THE ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION IN MULTIPLE LINEAR REGRESSION MODEL

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Closed convex hull of the family of multivalently close-to-convex functions of order β

In this paper the closed convex hulls of the compact familiesC _β(p), of multivalently close to convex functions of order β andV ₀ ^k (p), of multivalent functions of bounded boundary rotation, have been determined, respectively for β≥1 andk≥2(p+1)/p. Extreme points of these convex hulls are partially characterised. For a fixed pointz ₀∈D={z:|z|<1}, a new familyC _β(p, z₀) is defined through Montel normalisation and its closed convex hull is also foud. Sharp coefficient estimates for functions subordinate to or majorised by some function inC _β(p) orC' _β(p) are discussed for β>0. It is shown that iff is subordinate to some function inC _β(p) then each Taylor coefficient off is dominated by the corresponding coefficient of the function .     

Monotone perturbations of the laplacian inL 1(R N )

The semilinear perturbation of Poisson’s equation (E): −Δu+β(u)∋f, where β is a maximal monotone graph inR, has been investigated by Ph. Bénilan, H. Brézis and M. Crandall forf∈L ¹(R ^N ),N≧1, under the assumptions 0∈β(0) ifN≧3 and 0∈β(0) ∩ Int β(R) ifN=1,2. We discuss in this paper the solvability and well-posedness of (E) in terms of any maximal monotone graph β. In particular, if β takes only positive values andN≧3 we prove that no solution exists; ifN=2 we give necessary and sufficient conditions on β andf for (E) to be solvable in a natural sense.     

Rates of uniform convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions

Assume that {Xn} is a strictly stationary β-mixing random sequence with the β-mixing coefficient βk = O(k-r), 0 < r ≤1. Yu (1994) obtained convergence rates of empirical processes of strictly stationary β-mixing random sequence indexed by bounded classes of functions. Here, a new truncation method is proposed and used to study the convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions. The research results show that if the envelope of the index class of functions is in Lp, p > 2 or p > 4, uniform convergence rates of empirical processes of strictly stationary β-mixing random sequence over the index classes can reach O((nr/(l+r)/logn)-1/2) or O((nr/(1+r)/ log n)-3/4) and that the Central Limit Theorem does not always hold for the empirical processes.``     

Characterizations of BMO associated with Gauss measures via commutators of local fractional integrals

Let dγ(x) ≡ π ^−n/2 e ^{−|x| 2} dx for all x ∈ ℝ ⁿ be the Gauss measure on ℝ ⁿ . In this paper, the authors establish the characterizations of the space BMO(γ) of Mauceri and Meda via commutators of either local fractional integral operators or local fractional maximal operators. To this end, the authors first prove that such a local fractional integral operator of order β is bounded from L ^p (γ) to L ^p/(1−pβ)(γ), or from the Hardy space H ¹(γ) of Mauceri and Meda to L ^1/(1−β)(γ) or from L ^1/β (γ) to BMO(γ), where β ∈ (0, 1) and p ∈ (1, 1/β).     

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.     

A nucleation-and-growth model

Summary. We consider the following simple nucleation-and-growth model. On the lattice ℤ ^d , starting with all sites unoccupied, a site becomes occupied at rate e ^−ℬΓ if it has no occupied neighbors, at rate ɛ= e ^−βγ if it has 1 occupied neighbor, and at rate 1 if it has 2 or more occupied neighbors. Occupied sites remain occupied forever. The parameters Γ≧γ are fixed, and we are interested in the behavior of the system as β→∞. We show that the relaxation time of this system scales as e ^βκc , where κ _c = max {γ,( Γ + γ)/(d+1)}. Received: 20 February 1996 / In revised form: 15 June 1996     

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.     

Global existence and uniform decay for the coupled Klein-Gordon-Schr?dinger equations

This paper is concerned to the existence, uniqueness and uniform decay for the solutions of the coupled Klein-Gordon-Schr?dinger damped equations where ω is a bounded domain of R ⁿ , n≤ 3, F : R ²→R is a C ¹-function; γ, β; θ are constants such that γ, β > 0 and 1 ≤ 2θ≤ 2. Received January 1999 – Accepted October 1999     

On the intersection of infinite geometric and arithmetic progressions

We prove that the intersection G ∩ A of an infinite geometric progression G = u, uq, uq ², uq ³, ..., where u > 0 and q > 1 are real numbers, and an infinite arithmetic progression A contains at most 3 elements except for two kinds of ratios q. The first exception occurs for q = r ^1/d , where r > 1 is a rational number and d ∈ ℕ. Then this intersection can be of any cardinality s ∈ ℕ or infinite. The other (possible) exception may occur for q = β ^1/d , where β > 1 is a real cubic algebraic number with two nonreal conjugates of moduli distinct from β and d ∈ ℕ. In this (cubic) case, we prove that the intersection G ∩ A contains at most 6 elements.     

On weighted approximation by Bernstein-Durrmeyer operators

In this paper, we consider weighted approximation by Bernstein-Durrmeyer operators in L_p[0, 1] (1≤p≤∞), where the weight function w(x)=x^α(1−x)^β,−1/p<α, β<1-1/p. We obtain the direct and converse theorems. As an important tool we use appropriate K-functionals. Supported by Zhejiang Provincial Science Foundation.     

