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1.
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
*20c || f - Tn || £ \fracc( k,s )n2\upomega k( f",1 \mathord\vphantom 1 n n ) ( || f - Tn || £ \fracc( r + k,s )nr\upomega k( f(r),1 \mathord | / |
\vphantom 1 n n ), f ? C(r), r 3 2 ), \begin{array}{*{20}{c}} {\left\| {f - {T_n}} \right\| \leq \frac{{c\left( {k,s} \right)}}{{{n^2}}}{{{\upomega }}_k}\left( {f',{1 \mathord{\left/{\vphantom {1 n}} \right.} n}} \right)} \\ {\left( {\left\| {f - {T_n}} \right\| \leq \frac{{c\left( {r + k,s} \right)}}{{{n^r}}}{{{\upomega }}_k}\left( {{f^{(r)}},{1 \mathord{\left/{\vphantom {1 n}} \right.} n}} \right),\quad f \in {C^{(r)}},\quad r \geq 2} \right),} \\ \end{array} 相似文献
2.
An upper bound estimate in the law of the iterated logarithm for Σf(n
k ω) where nk+1∫nk≧ 1 + ck
-α (α≧0) is investigated. In the case α<1/2, an upper bound had been given by Takahashi [15], and the sharpness of the bound
was proved in our previous paper [8]. In this paper it is proved that the upper bound is still valid in case α≧1/2 if some
additional condition on {n
k} is assumed. As an application, the law of the iterated logarithm is proved when {n
k} is the arrangement in increasing order of the set B(τ)={1
i
1...qτ
i
τ|i1,...,iτ∈N
0}, where τ≧ 2, N
0=NU{0}, and q
1,...,q
τ are integers greater than 1 and relatively prime to each others.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
3.
We consider the differential operators Ψ
k
, defined by Ψ1(y) =y and Ψ
k+1(y)=yΨ
k
y+d/dz(Ψ
k
(y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ
k
F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z
2+β
z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ
k
(F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ
k
(f
′/f) =f
(k)/f, we deduce in particular that iff andf
(k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f
′/f :f ∈F} is normal.
The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999,
and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank
Günter Frank for helpful discussions. 相似文献
4.
Vincenzo De Filippis 《Israel Journal of Mathematics》2007,162(1):93-108
Let R be a prime ring with extended centroid C, g a nonzero generalized derivation of R, f (x
1,..., x
n) a multilinear polynomial over C, I a nonzero right ideal of R.
If [g(f(r
1,..., r
n)), f(r
1,..., r
n)] = 0, for all r
1, ..., r
n ∈ I, then either g(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:
5.
We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ
and expansive matrix M: Λ → Λ if ρ(M
−1) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M
−1. We shall prove further that if the polynomial f(x) = c
0 + c
1
x + ··· + c
k
x
k
∈ Z[x], c
k
= 1 satisfies the condition |c
0| > 2 Σ
i=1
k
|c
i
| then there is a suitable digit set D for which (Z
k
, M, D) is a number system, where M is the companion matrix of f(x).
The research was supported by OTKA-T043657 and Bolyai Fellowship Committee. 相似文献
6.
For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT
α=Σ
j=1
∞
a
j
T
j
, where {a
j
} are the coefficients in the power series expansion (1-t)α=1-Σ
j=1
∞
a
j
t
j
in the open unit disk, which satisfya
j
>0 anda
j
>0 and Σ
j=1
∞
a
j
=1. The operator calculus justifies the notation(I−T)
α
:=I−T
α
(e.g., (I−T
1/2)2=I−T). A vectory∈X is called an, α-fractional coboundary for
T if there is anx∈X such that(I−T)
α
x=y, i.e.,y is a coboundary forT
α
. The fractional Poisson equation forT is the Poisson equation forT
α
. We show that if(I−T)X is not closed, then(I−T)
α
X strictly contains(I−T)X (but has the same closure).
ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ
k=1
∞
T
k
y/k
1-α converges in norm, and conclude that lim
n
‖(1/n
1-α)Σ
k=1
n
T
k
y‖=0 for suchy.
