Convex polynomial and spline approximation inL p,O<p<∞

We prove that a convex functionf ∈ L _p[−1, 1], 0<p<∞, can be approximated by convex polynomials with an error not exceeding Cω ₃ ^ϕ (f,1/n)_p where ω ₃ ^ϕ (f,·) is the Ditzian-Totik modulus of smoothness of order three off. We are thus filling the gap between previously known estimates involving ω ₃ ^ϕ (f,1/n)_p, and the impossibility of having such estimates involving ω₄. We also give similar estimates for the approximation off by convexC ⁰ andC ¹ piecewise quadratics as well as convexC ² piecewise cubic polynomials. Communicated by Dietrich Braess     

On Lp-convergence of Grunwald interpolation

In this paper, the L_p-convergence of Grünwald interpolation G_n(f,x) based on the zeros of Jacobi polynomials J _n ^(α,β) (x)(−1<α,β<1) is considered. L_p-convergence (0<p<2) of Grünwald interpolation G_n(f,x) is proved for p·Max(α,β)<1. Moreover, L_p-convergence (p>0) of G_n(f,x) is obtained for −1<α,β≤0. Therefore, the results of [1] and [3–5] are improved.     

带振荡因子的粗双曲奇异积分算子

The singular integral operator J Ω,α, and the Marcinkiewicz integral operator (~μ)Ω,α are studied. The kernels of the operators behave like |y|-n-α(α＞0) near the origin, and contain an oscillating factor ei|y|-β(β＞0) and a distribution Ω on the unit sphere Sn-1 It is proved that, if Ω is in the Hardy space Hr (Sn-1) with 0＜r= (n-1)/(n-1 )(＞0), and satisfies certain cancellation condition,then J Ω,α and uΩ,α extend the bounded operator from Sobolev space Lpγ to Lebesgue space Lp for some p. The result improves and extends some known results.     

Two-dimensional incompressible viscous flow around a small obstacle

In this work we study the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity ω₀ and the circulation γ of the initial flow around the obstacle. We prove that, if γ is sufficiently small, the limit flow satisfies the full-plane Navier–Stokes system, with initial vorticity ω₀ + γδ, where δ is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors in Iftimie et al. (Comm. Part. Differ. Eqn. 28, 349–379 (2003)), where the effect of the small obstacle appears in the coefficients of the PDE and not only in the initial data. The main ingredients of the proof are L ^p − L ^q estimates for the Stokes operator in an exterior domain, a priori estimates inspired on Kato’s fixed point method, energy estimates, renormalization and interpolation.     

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.     

On a Banach space approximable by Jacobi polynomials

Let X represent either the space C[-1,1] L _p ^(α,β) (w), 1 ≦ p < ∞ on [-1, 1]. Then Xare Banach spaces under the sup or the p norms, respectively. We prove that there exists a normalized Banach subspace X ¹ _αβ of Xsuch that every f ∈ X ¹ _αβ can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Our method to prove such an approximation problem is Fourier–Jacobi analysis based on the convergence of Fourier–Jacobi expansions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Free discontinuity problems with unbounded data: The two dimensional case

We prove the existence of a minimizing pair for the functionalG defined for every closed setK ⊂R ² and for every functionu ∈C ¹(ω/K) by where ω is an open set inR ², λ, μ>0,q≥1,g ∈L ^q (ω) ∩L ^p (ω) withp>2q andH ¹ is the 1-dimensional Hausdorff measure.     

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 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.     

Representation ofL p-norms and isometric embedding inL p-spaces

For fixed 1≦p<∞ theL _p-semi-norms onR ⁿ are identified with positive linear functionals on the closed linear subspace ofC(R ⁿ ) spanned by the functions |<ξ, ·>| ^p , ξ∈R ⁿ . For every positive linear functional σ, on that space, the function Φ_σ:R ⁿ →R given by Φ_σ is anL _p-semi-norm and the mapping σ→Φ_σ is 1-1 and onto. The closed linear span of |<ξ, ·>| ^p , ξ∈R ⁿ is the space of all even continuous functions that are homogeneous of degreep, ifp is not an even integer and is the space of all homogeneous polynomials of degreep whenp is an even integer. This representation is used to prove that there is no finite list of norm inequalities that characterizes linear isometric embeddability, in anyL _p unlessp=2. Supported by the National Science Foundation MCS-79-06634 at U.C. Berkeley.     

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.     

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.     

