which is a quantitative form of the Luzin theorem on C-property: Let (X, d, μ) be a bounded metric space with metric d and regular Borel measure μ that are related to one another by the doubling condition. Then, for any function f measurable on X, there exist a positive increasing function η ∈ Ω (η(+0) = 0 and η(t)t
−a
decreases for a certain a > 0), a nonnegative function g measurable on X, and a set E ⊂ X, μE = 0 , for which
| f(x) - f(y) | \leqslant [ g(x) + g(y) ]h( d( x,y ) ), x,y ? X | / |
E \left| {f(x) - f(y)} \right| \leqslant \left[ {g(x) + g(y)} \right]\eta \left( {d\left( {x,y} \right)} \right),\,x,y \in {{X} \left/ {E} \right.} 相似文献
2.
Liangping Jiang 《Journal of Mathematical Sciences》2011,177(3):395-401
The classical criterion of asymptotic stability of the zero solution of equations x′ = f(t, x) is that there exists a function V (t, x), a(∥x∥) ≤ V (t, x) ≤ b(∥x∥) for some a, b ∈ K such that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) for some c ∈ K. In this paper, we prove that if V(m + 1) \mathop {V}\limits^{(m + {1})} (t, x) is bounded on some set [tk − T, tk + T] × BH(tk → +∞ as k → ∞), then the condition that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) can be weakened and replaced by that [(V)\dot] \dot{V} (t, x) ≤ 0 and − (−[(V)\dot] \dot{V} (tk, x)| + − [(V)\ddot] \ddot{V} (tk, x)| + ⋯ + − V(m) \mathop {V}\limits^{(m)} (tk, x)|) ≤ −c′(∥x∥) for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [1–10] 相似文献
3.
Given a (known) function f:[0,1]→(0,1), we consider the problem of simulating a coin with probability of heads f(p) by tossing a coin with unknown heads probability p, as well as a fair coin, N times each, where N may be random. The work of Keane and O’Brien (ACM Trans. Model. Comput. Simul. 4(2):213–219, 1994) implies that such a simulation scheme with the probability ℙ
p
(N<∞) equal to 1 exists if and only if f is continuous. Nacu and Peres (Ann. Appl. Probab. 15(1A):93–115, 2005) proved that f is real analytic in an open set S⊂(0,1) if and only if such a simulation scheme exists with the probability ℙ
p
(N>n) decaying exponentially in n for every p∈S. We prove that for α>0 noninteger, f is in the space C
α
[0,1] if and only if a simulation scheme as above exists with ℙ
p
(N>n)≤C(Δ
n
(p))
α
, where
\varDelta n(x):=max{?{x(1-x)/n},1/n}\varDelta _{n}(x):=\max\{\sqrt{x(1-x)/n},1/n\}. The key to the proof is a new result in approximation theory: Let B+n\mathcal{B}^{+}_{n} be the cone of univariate polynomials with nonnegative Bernstein coefficients of degree n. We show that a function f:[0,1]→(0,1) is in C
α
[0,1] if and only if f has a series representation ?n=1¥Fn\sum_{n=1}^{\infty}F_{n} with Fn ? B+nF_{n}\in \mathcal{B}^{+}_{n} and ∑
k>n
F
k
(x)≤C(Δ
n
(x))
α
for all x∈[0,1] and n≥1. We also provide a counterexample to a theorem stated without proof by Lorentz (Math. Ann. 151:239–251, 1963), who claimed that if some jn ? B+n\varphi_{n}\in\mathcal{B}^{+}_{n} satisfy |f(x)−φ
n
(x)|≤C(Δ
n
(x))
α
for all x∈[0,1] and n≥1, then f∈C
α
[0,1]. 相似文献
4.
A. I. Podvysotskaya 《Ukrainian Mathematical Journal》2009,61(5):847-853
We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than
( n - 1 ) \mathord | / | \vphantom ( n - 1 ) ?{1 - x2} ?{1 - x2} {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 - {x^2}} }} for all x ∈ (−1, 1) such that | x | ? èk = 0[ n \mathord | / |
\vphantom n 2 2 ] [ cos\frac2k + 12( n - 1 )p, cos\frac2k + 12np ] \left| x \right| \in \bigcup\nolimits_{k = 0}^{\left[ {{n \mathord{\left/{\vphantom {n 2}} \right.} 2}} \right]} {\left[ {\cos \frac{{2k + 1}}{{2\left( {n - 1} \right)}}\pi, \cos \frac{{2k + 1}}{{2n}}\pi } \right]} . 相似文献
5.