For a Dunford-Schwartz operatorT onL
1 of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T)
α
L
1 for some 0<α<1, then the one-sided Hilbert transform Σ
k=1
∞
T
k
f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T)
α
L
p
with α>1−1/p=1/q, then Σ
k=1
∞
T
k
f/k
1/p
converges a.e., and thus (1/n
1/p
) Σ
k=1
n
T
k
f converges a.e. to zero. Whenf∈(I−T)
1/q
L
p
(the case α=1/q), we prove that (1/n
1/p
(logn)1/q
)Σ
k=1
n
T
k
f converges a.e. to zero. 相似文献
7.
E. A. Zhizhina 《Theoretical and Mathematical Physics》1997,112(1):844-856
We consider the stochastic model of planar rotators x(t)={xk(t), k∈Zd}, t≥0, xk(t)∈T1, at high temperature. For the decay of correlations <fA(x(0)), gA+k(t) (x(t))>, the asymptotic formula is obtained at t→∞, k(t)→∞, k(t)∈Zd. The basic methods we used are the spectral analysis of the Markov semigroup generator and the saddle-point method.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 67–80. 相似文献
8.
Copositive approximation of periodic functions 总被引:1,自引:0,他引:1
Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y
i
∈ Y:= {y
i
}
i∈ℤ such that for x ∈ [y
i
, y
i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each n ≧ N(Y) we construct a trigonometric polynomial P
n
of order ≦ n, changing its sign at the same points y
i
∈ Y as f, and
9.
Let Γ be a regular curve and Lp(Γ),1<p<+∞, be the class of all complex-valued functions f defined on Γ which are such that |f|p is integrable in sense of Lebesgue. In this work, we define the kth p-Faber polynomial Fk.p(z), the kth p-Faber principle part ≈Fk.p(1/z) for Γ, and defined the nth p-Faber-Laurent rational function Rn,p(f, z) and p-generalized modulus of continuity Ωp of a function f of Lp(Γ). We investigate some properties of Fk.p(z) and ≈Fk.p(1/z). And then we prove a direct theorem characterizing the degree of approximation with respect to Ωp in the mean of functions of Lp(Γ) by the rational functions Rn.p(.,z). 相似文献
10.
Let r, k, s be three integers such that , or We prove the following:
Proposition.
Let Y:={y
i
}
i=1
s
be a fixed collection of distinct points y
i
∈ (-1,1) and Π (x):= (x-y
1
). ... .(x-y
s
). Let I:=[-1,1]. If f ∈ C
(r)
(I) and f'(x)Π(x) ≥ 0, x ∈ I, then for each integer n ≥ k+r-1 there is an algebraic polynomial P
n
=P
n
(x) of degree ≤ n such that P
n
'(x) Π (x) ≥ 0 and
$$ \vert f(x)-P_n(x) \vert \le B\left(\frac{1}{n^2}+\frac{1}{n}\sqrt{1-x^2}\right)^r \omega_k \left(f^{(r)};\frac{1}{n^2}+\frac{1}{n}\sqrt{1-x^2}\right)
\legno{(1)}$$
for all x∈ I, where ω
k
(f
(r)
;t) is the modulus of smoothness of the k -th order of the function f
(r)
and B is a constant depending only on r , k , and Y. If s=1, the constant B does not depend on Y except in the case
(r=1, k=3).
In addition it is shown that (1) does not hold for r=1, k>3.
March 20, 1995. Dates revised: March 11, 1996; December 20, 1996; and August 7, 1997. 相似文献
11.
M. A. Nalbandyan 《Russian Mathematics (Iz VUZ)》2009,53(10):45-56
For any sequence {ω(n)}
n∈ℕ tending to infinity we construct a “quasiquadratic” representation spectrum Λ = {n
2 + o(ω(n))}
n∈ℕ: for any almost everywhere (a. e.) finite measurable function f(x) there exists a series in the form $
\mathop \sum \limits_{k \in \Lambda }
$
\mathop \sum \limits_{k \in \Lambda }
α
k
ω
k
(x) that converges a. e. to this function, where {w
k
(x)}
k∈ℕ is the Walsh system. We find representation spectra in the form {n
l
+ o(n
l
)}
n∈ℕ, where l ∈ {2
k
}
k∈ℕ. 相似文献
12.
