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1.
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
where c(α) is a constant depending only on α. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (−1,1) is also investigated. It turns out that there are substantial differences between the cases
s≦ 1 and s ≧ 2.
Dedicated to Jóska Szabados on his 70th birthday 相似文献
2.
Letc
n
(A) denote the codimensions of a P.I. algebraA, and assumec
n
(A) has a polynomial growth:
. Then, necessarily,q∈ℚ [D3]. If 1∈A, we show that
, wheree=2.71…. In the non-unitary case, for any 0<q∈ℚ, we constructA, with a suitablek, such that
.
In memory of S. A. Amitsur, our teacher and friend
Partially supported by Grant MM404/94 of Ministry of Education and Science, Bulgaria and by a Bulgarian-American Grant of
NSF.
Partially supported by NSF grant DMS-9101488. 相似文献
3.
In this paper we consider special elements of the Fock space #x2131;
n
. That is the space of entire functionsf:ℂ:
n
→ℂ, such that the followingL
2- condition is satisfied:
. Here we show that there exists an entire functiong:ℂ
n
→ℂ such that for every one-dimensional subspace Π⊂ℂ
n
and for all 0<∈<2 we have
, but in the limit case ∈=0 we have
. This result is analogue to a result from [1]. There holomorphic functions on the unit-ball are investigated. Furthermore
the proof — as the one in [1] — uses a theorem from [2]. Therefore we give another application of the results from [2] — namely
for spaces of entire functions. 相似文献
4.
刘新和 《高校应用数学学报(英文版)》2003,18(2):129-137
§ 1 IntroductionThe Feigenbaum functional equation plays an importantrole in the theory concerninguniversal properties of one-parameter families of maps of the interval that has the formf2 (λx) +λf(x) =0 ,0 <λ=-f(1 ) <1 ,f(0 ) =1 ,(1 .1 )where f is a map ofthe interval[-1 ,1 ] into itself.Lanford[1 ] exhibited a computer-assist-ed proof for the existence of an even analytic solution to Eq.(1 .1 ) .It was shown in[2 ]that Eq.(1 .1 ) does not have an entire solution.Si[3] discussed the it… 相似文献
5.
LetH be any complex inner product space with inner product <·,·>. We say thatf: ℂ→ℂ is Hermitian positive definite onH if the matrix
is Hermitian positive definite for all choice ofz
1,…,z
n inH for alln. It is strictly Hermitian positive definite if the matrix (*) is also non-singular for any choice of distinctz
1,…,z
n inH. In this article, we prove that if dimH≥3, thenf is Hermitian positive definite onH if and only if
whereb
k,l
≥0 for allk, l in ℤ, and the series converges for allz in ℂ. We also prove thatf of the form (**) is strictly Hermitian positive definite on anyH if and only if the setJ={(k,l):b
k,l
>0} is such that (0,0)∈J, and every arithmetic sequence in ℤ intersects the values {k−l: (k, l)∈J} an infinite number of times. 相似文献
(1) |
(1) |
6.
In the same spirit of the classical Leau-Fatou flower theorem, we prove the existence of a petal, with vertex at the Wolff
point, for a holomorphic self-map f of the open unit disc Δ ⊂ ℂ of parabolic type. The result is obtained in the framework of two interesting dynamical situations
which require different kinds of regularity of f at the Wolff point τ: f of non-automorphism type and
or f injective of automorphism type, f∈C
3+ɛ(τ) and
.
Partially supported by PRIN Proprietà geometriche delle varietà reali e complesse.
Partially supported by GNSAGA of the Istituto Nazionale di Alta Matematica, Rome. 相似文献
7.
In this paper, a natural R
+
n+1
extension of singular integrals, i.e.,T
κ:f→K*φ
t
*t with K a standard C-Z kernel and ϕ usual one, is investigated. One of the main results is: Let (dμ, udx) ∈C1 and u-Mw, w∈A∞, then Tk is of type (Lp(udx), Lp(dμ)). As a related topic, a maximal operator
is proved to be of type
, where
, provided (dμ, udx) ∈C1 and u∈ A∞.
Supported by National Science Foundation of China 相似文献
8.
The wave equation, ∂
tt
u=Δu, in ℝ
n+1, considered with initial data u(x,0)=f∈H
s
(ℝ
n
) and u’(x,0)=0, has a solution which we denote by . We give almost sharp conditions under which and are bounded from H
s
(ℝ
n
) to L
q
(ℝ
n
). 相似文献
9.
B. P. Duggal 《Integral Equations and Operator Theory》2009,63(1):17-28
A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower
semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M
C
denote the operator matrix . If A is polaroid on , M
0 satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A* has SVEP at points and B
* has SVEP at points , then . Here the hypothesis that λ ∈ π0(M
C
) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A.
