Uniform rates of approximation by short asymptotic expansions in the CLT for quadratic forms

Let X, X ₁, X ₂,… be i.i.d. \mathbbR^d {\mathbb{R}^d} -valued real random vectors. Assume that E X = 0 and that X has a nondegenerate distribution. Let G be a mean zero Gaussian random vector with the same covariance operator as that of X. We study the distributions of nondegenerate quadratic forms \mathbbQ[ S_N ] \mathbb{Q}\left[ {{S_N}} \right] of the normalized sums S _N  = N ^−1/2 (X ₁ + ⋯ + X _N ) and show that, without any additional conditions,

On conjugacy of some systems of functions

Summary. We investigate the bounded solutions j:[0,1]? X \varphi:[0,1]\to X of the system of functional equations¶¶j(f_k(x))=F_k(j(x)),    k=0,?,n-1,x ? [0,1] \varphi(f_k(x))=F_k(\varphi(x)),\;\;k=0,\ldots,n-1,x\in[0,1] ,(*)¶where X is a complete metric space, f₀,?,f_n-1:[0,1]?[0,1] f_0,\ldots,f_{n-1}:[0,1]\to[0,1] and F₀,...,F_n-1:X? X F_0,...,F_{n-1}:X\to X are continuous functions fulfilling the boundary conditions f₀(0) = 0, f_n-1(1) = 1, f_k+1(0) = f_k(1), F₀(a) = a,F_n-1(b) = b,F_k+1(a) = F_k(b), k = 0,?,n-2 f_{0}(0) = 0, f_{n-1}(1) = 1, f_{k+1}(0) = f_{k}(1), F_{0}(a) = a,F_{n-1}(b) = b,F_{k+1}(a) = F_{k}(b),\,k = 0,\ldots,n-2 , for some a,b ? X a,b\in X . We give assumptions on the functions f_k and F_k which imply the existence, uniqueness and continuity of bounded solutions of the system (*). In the case X = \Bbb C X= \Bbb C we consider some particular systems (*) of which the solutions determine some peculiar curves generating some fractals. If X is a closed interval we give a collection of conditions which imply respectively the existence of homeomorphic solutions, singular solutions and a.e. nondifferentiable solutions of (*).     

Commutators of BMO functions and degenerate Schr?dinger operators with certain nonnegative potentials

Let Lf(x)=-\frac1w?_i,j ?_i(a_i,j(·)?_jf)(x)+V(x)f(x){\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)} with the non-negative potential V belonging to reverse H?lder class with respect to the measure ω(x)dx, where ω(x) satisfies the A ₂ condition of Muckenhoupt and a _i,j (x) is a real symmetric matrix satisfying l^-1w(x)|x|² ￡ ?ⁿ_i,j=1a_i,j(x)x_ix_j ￡ lw(x)|x|².{\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. } We obtain some estimates for V^aL^-a{V^{\alpha}\mathcal{L}^{-\alpha}} on the weighted L ^p spaces and we study the weighted L ^p boundedness of the commutator [b, V^a L^-a]{[b, V^{\alpha} \mathcal{L}^{-\alpha}]} when b ? BMO_w{b\in BMO_\omega} and 0 < α ≤ 1.     

Linear Maps Preserving Operators of Local Spectral Radius Zero

Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let r_T(x) = limsup_{n ? ￥} || Tⁿx|| ^1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r _T (x) = 0 if and only if r_j(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r _T (By) = 0 if and only if r_{j( T)} (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB^-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}.     

Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth

The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C ¹ in \mathbbRⁿ{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)e^ahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and lim_{h? +￥}h^b-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.     

Quantitative form of the Luzin <Emphasis Type="Italic">C</Emphasis>-property

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

Convergence rates of the LIL for random fields in Hilbert spaces

Considering the positive d-dimensional lattice point Z ₊ ^d (d ≥ 2) with partial ordering ≤, let {X _k: k ∈ Z ₊ ^d } be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $ S_n = \sum\limits_{k \leqslant n} {X_k } $ S_n = \sum\limits_{k \leqslant n} {X_k } , n ∈ Z ₊ ^d . Let σ _i ², i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ ². Let logx = ln(x ∨ e), x ≥ 0. This paper studies the convergence rates for $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) . We show that when l ≥ 2 and b > −l/2, E[‖X‖²(log ‖X‖) ^d−2(log log ‖X‖) ^b+4] < ∞ implies $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} , where Γ(·) is the Gamma function and $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } .     

Precise asymptotics in self-normalized sums of iterated logarithm for multidimensionally indexed random variables

In the case of Zd (d ≥ 2)-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k ∈ Zd } i.i.d. random variables with mean 0, Sn = ∑k≤nXk and Vn2 = ∑j≤nX2j, the precise asymptotics for ∑n1/|n|(log|n|)dP(|Sn/vn|≥ ε√loglog|n|) and ∑n(logn|)δ/|n|(log|n|)d-1 P(|Sn/Vn| ≥ ε√log n), as ε ↘ 0, is established.     

