A limit theorem for random coverings of a circle

LetN _{α, m} equal the number of randomly placed arcs of length α (0<α<1) required to cover a circleC of unit circumferencem times. We prove that lim_α→0 P(N_α,m≦(1/α) (log (1/α)+mlog log(1/α)+x)=exp ((−1/(m−1)!) exp (−x)). Using this result for m=1, we obtain another derivation of Steutel's resultE(N_α,1)=(1/α) (log(1/α)+log log(1/α)+γ+o(1)) as α→0, γ denoting Euler's constant.     

Convergence of weighted sums of independent random variables

Let {X_k} be a sequence of i.i.d. random variables with d.f. F(x). In the first part of the paper the weak convergence of the d.f.'s F_n(x) of sums is studied, where 0<α≤2, a_nk>0, 1≤k≤m_n, and, as n→∞, bothmax _1≤k≤mn_{a nk}→0 and . It is shown that such convergence, with suitably chosen A_n's and necessarily stable limit laws, holds for all such arrays {α_nk} provided it holds for the special case α_nk=1/n, 1≤k≤n. Necessary and sufficient conditions for such convergence are classical. Conditions are given for the convergence of the moments of the sequence {F_n(x)}, as well as for its convergence in mean. The second part of the paper deals with the almost sure convergence of sums , where a_n≠0, b_n>0, andmax _1≤k≤n a_k/b_n→0. The strong law is said to hold if there are constants A_n for which S_n→0 almost surely. Let N(0)=0 and N(x) equal the number of n≥1 for which b_n/|a_n|<x if x>0. The main result is as follows. If the strong law holds,EN (|X₁|)<∞. If for some 0<p≤2, then the strong law holds with if 1≤p≤2 and A_n=0 if 0<p<1. This extends the results of Heyde and of Jamison, Orey, and Pruitt. The strong law is shown to hold under various conditions imposed on F(x), the coefficients a_n and b_n, and the function N(x). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

On degrees of maps between Grassmannians

Let G_n,k denote the oriented grassmann manifold of orientedk-planes in ℝⁿ. It is shown that for any continuous mapf: G_n,k → G_n,k, dim G_n,k = dim G_m,l = l(m −l), the Brouwer’s degree is zero, providedl > 1,n ≠ m. Similar results for continuous mapsg: ℂG_m,l → ℂG_n,k,h: ℍG_m,l → ℍG_n,k, 1 ≤ l < k ≤ n/2, k(n — k) = l(m — l) are also obtained.     

On the maximal operator of (<Emphasis Type="Italic">C</Emphasis>, α)-means of Walsh–Kaczmarz–Fourier series

Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator σ^α,κ,* of the (C, α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space H _p to the space L _p for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved that this boundedness result does not hold if p ≤ 1 / (1 + α). However, in the endpoint case p = 1 / (1 + α ), the maximal operator σ^α,κ,* is bounded from the martingale Hardy space H _1/(1+α) to the space weak- L _1/(1+α). The main aim of this paper is to prove a stronger result, namely, that, for any 0 < p ≤ 1 / (1 + α), there exists a martingale f ∈ H _p such that the maximal operator σ^α,κ,* f does not belong to the space L _p .     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt Foundation.     

Optimal interpolation in spaces of Lorentz-Zygmund type

The Lorentz-Zygmund spaces, introduced by C. Bennett and K. Rudnick in [BR], are generalized by taking the exterior norm in arbitrary rearrangement invariant spaceE instead of onlyL _r-spaces. On the spacesL _p,α,E thus obtained, we study all operatorsT of two weak types (a, b) and (p, q) with 1≤a<p≤∞, 1≤b<q≤∞, and prove thatT:L _p,α,E→L _q,α−1,E. Moreover, for any set of parametersp, q, α, E, we construct the smallest possible spaceB _q,α,E such thatT:L _p,α,E→B _q,α,E and the largest possible spaceA _p,α,E such thatT:A _p,α,E→L _q,α−1,E. For spaces of all three types, we find their fundamental functions and Boyd indices, state various embeddings, equivalences and other properties. The research was supported by the Center of Scientific Absorption of the Ministry of Absorption of the State of Israel.     

Exchangeable Gibbs partitions and Stirling triangles

For two collections of nonnegative and suitably normalized weights W = (W_j) and V = (V_n,k), a probability distribution on the set of partitions of the set {1, …, n} is defined by assigning to a generic partition {A_j, j ≤ k} the probability V_n,k , where |A_j| is the number of elements of A_j. We impose constraints on the weights by assuming that the resulting random partitions Π _n of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights W must be of a very special form depending on a single parameter α ∈ [− ∞, 1]. The case α = 1 is trivial, and for each value of α ≠ = 1 the set of possible V-weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalized Stirling triangle. In particular, we show that the boundary is discrete for − ∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by (α,θ), while for 0 < α < 1 the extremes are obtained by conditioning an (α,θ)-partition on the asymptotics of the number of blocks of Π_n as n tends to infinity. Bibliography: 29 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 83–102.     

Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems

We present existence principles for the nonlocal boundary-value problem (φ(u^(p−1)))′=g(t,u,...,u^(p−1), α_k(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α _k: C ^p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)ⁿ(φ(u⁽²ⁿ⁻⁾))′=f(t,u,...,u⁽²ⁿ⁻¹⁾), u^(2k)(0)=0, α_ku^(2k)(T)+b_ku^(2k=1)(T)=0, 0≤k≤n−1, is given. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008.     

L p − Lq estimates for convolution operators with n-Dimensional singular measures

Let S ⊂ ℜⁿ⁺¹ be the graph of the function ϕ :[−1, 1] ⁿ → ℜ defined by ϕ (x ₁ , …, x_n) = ∑ _j=1 ⁿ |x_j|^αj, with1<α ₁ ≤ … ≤ α_n, let σ the Euclidean area measure on S. In this article we study the L^p − L^q boundedness of convolution operators with the singular Borel measure on Rⁿ⁺¹ given by μ (E)=σ (E ∩ S)     

On the law of the iterated logarithm for subsequences for a stable subordinator

Let {X(t): 0≤t<∞) be a stable subordinator with α∈(0,1). For increasing sequences t_k we give normalizing constants a_k such thatliminf _k→∞ a _k ⁻¹ X(t_k) is a.s constant. We also derive a.s. upper bounds. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models Part I, Eger, Hungary, 1994.     

Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

This paper studies the convergence of the sequence defined by x0∈C,xn 1=αnu (1-αn)Txn,n=0,1,2,…, where 0 ≤αn ≤ 1, limn→∞αn = 0, ∑∞n=0 αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.     

On the moments of elements of continued fractions for some rational numbers

Let p be a prime and let 1 ≤ a ≤ p − 1. In the paper, an asymptotics for the sum over a of the moments of order α (0 < α < 1) of the sequence of elements of the expansion of a/p into a continued fraction is obtained. As a corollary, an upper bound for the number of those a whose expansions contain at least one element larger than log ^λ p (λ > 1) is derived. Note that in the case considered, the set of elements has no limiting distribution as p → ∞, which is in contrast with the case of rational fractions b/c, where (b, c) = 1 and b² + c² ≤ R² (R → ∞). Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 13–22.     

非Lipschitz条件下严格伪压缩映射不动点的逼近

Abstract. Without the Lipschitz assumption and boundedness of K in arbitrary Banach spaces,the Ishikawa iteration     

Global asymptotics for Laguerre polynomials with large negative parameter—a Riemann-Hilbert approach

In this paper, we study the asymptotic behavior of the Laguerre polynomials as n→∞. Here α _n is a sequence of negative numbers and −α _n /n tends to a limit A>1 as n→∞. An asymptotic expansion is obtained, which is uniformly valid in the upper half plane ℂ₊={z:Im z≥0}. A corresponding expansion is also given for the lower half plane ℂ₋={z:Im z≤0}. The two expansions hold, in particular, in regions containing the curve Γ in the complex plane, on which these polynomials are orthogonal. Our method is based on the Riemann-Hilbert approach introduced by Deift and Zhou. The work of R. Wong is partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 102504).     

Non-commutative Clarkson Inequalities for n-Tuples of Operators

Let A ₀, ... , A _n−1 be operators on a separable complex Hilbert space , and let α₀,..., α _n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,

Singular solutions of some nonlinear parabolic equations

We consider the existence and uniqueness of singular solutions for equations of the formu ₁=div(|Du|^p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2. Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫_|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result. In the case ϕ(u)=u ^q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal. Dedicated to Professor Shmuel Agmon     

Weighted Integrals and Bloch Spaces of n-Harmonic Functions on the Polydisc

We study anisotropic mixed norm spaces h(p,q,α) consisting of n-harmonic functions on the unit polydisc of by means of fractional integro-differentiation including small 0 < p < 1 and multi-indices α = (α ₁,...,α _n ) with non-positive α _j  ≤ 0. As an application, two different Bloch spaces of n-harmonic functions are characterized.      

On the Asymptotics of the Meixner—Pollaczek Polynomials and Their Zeros

An infinite asymptotic expansion is derived for the Meixner—Pollaczek polynomials M _n (nα;δ, η) as n→∞ , which holds uniformly for -M≤α≤ M , where M can be any positive number. This expansion involves the parabolic cylinder function and its derivative. If α _{n, s} denotes the s th zero of M _n (nα;δ, η) , counted from the right, and if α˜ _n,s denotes its s th zero counted from the left, then for each fixed s , three-term asymptotic approximations are obtained for both α _n,s and α˜ _n,s as n→∞ . December 28, 1998. Date revised: June 4, 1999. Date accepted: September 6, 1999.     

Commutators of integral operators with variable kernels on Hardy spaces

LetT _Ω,α (0 ≤ α< n) be the singular and fractional integrals with variable kernel Ω(x, z), and [b, T_Ω,α] be the commutator generated by T_Ω,α and a Lipschitz functionb. In this paper, the authors study the boundedness of [b, T_Ω,α] on the Hardy spaces, under some assumptions such as theL ^r -Dini condition. Similar results and the weak type estimates at the end-point cases are also given for the homogeneous convolution operators . The smoothness conditions imposed on are weaker than the corresponding known results.