For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π
a
0(M
C) and B is polaroid on π
a
0(B), then .
相似文献
10.
Letf(z, t) be a subordination chain fort ∈ [0, α], α>0, on the Euclidean unit ballB inC
n. Assume thatf(z) =f(z, 0) is quasiconformal. In this paper, we give a sufficient condition forf to be extendible to a quasiconformal homeomorphism on a neighbourhood of
. We also show that, under this condition,f can be extended to a quasiconformal homeomorphism of
onto itself and give some applications.
Partially supported by Grant-in-Aid for Scientific Research (C) no. 14540195 from Japan Society for the Promotion of Science,
2004. 相似文献
11.
Gabriele H. Greco 《Annali dell'Universita di Ferrara》1981,27(1):13-19
Riassunto In questo articolo si danno delle condizioni necessarie e sufficienti affinchè per una fissata coppia di funzioni d’insieme
ν, μ crescenti esista una funzionef tale che ν=∫fdμ. Si ottiene cosi una proposizione comprendente il teorema di R-N. classico e dei teoremi di R-N., presentati da altri autori,
riguardanti le funzioni d’insieme finitamente additive e le funzioni d’insieme subadditive e continue per successioni crescenti.
Résumé Soient ν, μ:A→[0,+∞) deux fonctions d’ensemble croissantes sur une σ-algèbre d’ensemblesA⊂T(X), telles que pour chaqueA∈A avec ν(A)=μ(A)=0 on a l’égalité μ(A)=μ(A∪S) ∀S∈A (c’est le cas des fonctions sousadditives!). Dans cet article on démontre qu’il existe une fonctionf A-measurable telle que ν=∫fdμ si et seulement si pour chaquer∈(0, + ∞) il y a un ensembleA r∈A qui vérifie les trois conditions suivantes: (1) ,B∈A avecB⊂A; (2) A (3) limν(A r)=0. On déduit ainsi une proposition qui a été donnée parI. Forana: ?Si ν, μ sont simplement additives, il existe une fonctionf telle que ν=∫fdμ si et seulement si ν≪μ et la fonction d’ensemble additive a une decomposition de Hahn pour chaquer∈(0, + ∞), c’est-á-dire il y aA r∈A tel que ?.相似文献
12.
E. Amar 《Journal of Geometric Analysis》1991,1(4):291-305
We show that if f1, f2 are bounded holomorphic functions in the unit ball
of ℂn such that
, |f1(z)|2 + |f2(z)2|2 ≥ δ2 >; 0, then any functionh in the Hardy space
,p < +∞ can be decomposed ash = f1h1
+ f2h2 with
. The Corona theorem in
would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction. 相似文献
13.
The generalized Roper-Suffridge extension operator Ф(f) on the bounded complete Reinhardt domain Ω in Cn with n ≥ 2 is defined by Φrn,β2,γ2,…,βn,γn(f)(z)=(rf(z1/r),(rf(z1/r)/z1)β2(f'(z1/r))γ2z2,…,(rf(z1/r)/z1)βn(f'(z1/r)γnzn) for (z1,z2,…,zn) ∈Ω, where r = r(Ω) = sup{|z1| (z1,z2,…,zn) ∈ Ω},0 ≤ γj ≤ 1 -βj,0 ≤ βj ≤ 1,and we choose the branch of the power functions such that (f(z1)/z1)βj |z1=0 = 1 and (f′(z1))γj |z1=0 =1,j = 2,…,n. In this paper, we prove that the operator Фrn,β2,γ2,…,βn,γn(f) is from the subset of S*α(U) to S*α(Ω)(0 ≤ α < 1) on Ω and the operator Фrn,β2,γ2,…, βn,γn(f) preserves the starlikeness of order α or the spirallikeness of type β on Dp for some suitable constantsβj,γj,pj, where Dp ={(z1,z2,…,zn) ∈ Cn ∑nj=1|zj|pj < 1} (pj > 0, j = 1,2,…,n), U is the unit disc in the complex plane C, and Sα* (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that Φrn,β2,γ2,…,γn(f) ∈ S*α(Dp) if and only if f ∈ S*a(U) for 0 ≤ α < 1 and some suitable constants βj,γj,pj. 相似文献
14.
Kâzim Ilhan Ikeda 《Proceedings Mathematical Sciences》2003,113(2):99-137
This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. LetK be a local field with finite residue class fieldK
k. We first define (cf. Definition 2.4) the conductorf(E/K) of an arbitrary finite Galois extensionE/K in the sense of non-abelian local class field theory as wheren
G is the break in the upper ramification filtration ofG = Gal(E/K) defined by
. Next, we study the basic properties of the idealf(E/K) inO
k in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]).