On the regular growth of Dirichlet series absolutely convergent in a half-plane

For the Dirichlet series F(s) = ?_{n = 1}^￥ a_nexp{ sl_n } F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}\exp \left\{ {s{\lambda_n}} \right\}} with abscissa of absolute convergence σ _a =0, we establish conditions for (λ _n ) and (a _n ) under which lnM( s, F ) = T_R( 1 + o(1) )exp{ r_R

Implicatively selector sets

Let A⊆N={0,1,2,...} and β be an n-ary Boolean function. We call A a β-implicatively selector (β-IS) set if there exists an n-ary selector general recursive function f such that (∀x₁,...,x_n)(β(χ(x₁),...,χ(x_n))=1⟹f(x₁,...,x_n)∈A), where χ is the characteristic function of A. Let F^(m), m≥1, be the family of all d _m+1 ^* -IS sets, where , F⁽⁰⁾=N, and F^(∞) is the class of all subsets in N. The basic result of the article says that the family of all β-IS sets coincides with one of F^(m), m≥0, or F^(∞), and, moreover, the inclusions F⁽⁰⁾⊂F⁽¹⁾⊂...⊂F^(∞) hold. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 145–153, March–April, 1996.     

Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, u^￠_n(t) = ?_{j ? \mathbbZ^d} J_n-ju_j(t)-u_n(t)u^{\prime}_{n}(t) = \sum_{{j\in}{{{\mathbb{Z}}}^{d}}} J_{n-j}u_{j}(t)-u_{n}(t) with t ≥ 0 and n ? \mathbbZ^dn \in {\mathbb{Z}}^{d}. We assume that J is nonnegative and verifies ?_{n ? \mathbbZ^d}J_n = 1\sum_{{n \in {\mathbb{Z}}}^{d}}J_{n}= 1. We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.     

Properties of Riesz fractional derivatives for Korteweg–de Vries solitons

Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (?Δ) ^α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (−Δ) ^α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u ₀(X) with X = x − 4t, these derivatives, u _α (X) = D ^α u ₀(X), and their Hilbert transforms, v _α (X) = −HD ^α u ₀(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ₊(s, a) and ζ₋(s, a), respectively. New properties are established for u _α (X) and v _α (X). It is proved that the functions w _α (X) = u _α (X) + iv _α (X) with α > −1 are solutions of the differential equation

-\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},      13. The density of rational points on non-singular hypersurfaces      D. R. Heath-Brown 《Proceedings Mathematical Sciences》1994,104(1):13-29 LetF(x) =F[x1,…,xn]∈ℤ[x1,…,xn] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ n ;F(x)=0, |x|⩽X}, where . It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X n−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process. Dedicated to the memory of Professor K G Ramanathan      14. Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications      Wei-Feng Xia  Yu-Ming Chu 《Ukrainian Mathematical Journal》2009,61(10):1541-1555 For x = (x 1, x 2, …, x n ) ∈ (0, 1 ] n and r ∈ { 1, 2, … , n}, a symmetric function F n (x, r) is defined by the relation Fn( x,r ) = Fn( x1,x2, ?, xn;r ) = ?1 \leqslant1 < i2 ?ir \leqslant n ?j = 1r \frac1 - xijxij , {F_n}\left( {x,r} \right) = {F_n}\left( {{x_1},{x_2}, \ldots, {x_n};r} \right) = \sum\limits_{1{ \leqslant_1} < {i_2} \ldots {i_r} \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 - {x_{{i_j}}}}}{{{x_{{i_j}}}}}} },      15. On the Problem of Parameter Estimation in Exponential Sums      F. Filbir  H. N. Mhaskar  J. Prestin 《Constructive Approximation》2012,35(3):323-343 Let I≥1 be an integer, ω 0=0<ω 1<⋯<ω I ≤π, and for j=0,…,I, a j ∈ℂ, a-j=[`(aj)]a_{-j}={\overline{{a_{j}}}}, ω −j =−ω j , and aj 1 0a_{j}\not=0 if j 1 0j\not=0. We consider the following problem: Given finitely many noisy samples of an exponential sum of the form [(x)\tilde](k) = ?j=-II ajexp(-iwjk) +e(k),     k=-2N,?,2N,\tilde{x}(k)= \sum_{j=-I}^I a_j\exp(-i\omega _jk) +\epsilon (k), \quad k=-2N,\ldots,2N,      16. Limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival function      Zheng Zukang 《数学学报(英文版)》1994,10(4):337-347 LetX 1,X 2, ...,X n be a sequence of nonnegative independent random variables with a common continuous distribution functionF. LetY 1,Y 2, ...,Y n be another sequence of nonnegative independent random variables with a common continuous distribution functionG, also independent of {X i }. We can only observeZ i =min(X i ,Y i ), and . LetH=1−(1−F)(1−G) be the distribution function ofZ. In this paper, the limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival functionS(t) are given. It is shown that (1)P(δ(n)=1 i.o.)=0 ifF(τ H ) < 1 andP(δ n =0 i.o. )=0 ifG(τH) > 1 where δ(n) is the corresponding indicator function of and have the same order a.s., where {T n } is a sequence of constants such that 1−H(T n )=n −α(logn)β(log logn)γ.      17. Onp-absolutely summing constants of banach spaces      Yehoram Gordon 《Israel Journal of Mathematics》1969,7(2):151-163 Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ p(X) as inf{Σ i =1/m |x*(x i)|p p Σ i =1/m ‖x i‖p p]1 p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x 1,x 2, …,x m} ⊂X such that Σ i =1/m ‖x i‖>0. It follows immediately from [2] thatμ p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ p(X) for various spaces, and obtain some asymptotic estimates ofμ p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.      18. A topological position of the set of continuous maps in the set of upper semicontinuous maps      ZhongQiang Yang  NaDa Wu 《中国科学A辑(英文版)》2009,52(8):1815-1828 Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q \ Σ)), i.e., there is a homeomorphism h :↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q \ Σ...      19. Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces      Markus Haase 《Integral Equations and Operator Theory》2006,56(2):197-228 We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( Aa ) ?R( Aa ) )q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as { x  ?  X|t - q\textRea y1 ( tA )x, t - q\textRea y2 ( tA )x ? L*p ( ( 0,￥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}}      20. On the growth of functions represented by Dirichlet series with complex coefficients on the real axis      B. V. Vynnyts’kyi  O. V. Shapovalovs’kyi 《Ukrainian Mathematical Journal》1997,49(12):1810-1818 We establish conditions under which, for a Dirichlet series F(z) = Σ n = 1 ∞ d n exp(λ n z), the inequality ⋎F(x)⋎≤y(x),x≥x o, implies the relation Σ n = 1 ∞ |d n exp(λ n z)| ⪯ γ((1 + o(1))x) as x→+∞, where γ is a nondecreasing function on (−∞,+∞). Franko Drohobych State Pedagogic Institute, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1610–1616. December, 1997      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