Existence of Solutions to a Singular Elliptic Equation

We study the equation ${{-{\Delta}u = (-\frac{1}{u^{\beta}}+\lambda{u}^{p})\chi\{u >0 }\}}${{-{\Delta}u = (-\frac{1}{u^{\beta}}+\lambda{u}^{p})\chi\{u >0 }\}} in Ω with Dirichlet boundary condition, where 0 < p < 1 and 0 < β < 1. We regularize the term 1/u ^β near u ~ 0 by using a function g _ε (u) which pointwisely tends to 1/u ^β as ε → 0. When the parameter λ > 0 is large enough, the corresponding energy functional has critical points u _ε . Letting ε → 0, then u _ε converges to a solution of the original problem, which is nontrivial, nonnegative and vanishes at some portion of Ω. There are two nontrivial solutions.     

Asymptotic behavior of small deviations for Bogoliubov’s Gaussian measure in the L p norm, 2 ≤ p ≤ ∞

We prove several results on exact asymptotic formulas for small deviations in the L^p-norm with 2 ～ p ～ ∞ for Bogoliubov’s stationary Gaussian process ξ(t). We prove the property of mutual absolute continuity for the conditional Bogoliubov measure and the conditional Wiener measure and calculate the Radon-Nikodym derivative.     

On an inclusion between operator ideals

Let 1 ⩽ q < p < ∞ and 1/r:= 1/p max(q/2, 1). We prove that L _r,p ^(c), the ideal of operators of Gel’fand type l _r,p , is contained in the ideal Π _p,q of (p, q)-absolutely summing operators. For q > 2 this generalizes a result of G. Bennett given for operators on a Hilbert space.     

Subspaces ofL p (G) spanned by characters: 0<p<1

LetG be an infinite compact abelian group,μ a Borel measure onG with spectrumE, and 0<p<1. We show that ifμ is not absolutely continuous with respect to Haar measure, thenL _E ^P (G), the closure inL ^p (G) of theE-trigonometric polynomials, does not have enough continuous linear functionals to separate points. Ifμ is actually singular, thenL _E ^p (G) does not have any nontrivial continuous linear functionals at all. Our methods recover the classical F. and M. Riesz theorem, and a related several variable result of Bochner; they reveal the existence of small sets of characters that spanL ^P (T), where T is the unit circle; and they show that theH ^p spaces of the “big disc algebra” have one-dimensional dual.     

Perturbation theory series in quantum mechanics: Phase transition and exact asymptotic forms for the expansion coefficients

We consider the model of a harmonic oscillator with a power-law potential and derive new asymptotic formulas for the coefficients of the perturbation theory series in powers of the coupling constant in the case of a power-law perturbing potential |x|^p, p > 0. We prove the existence of a critical value p ₀ and compute it. It is a threshold in the sense that the asymptotic forms of the studied coefficients for 0 < p < p ₀ and for p > p ₀ differ qualitatively. We note that the considered physical system undergoes a phase transition at p = p ₀ . The analysis uses the Laplace method for functional integrals with Gaussian measures.     

A new inductive approach to the lace expansion for self-avoiding walks

Summary. We introduce a new inductive approach to the lace expansion, and apply it to prove Gaussian behaviour for the weakly self-avoiding walk on ℤ ^d where loops of length m are penalised by a factor e ^{−β/m p} (0<β≪1) when: (1) d>4, p≥0; (2) d≤4, . In particular, we derive results first obtained by Brydges and Spencer (and revisited by other authors) for the case d>4, p=0. In addition, we prove a local central limit theorem, with the exception of the case d>4, p=0. Received: 29 October 1997 / In revised form: 15 January 1998     

Weak type estimates for some maximal operators on the weighted Hardy spaces

Weighted weak type estimates are proved for some maximal operators on the weighted Hardy spacesH _ω ^p (0 <p < 1, ω ∈A ₁) (0<p<1, ω∞A₁); in particular, weighted weak type endpoint estimates are obtained for the maximal operators arising from the Bochner-Riesz means and the spherical means onH _ω ^p .     

Strong Arens irregularity of Beurling algebras with a locally convex topology

Let G be a locally compact group with a weight function ω. Recently, we have shown that the Banach space L₀^∞ (G,1/ω) can be identified with the strong dual of L¹(G, ω)equipped with some locally convex topologies τ. Here we use this duality to introduce an Arens multiplication on (L¹(G, ω), τ)**, and prove that the topological center of (L¹(G, ω), τ)** is (L¹(G, ω); this enables us to conclude that (L¹(G, ω), τ) is Arens regular if and only if G is discrete. We also give a characterization for Arens regularity of L₀^∞ (G, 1/ω)¹. Received: 8 March 2005