On Mean Convergence of Lagrange Interpolation for General Arrays

For n1, let {x_jn}ⁿ_j=1 be n distinct points in a compact set K and letL_n[·] denote the corresponding Lagrange interpolation operator. Let v be a suitably restricted function on K. What conditions on the array {x_jn}_1jn, n1 ensure the existence of p>0 such that lim_n→∞ (f−L_n[f]) v_Lp(K)=0 for very continuous f: K→ ? We show that it is necessary and sufficient that there exists r>0 with sup_n1 π_nv_Lr(K) ∑ⁿ_j=1 (1/|π′_n| (x_jn))<∞. Here for n1, π_n is a polynomial of degree n having {x_jn}ⁿ_j=1 as zeros. The necessity of this condition is due to Ying Guang Shi.     

Le lemme de Schur pour les représentations orthogonales

Let σ be an orthogonal representation of a group G on a real Hilbert space. We show that σ is irreducible if and only if its commutant σ(G)' is isomorphic to , or . This result is an analogue of the classical Schur lemma for unitary representations. In both cases (orthogonal and unitary), a representation is irreducible if and only if its commutant is a field. If σ is irreducible, we show that there exists a unitary irreducible representation π of G such that the complexification σ is unitarily equivalent to π if σ(G)' , to π π̄ if σ(G)' , and to π π if σ(G)' (here π̄ denotes the contragredient representation of π). These results are classical for a finite-dimensional σ, but seem to be new in the general case.     

Enveloppes convexes des processus gaussiens

Let X={X(t), t[0,1]} be a process on [0,1] and V_X=Conv{(t,x)t[0,1], x=X(t)} be the convex hull of its path.The structure of the set ext(V_X) of extreme points of V_X is studied. For a Gaussian process X with stationary increments it is proved that:

• The set ext(V_X) is negligible if X is non-differentiable.

• If X is absolutely continuous process and its derivative X′ is continuous but non-differentiable, then ext(V_X) is also negligible and moreover it is a Cantor set.

It is proved also that these properties are stable under the transformations of the type Y(t)=f(X(t)), if f is a sufficiently smooth function.     

CONVEX CONCENTRATION INEQUALITIES FOR CONTINUOUS GAS AND STOCHASTIC DOMINATION

In this article, we consider the continuous gas in a bounded domain ∧ of R^＋ or R^d described by a Gibbsian probability measure μη∧ associated with a pair interaction φ, the inverse temperature β, the activity z 〉 0, and the boundary condition η. Define F ∫ωf（s）wA（ds）. Applying the generalized Ito＇s formula for forward-backward martingales （see Klein et M. [5]）, we obtain convex concentration inequalities for F with respect to the Gibbs measure μη∧. On the other hand, by FKG inequality on the Poisson space, we also give a new simple argument for the stochastic domination for the Gibbs measure.     

On the similarity of dual operators

Let k be a field with an involution σ and a non-degenerate sesquilinear form, where V,W are n-dimensional k-spaces. Assume that ΛEnd(V) and Λ^*End(W) are dual operators. We show that if Λ and Λ^* are similar, then Λ^*=Λ^-1, where :V→W is Hermitian.     

Regularity results for stable-like operators

For α[1,2) we consider operators of the form

and for α(0,1) we consider the same operator but where the f term is omitted. We prove, under appropriate conditions on A(x,h), that any solution u to will be in C^α+β if fC^β.     

Zero product preserving maps on

The main result of the paper characterizes continuous bilinear maps from C¹[0,1]×C¹[0,1] into a Banach space X with the property that (f,g)=0 whenever fg=0. This is applied to the study of zero product preserving operators on C¹[0,1], and operators on C¹[0,1] satisfying a version of the condition of the locality of an operator.     

Largest cliques in connected supermagic graphs

A graph G=(V,E) is said to be magic if there exists an integer labeling f:VE[1,|VE|] such that f(x)+f(y)+f(xy) is constant for all edges xyE.Enomoto, Masuda and Nakamigawa proved that there are magic graphs of order at most 3n²+o(n²) which contain a complete graph of order n. Bounds on Sidon sets show that the order of such a graph is at least n²+o(n²). We close the gap between those two bounds by showing that, for any given connected graph H of order n, there is a connected magic graph G of order n²+o(n²) containing H as an induced subgraph. Moreover G admits a supermagic labeling f, which satisfies the additional condition f(V)=[1,|V|].     

On constants in some inequalities for intermediate derivatives on a finite interval

Let L_q (1q<∞) be the space of functions f measurable on I=[−1,1] and integrable to the power q, with normL_∞ is the space of functions measurable on I with normWe denote by AC the set of all functions absolutely continuous on I. For nN, q[1,∞] we setW_n,q={f:f⁽ⁿ⁻¹⁾AC, f⁽ⁿ⁾L_q}.In this paper, we consider the problem of accuracy of constants A, B in the inequalities (1)|| f^(m)||_qA|| f||_p+B|| f^(m+k+1)||_r, mN, kW; p,q,r[1,∞], fW_m+k+1,r.     

NONTRIVIAL SOLUTIONS FOR SEMILINEAR SCHR(O)DINGER EQUATIONS

The authors prove the existence of nontrivial solutions for the SchrSdinger equation -△u ＋ V（x）u =λf（x, u） in R^N, where f is superlinear, subcritical and critical at infinity, respectively, V is periodic.     