Robert S. Strichartz 《Journal of Geometric Analysis》1991,1(3):269-289
Let μ be a measure on ℝn that satisfies the estimate μ(B
r(x))≤cr
α for allx ∈ ℝn and allr ≤ 1 (B
r(x) denotes the ball of radius r centered atx. Let ϕ
j,k
(ɛ)
(x)=2
nj2ϕ(ɛ)(2
j
x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤn, and ∈ ∈E, a finite set, and letP
j
(T)=Σɛ,k
<T,ϕ
j,k
(ɛ)
>ϕ
j,k
(ɛ)
denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls
p(dμ) for 1 ≤p ≤ ∞, we show thatP
j(f dμ) ∈ Lp(dx) and ||P
j
(fdμ)||L
p(dx)≤c2
j((n-α)/p′))||f||L
p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P
j
(fdμ)||L
p(dx) under more restrictive hypotheses.
Communicated by Guido Weiss 相似文献
13.
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
14.
Let T be a tree and let Ω ( f ) be the set of non-wandering points of a continuous map f: T→ T. We prove that for a continuous
map f: T→ T of a tree T: ( i) if x∈ Ω( f) has an infinite orbit, then x∈ Ω( fn) for each n∈ ℕ; (ii) if the topological entropy of f is zero, then Ω( f) = Ω( fn) for each n∈ ℕ. Furthermore, for each k∈ ℕ we characterize those natural numbers n with the property that Ω(fk) = Ω(fkn) for each continuous map f of T. 相似文献
15.
Let f∈C
[−1,1]
″
(r≥1) and Rn(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions Rn(f,α,β,xk)=f(xk),Rn
′(f,α,β,xk)=f′(xk)(k=1,2,…,n), where {xk} are the roots of n-th Jacobi polynomial Pn(α,β,x),α,β>−1 and {x
k
″
} are the roots of (1−x2)Pn″(α,β,x). In this paper, we prove that
holds uniformly on [0,1].
In Memory of Professor M. T. Cheng
Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang. 相似文献
16.
O. M. Fomenko 《Journal of Mathematical Sciences》2006,133(6):1733-1748
Let Sk(Γ) be the space of holomorphic Γ-cusp forms f(z) of even weight k ≥ 12 for Γ = SL(2, ℤ), and let Sk(Γ)+ be the set of all Hecke eigenforms from this space with the first Fourier coefficient af(1) = 1. For f ∈ Sk(Γ)+, consider the Hecke L-function L(s, f). Let
17.
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
18.
Alfred Lehman 《Israel Journal of Mathematics》1963,1(1):22-28
Circular symmetry is defined for ordered sets ofn real numbers: (y)=(y
1,...,y
n). Letf(x) be non-decreasing and convex forx≧0 and let (y) be given except in arrangement. The Σ
i
=1n
f(|y
i−y
i+1|) (wherey
n+1=y
1) is minimal if (and under some additional assumptions only if) (y) is arranged in circular symmetrical order.
Sponsored by the Mathematics Research Center, United States Army under Contract No. DA-11-022-ORD-2059, University of Wisconsin,
Madison. 相似文献
19.
Let ρ be a triangulation of a polygonal domain D⊂R2 with vertices V={vi:l≤i≤Nv} and RSk(D, ρ)={u∈Ck(D): ≠ T∈ρ, u/T is a rational function}. The purpose of this paper is to study the existence and construction of Cμ-rational spline functions on any triangulation ρ for CAGD. The Hermite problem Hμ(V,U)={find u∈U: Dαu(vi)=Dαf(vi),|α|≤μ} is solved by the generalized wedge function method in rational spline function family, i.e. U=RSμ. this solution needs only the knowledge of partial derivatives of order≤μ at vi. The explicit repesentations of all Cμ-GWF(generalized wedge functions)and the interpolating operator with degree of precision at least 2μ+1 for any triangulation
are given. 相似文献
20.
Vladimir Protasov 《Journal of Fourier Analysis and Applications》2000,6(1):55-78
In this paper we analyze solutions of the n-scale functional equation Ф(x) = Σk∈ℤ
Pk Ф(nx−k), where n≥2 is an integer, the coefficients {Pk} are nonnegative and Σpk = 1. We construct a sharp criterion for the existence of absolutely continuous solutions of bounded
variation. This criterion implies several results concerning the problem of integrable solutions of n-scale refinement equations
and the problem of absolutely continuity of distribution function of one random series. Further we obtain a complete classification
of refinement equations with positive coefficients (in the case of finitely many terms) with respect to the existence of continuous
or integrable compactly supported solutions. 相似文献
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