Lou Yuanren 《分析论及其应用》1990,6(1):46-64
Let f∈Ap. For any positive integer l, the quantity Δ1,n−1(f:z) has been studied extensively. Here we give some quantitative estimates for
and investigate some pointwise estimates of Δ
l,n−1
(r)
(f;z).
Supported by National Science Foundation of China 相似文献
6.
For β > 0 and an integer r ≥ 2, denote by [(H)\tilde]¥,br\tilde H_{\infty ,\beta }^r those 2π-periodic, real-valued functions f on ℝ, which are analytic in S
β
:= {z ∈ ℂ: |Im z| < β} and satisfy the restriction |f
(r)(z)|≤1, z ∈ S
β
. The optimal quadrature formulae about information composed of the values of a function and its kth (k = 1, ..., r − 1) derivatives on free knots for the classes [(H)\tilde]¥,br\tilde H_{\infty ,\beta }^r are obtained, and the error estimates of the optimal quadrature formulae are exactly determined. 相似文献
7.
V. V. Zhuk 《Journal of Mathematical Sciences》2010,166(2):167-185
Let L
p
, 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm
|| f ||p = ( ò - pp | f |p )1 \mathord | / |
\vphantom 1 p p {\left\| f \right\|_p} = {\left( {\int\limits_{ - \pi }^\pi {{{\left| f \right|}^p}} } \right)^{{1 \mathord{\left/{\vphantom {1 p}} \right.} p}}} , and let C = L
∞ be the space of continuous 2π-periodic functions with the norm
|| f ||¥ = || f || = maxe ? \mathbbR | f(x) | {\left\| f \right\|_\infty } = \left\| f \right\| = \mathop {\max }\limits_{e \in \mathbb{R}} \left| {f(x)} \right| . Let CP be the subspace of C with a seminorm P invariant with respect to translation and such that
P(f) \leqslant M|| f || P(f) \leqslant M\left\| f \right\| for every f ∈ C. By ?k = 0¥ Ak (f) \sum\limits_{k = 0}^\infty {{A_k}} (f) denote the Fourier series of the function f, and let l = { lk }k = 0¥ \lambda = \left\{ {{\lambda_k}} \right\}_{k = 0}^\infty be a sequence of real numbers for which ?k = 0¥ lk Ak(f) \sum\limits_{k = 0}^\infty {{\lambda_k}} {A_k}(f) is the Fourier series of a certain function f
λ ∈ L
p
. The paper considers questions related to approximating the function f
λ by its Fourier sums S
n
(f
λ) on a point set and in the spaces L
p
and CP. Estimates for || fl - Sn( fl ) ||p {\left\| {{f_\lambda } - {S_n}\left( {{f_\lambda }} \right)} \right\|_p} and P(f
λ − S
n
(f
λ)) are obtained by using the structural characteristics (the best approximations and the moduli of continuity) of the functions
f and f
λ. As a rule, the essential part of deviation is estimated with the use of the structural characteristics of the function f.
Bibliography: 11 titles. 相似文献
8.
S. P. Zhou 《Israel Journal of Mathematics》1992,78(1):75-83
The present paper gives a converse result by showing that there exists a functionf ∈C
[−1,1], which satisfies that sgn(x)f(x) ≥ 0 forx ∈ [−1, 1], such that {fx75-1} whereE
n
(0)
(f, 1) is the best approximation of degreen tof by polynomials which are copositive with it, that is, polynomialsP withP(x(f(x) ≥ 0 for allx ∈ [−1, 1],E
n(f) is the ordinary best polynomial approximation off of degreen. 相似文献
9.
H. A. Dzyubenko 《Ukrainian Mathematical Journal》2009,61(4):519-540
In the case where a 2π-periodic function f is twice continuously differentiable on the real axis ℝ and changes its monotonicity at different fixed points y
i
∈ [− π, π), i = 1,…, 2s, s ∈ ℕ (i.e., on ℝ, there exists a set Y := {y
i
}
i∈ℤ of points y
i
= y
i+2s
+ 2π such that the function f does not decrease on [y
i
, y
i−1] if i is odd and does not increase if i is even), for any natural k and n, n ≥ N(Y, k) = const, we construct a trigonometric polynomial T
n
of order ≤n that changes its monotonicity at the same points y
i
∈ Y as f and is such that
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