Sharp estimates of bounded solutions to some semilinear second order dissipative equations

Let H, V be two real Hilbert spaces such that VH with continuous and dense imbedding, and let FC¹(V) be convex. By using differential inequalities, a close-to-optimal ultimate bound of the energy is obtained for solutions in to u^″+cu^′+bu+F(u)=f(t) whenever .     

The Approximation Functional and Interpolation with Perturbed Continuity

The continuity conditions at the endpoints of interpolation theorems, Ta_BjM_j a_Aj for j=0, 1, can be written with the help of the approximation functional: E(t, Ta; B₁, B₀)_L^∞M₀ a_A0 and E(t, Ta; B₀, B₁)_L^∞M₁ a_A1. As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the L^∞ norms can be replaced by BMO( ₊) norms. This leads to a strong version of the Stein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition fL⁰, i.e., f*L^∞, where f*(γ)=μ{|f|>γ} is the distribution function of f, can be replaced in interpolation with L(p, q) spaces by the weaker f*BMO( ₊).     

Asymptotic behavior of large solutions to -Laplacian of Bieberbach–Rademacher type

By the Karamata regular variation theory and the method of lower and upper solutions, we establish the asymptotic behavior of boundary blow-up solutions of the quasilinear elliptic equation div(|u|^p−2u)=b(x)f(u) in a bounded ΩR^N subject to the singular boundary condition u(x)=∞, where the weight b(x) is non-negative and non-trivial in Ω, which may be vanishing on the boundary or go to unbounded, the nonlinear term f is a Γ-varying function at infinity, whose variation at infinity is not regular.     

LOCAL TIME ANALYSIS OF ADDITIVE LEVY PROCESSES WITH DIFFERENT L(E)VY EXPONENTS

Let X1 XN be independent, classical Levy processes on R^d with Levy exponents ψ1,…, ψN, respectively. The corresponding additive Levy process is defined as the following N-parameter random field on R^d, X（t） △= X1（t1） ＋ ... ＋ XN（tN）, At∈N. Under mild regularity conditions on the ψi＇s, we derive estimate for the local and uniform moduli of continuity of local times of X = {X（t）; t ∈R^N}.     

Non-linear maps preserving solvability

Let M_n be the algebra of all n×n complex matrices and let L be the general linear Lie algebra gl(n,C) or the special linear Lie algebra sl(n,C). A bijective (not necessarily linear) map preserves solvability in both directions if both and ⁻¹ map every solvable Lie subalgebra of L into some solvable Lie subalgebra. If n3 then every such map is either a composition of a bijective lattice preserving map with a similarity transformation and a map [a_ij][f(a_ij)] induced by a field automorphism , or a map of this type composed with the transposition. We also describe the general form of such maps in the case when n=2. Using Lie's theorem we will reduce the proof of this statement to the problem of characterizing bijective maps on M_n preserving triangularizability of matrix pairs in both directions. As a byproduct we will characterize bijective maps on M_n that preserve inclusion for lattices of invariant subspaces in both directions.     

On Polynomials over Prime Fields Taking Only Two Values on the Multiplicative Group

Let p>2 be a prime, denote by F_p the field with F_p=p, and let F^*_p=F_p\{0}. We prove that if fεF_p[x] and f takes only two values on F^*_p, then (excluding some exceptional cases) the degree of f is at least (p−1).     

LOWER BOUNDS FOR SUP＋INF AND SUP＊INF AND AN EXTENSION OF CHEN-LIN RESULT IN DIMENSION 3

We give two results about Harnack type inequalities. First, on Riemannian surfaces, we have an estimate of type sup ＋ inf. The second result concern the solutions of prescribed scalar curvature equation on the unit ball of Rn with Dirichlet condition. Next, we give an inequality of type （supK ^u）^2s-1 × infπu ≤ c for positive solutions of △u = V u^5 on Ω belong toR^3, where K is a compact set of Ω and V is s-Holderian, s ∈] - 1/2, 1]. For the case s = 1/2 and Ω = S3, we prove that, if minΩ u 〉 m 〉 0 （for some particular constant m 〉 0）, and the H¨olderian constant A of V tends to 0 （in certain meaning）, we have the uniform boundedness of the supremum of the solutions of the previous equation on any compact set of Ω.     

Best constants of harmonic approximation on classes associated with the Laplace operator

We compute the best constants of approximation by entire functions of spherical type and by trigonometric polynomials of spherical degree on classes of functions f satisfying the condition Δ^kf_Lp1, where p=1 or 2 and Δ is the Laplace operator.     

Deformation of Minimal Polynomials and Approximation of Several Intervals by an Inverse Polynomial Mapping

In this paper we show that for a given set of l real disjoint intervals E_l=^l_j=1 [a_2j−1, a_2j] and given >0 there exists a real polynomial and a set of l disjoint intervals E_l=^l_j=1 [ã_2j−1, ã_2j] with E_lE_l and (ã₁, …, ã_2l)−(a₁, …, a_2l)_max<, such that ⁻¹([−1, 1])=E_l. The statement follows by showing how to get in a constructive way by a continuous deformation procedure from a minimal polynomial on E_l with respect to the maximum norm a polynomial mapping of E_l.     

Double piling structure of matrix monotone functions and of matrix convex functions

There are basic equivalent assertions known for operator monotone functions and operator convex functions in two papers by Hansen and Pedersen. In this note we consider their results as correlation problem between two sequences of matrix n-monotone functions and matrix n-convex functions, and we focus the following three assertions at each label n among them:

(i) f(0)0 and f is n-convex in [0,α),

(ii) For each matrix a with its spectrum in [0,α) and a contraction c in the matrix algebra M_n,

f(cac)cf(a)c,