The density of rational points on non-singular hypersurfaces

LetF(x) =F[x₁,…,x_n]∈ℤ[x₁,…,x_n] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ ⁿ ;F(x)=0, |x|⩽X}, where . It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X ^n−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process. Dedicated to the memory of Professor K G Ramanathan     

Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

For x = (x ₁, x ₂, …, x _n ) ∈ (0, 1 ] ⁿ and r ∈ { 1, 2, … , n}, a symmetric function F _n (x, r) is defined by the relation

On the Problem of Parameter Estimation in Exponential Sums

Let I≥1 be an integer, ω ₀=0<ω ₁<⋯<ω _I ≤π, and for j=0,…,I, a _j ∈ℂ, a_-j=[`(a_j)]a_{-j}={\overline{{a_{j}}}}, ω _−j =−ω _j , and a_j 1 0a_{j}\not=0 if j 1 0j\not=0. We consider the following problem: Given finitely many noisy samples of an exponential sum of the form

Limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival function

LetX ₁,X ₂, ...,X _n be a sequence of nonnegative independent random variables with a common continuous distribution functionF. LetY ₁,Y ₂, ...,Y _n be another sequence of nonnegative independent random variables with a common continuous distribution functionG, also independent of {X _i }. We can only observeZ _i =min(X _i ,Y _i ), and . LetH=1−(1−F)(1−G) be the distribution function ofZ. In this paper, the limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival functionS(t) are given. It is shown that (1)P(δ_(n)=1 i.o.)=0 ifF(τ _H ) < 1 andP(δ _n =0 i.o. )=0 ifG(τH) > 1 where δ_(n) is the corresponding indicator function of and have the same order a.s., where {T _n } is a sequence of constants such that 1−H(T _n )=n ^−α(logn)^β(log logn)^γ.     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

A topological position of the set of continuous maps in the set of upper semicontinuous maps

Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q \ Σ)), i.e., there is a homeomorphism h :↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q \ Σ...     

Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces

We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( A^a ) ?R( A^a ) )_q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as

On the growth of functions represented by Dirichlet series with complex coefficients on the real axis

We establish conditions under which, for a Dirichlet series F(z) = Σ _{n = 1} ^∞ d _n exp(λ _n z), the inequality ⋎F(x)⋎≤y(x),x≥x _o, implies the relation Σ _{n = 1} ^∞ |d _n exp(λ _n z)| ⪯ γ((1 + o(1))x) as x→+∞, where γ is a nondecreasing function on (−∞,+∞). Franko Drohobych State Pedagogic Institute, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1610–1616. December, 1997