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1.
This note is devoted to a generalization of the Strassen converse. Let gn:R→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,yR, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X1,X2,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X1,X2,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.  相似文献   

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Let E be a real uniformly convex Banach space whose dual space E satisfies the Kadec-Klee property, K be a closed convex nonempty subset of E. Let be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . For arbitrary ?∈(0,1), let be a sequence in [?,1−?], for each i∈{1,2,…,m} (respectively). Let {xn} be a sequence generated for m?2 by
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Let be a sequence of i.i.d. random variables with EX=0 and EX2=σ2<∞. Set , Mn=maxk?n|Sk|, n?1. Let r>1, then we obtain
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Let E be a real uniformly convex Banach space, K be a closed convex nonempty subset of E which is also a nonexpansive retract with retraction P. Let be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . Let be a sequence in [?,1−?],?∈(0,1), for each i∈{1,2,…,m} (respectively). Let {xn} be a sequence generated for m?2 by
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10.
We study certain Hardy-type sequence spaces Hp and , 1?p?∞, which are analogues of ? and c0, respectively. We show that the Mazur product is not onto for every p∈(1,∞) with q=p−1(p−1). We present corollaries for spaces defined via weighted ?p seminorms and for c0. The latter corollary provides a new solution of Mazur's Problem 8 in the Scottish Book.  相似文献   

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Fix a sequence of positive integers (mn) and a sequence of positive real numbers (wn). Two closely related sequences of linear operators (Tn) are considered. One sequence has given by the Lebesgue derivatives . The other sequence has given by the dyadic martingale when (l−1)/n2?x<l/n2 for l=1,…,n2. We prove both positive and negative results concerning the convergence of .  相似文献   

12.
Let f(z) be a normalized convex (starlike) function on the unit disc D. Let , where z=(z1,z2,…,zn), z1D, , pi?1, i=2,…,n, are real numbers. In this note, we prove that Φ(f)(z)=(f(z1),f′(z1)1/p2z2,…,f′(z1)1/pnzn) is a normalized convex (starlike) mapping on Ω, where we choose the power function such that (f′(z1))1/pi|z1=0=1, i=2,…,n. Some other related results are proved.  相似文献   

13.
Suppose f is a spirallike function of type β (or starlike function of order α) on the unit disk D in C. Let , where 1?p1?2 (or 0<p1?2), pj?1, j=2,…,n, are real numbers. In this paper, we prove that
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Given a basis of solutions to k ordinary linear differential equations ?j[y]=0(j=1,2,…,k), we show how Green's functions can be used to construct a basis of solutions to the homogeneous differential equation ?[y]=0, where ? is the composite product ?=?1?2?k. The construction of these solutions is elementary and classical. In particular, we consider the special case when . Remarkably, in this case, if {y1,y2,…,yn} is a basis of ?1[y]=0, then our method produces a basis of for any kN. We illustrate our results with several classical differential equations and their special function solutions.  相似文献   

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Let n?2, Sn−1 be the unit sphere in Rn. For 0?α<1, mN0, 1<p?2, and ΩL(RnHr(Sn−1) with (where Hr is the Hardy space if r?1 and Hr=Lr if 1<r<∞), we study the singular integral operator, for r?1, defined by
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17.
By constructing the comparison functions and the perturbed method, it is showed that any solution uC2(Ω) to the semilinear elliptic problems Δu=k(x)g(u), xΩ, u|Ω=+∞ satisfies , where Ω is a bounded domain with smooth boundary in RN; , −2<σ, c0>0, ; gC1[0,∞), g?0 and is increasing on (0,∞), there exists ρ>0 such that , ∀ξ>0, , .  相似文献   

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In contrast to the famous Henkin-Skoda theorem concerning the zero varieties of holomorphic functions in the Nevanlinna class on the open unit ball Bn in , n?2, it is proved in this article that for any nonnegative, increasing, convex function ?(t) defined on , there exists satisfying such that there is no fHp(Bn), 0<p<∞, with . Here Ng(ζ,1) denotes the integrated zero counting function associated with the slice function gζ. This means that the zero sets of holomorphic functions belonging to the Hardy spaces Hp(Bn), 0<p<∞, unlike that of the holomorphic functions in the Nevanlinna class, cannot be characterized in the above manner.  相似文献   

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Suppose that K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E. Let be two nonself asymptotically nonexpansive mappings with sequences {kn},{ln}⊂[1,∞), limn→∞kn=1, limn→∞ln=1, , respectively. Suppose {xn} is generated iteratively by
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20.
Let X1,X2,… be a strictly stationary sequence of ρ-mixing random variables with mean zeros and positive, finite variances, set Sn=X1+?+Xn. Suppose that , , where q>2δ+2. We prove that, if for any 0<δ?1, then
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