Lattice-universal Orlicz spaces on probability spaces

Lattice-universal Orlicz function spacesL ^F _α,β[0, 1] with prefixed Boyd indices are constructed. Namely, given 0<α<β<∞ arbitrary there exists Orlicz function spacesL ^F _α,β[0, 1] with indices α and β such that every Orlicz function spaceL ^G [0, 1] with indices between α and β is lattice-isomorphic to a sublattice ofL ^F _α,β[0, 1]. The existence of classes of universal Orlicz spacesl ^Fα,β(I) with uncountable symmetric basis and prefixed indices α and β is also proved in the uncountable discrete case. Partially supported by BFM2001-1284.     

Mollification formulas and implicit smoothing

This paper develops some mollification formulas involving convolutions between popular radial basis function (RBF) basic functions Φ, and suitable mollifiers. Polyharmonic splines, scaled Bessel kernels (Matern functions) and compactly supported basic functions are considered. A typical result is that in ℛ^d the convolution of |{•}|^β and (•²+c ²)^−(β+2d)/2 is the generalized multiquadric (•²+c ²)^β/2 up to a multiplicative constant. The constant depends on c>0, β, where ℜ(β)>−d, and d. An application which motivated the development of the formulas is a technique called implicit smoothing. This computationally efficient technique smooths a previously obtained RBF fit by replacing the basic function Φ with a smoother version Ψ during evaluation.     

On lower bounds for the approximation of individual functions by local splines with nonfixed nodes

For functions with the integrable βth power, where β = (r + 1 + 1/p)⁻¹, we obtain asymptotically exact lower bounds for the approximation by local splines of degreer and defectk ≥r/2 in the metric ofL _p. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 12, pp. 1628–1637, December, 1999.     

Diffusive scaling of the spectral gap for the dilute Ising lattice-gas dynamics below the percolation threshold

We consider a conservative stochastic lattice-gas dynamics reversible with respect to the canonical Gibbs measure of the bond dilute Ising model on ℤ ^d at inverse temperature β. When the bond dilution density p is below the percolation threshold we prove that for any particle density and any β, with probability one, the spectral gap of the generator of the dyamics in a box of side L centered at the origin scales like L ⁻². Such an estimate is then used to prove a decay to equilibrium for local functions of the form where ε is positive and arbitrarily small and α = ? for d = 1, α=1 for d≥2. In particular our result shows that, contrary to what happes for the Glauber dynamics, there is no dynamical phase transition when β crosses the critical value β _c of the pure system. Received: 10 April 2000 / Revised version: 23 October 2000 / Published online: 5 June 2001     

On double sine and cosine transforms,Lipschitz and Zygmund classes

We consider complex-valued functions f ∈ L 1 (R+2),where R +:= [0,∞),and prove sufficient conditions under which the double sine Fourier transform f ss and the double cosine Fourier transform f cc belong to one of the two-dimensional Lipschitz classes Lip(α,β) for some 0 α,β≤ 1;or to one of the Zygmund classes Zyg(α,β) for some 0 α,β≤ 2.These sufficient conditions are best possible in the sense that they are also necessary for nonnegative-valued functions f ∈ L 1 (R+2).     

Necessary and sufficient conditions for boundedness of commutators of strongly singular integral operators with weighted Lipschitz functions

In this paper, we prove the commutator T _b generated by the strongly singular integral operator T and the function b is bounded from L ^p (w) to L ^q (w ^1−q ) if and only if b ∈ Lip _β (w), where w ∈ A ₁, 0 < β < 1, 1 < p < n/β and 1/q = 1/p − β/n. To do this, we first show a maximal function estimate for the commutator.     

The maximal (C, α, β) operator of two-parameter walsh-fourier series

The two-parameter dyadic martingale Hardy spacesH _p are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from H_p to L_p (1/(α+1), 1/(β+1)<p<∞) and is of weak type (H ₁ ^# , L₁), where the Hardy space H ₁ ^# is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f∈H ₁ ^# converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on H_p whenever 1/(α+1), 1/(β+1)<p<∞. Thus in case f∈H_p, the (C, α, β) means converge to f in H_p norm. The same results are proved for the conjugate (C, α, β) means, too.     

Universality Limits for Exponential Weights

We establish universality in the bulk for fixed exponential weights on the whole real line. Our methods involve first-order asymptotics for orthogonal polynomials and localization techniques. In particular, we allow exponential weights such as | x | ^2β g ²(x)exp (−2Q(x)), where β>−1/2, Q is convex and Q ^′′ satisfies some regularity conditions, while g is positive, and has a uniformly continuous and slowly growing or decaying logarithm.      

Asymptotically most powerful rank tests for regression parameters in manova

Summary Srivastava [5] proposed a class of rank score tests for testing the hypothesis that β₁=⋯β _p =0 in the linear regression modely _i =β₁ x _1i +β₂ x _2i +⋯+β _p +x _pi +ɛ _i under weaker conditions than Hájek [2]. In this paper, under the same weak conditions, a class of rank score tests is proposed for testing β₁=⋯β _q =0 in the multivariate linear regression modely _i =β₁ x _1i +β₂ x _2i +⋯+β _p +x _pi +ɛ _i ,q≦p, where β _i ’s arek-vectors. The limiting distribution of the test statistic is shown to be central χ _qk ² underH and non-central χ _qk ² under a sequence of alternatives tending to the hypothesis at a suitable rate. Research supported by Canada Council and National Research Council of Canada.