After reviewing the Artin charactera
G : G → ℂ ofG := Gal(E/K) and Artin representationsA
g G → G →GL(V) corresponding toa
G : G → ℂ, we prove that (Proposition 3.2 and Corollary 3.5)
where Χgr
: G → ℂ is the character associated to an irreducible representation ρ: G → GL(V) ofG (over ℂ). The first main result (Theorem 1.2) of the paper states that, if in particular,ρ : G → GL(V) is an irreducible representation ofG(over ℂ) with metabelian image, then
where Gal(Eker(ρ)/Eker(ρ)•) is any maximal abelian normal subgroup of Gal(Eker(ρ)/K) containing Gal(Eker(ρ)
/K)′, and the break nG/ker(ρ) in the upper ramification filtration of G/ker(ρ) can be computed and located by metabelian local class field theory. The
proof utilizes Basmaji’s theory on the structure of irreducible faithful representations of finite metabelian groups (cf.
[1]) and on metabelian local class field theory (cf. [8]).
We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a ‘natural’A
G ofG over ℂ (Problem 1.3). More precisely, we prove in Theorem 1.4 that ifE/K is a metabelian extension with Galois group G, then
Kazim İlhan ikeda whereN runs over all normal subgroups of G, and for such anN, V
n denotes the collection of all ∼-equivalence classes [ω]∼, where ‘∼’ denotes the equivalence relation on the set of all representations
ω : (G/N)• → ℂΧ satisfying the conditions Inert(ω) = {δ ∈ G/N : ℂδ} = ω =(G/N) and
where δ runs over R((G/N)•/(G/N)), a fixed given complete system of representatives of (G/N)•/(G/N), by declaring that ω1 ∼ ω2 if and only if ω1
= ω
2,δ for some δ ∈ R((G/N)•/(G/N)).
Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3. 相似文献
15.
S. P. Sidorov 《Numerical Algorithms》2007,44(3):273-279
Let , –1<x
1<...<x
n
<1. Denote , t∈(–1,1). Given a function f∈W we try to recover f(ζ) at fixed point ζ∈(–1,1) by an algorithm A on the basis of the information f(x
1),...,f(x
n
). We find the intrinsic error of recovery .
This work is supported by RFBR (grant 07-01-00167-a and grant 06-01-00003). 相似文献
16.
In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation
has a positive and bounded solution, where q, h, f ∈ C ([0, ∞), ℝ) such that q(t) ≥ 0, but ≢ 0, h(t) ≤ t, h(t) → ∞ as t → ∞, r ∈ C
(1) ([0, ∞), (0, ∞)), p ∈ C
(2) [0, ∞), ℝ), G ∈ C(ℝ, ℝ) and τ ∈ ℝ+. In our work r(t) ≡ 1 is admissible and neither we assume G is non-decreasing, xG(x) > 0 for x ≠ 0, nor we take G is Lipschitzian. Hence the results of this paper improve many recent results.
相似文献
17.
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
18.
S. Norvidas 《Lithuanian Mathematical Journal》2009,49(2):185-189
For a compact set K in ℝ
n
, let B
2
K
be the set of all functions f ∈ L
2(ℝ2) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B
2
K
by finite exponential sums
in the space , as τ → ∞. 相似文献
19.
Riccardo De Arcangelis 《Annali dell'Universita di Ferrara》1989,35(1):135-145
Summary Letf: (x, z)∈R
n×Rn→f(x, z)∈[0, +∞] be measurable inx and convex inz.
It is proved, by an example, that even iff verifies a condition as|z|
p≤f(x, z)≤Λ(a(x)+|z|q) with 1<p<q,a∈L
loc
s
(R
n),s>1, the functional
that isL
1(Ω)-lower semicontinuous onW
1,1(Ω), does not agree onW
1,1(Ω) with its relaxed functional in the topologyL
1(Ω) given by inf
Riassunto Siaf: (x, z)∈R n×Rn→f(x, z)∈[0, +∞] misurabile inx e convessa inz. Si mostra con un esempio che anche sef verifica una condizione del tipo|z| p≤f(x, z)≤Λ(a(x)+|z|q) con 1<p<q,a∈L loc s (R n),s>1, il funzionale , che èL 1(Ω)-semicontinuo inferiormente suW 1,1(Ω), non coincide suW 1,1(Ω) con il suo funzionale rilassato nella topologiaL 1(Ω) definito da inf相似文献
20.
E. S. Dubtsov 《Journal of Mathematical Sciences》2007,141(5):1531-1537
Let
and τ denote the invariant gradient and invariant measure on the unit ball B of ℂn, respectively. Assume that f is a holomorphic function on B and ϕ ∈ C2(ℝ) is a nonnegative, nondecreasing, convex function. Then f belongs to the Hardy-Orlicz space H
ϕ(B>) if and only if
Analogous characterizations of Bergman-Orlicz spaces are obtained. Bibliography: 9 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 43–53. 相似文献