The natural operators lifting connections to higher order cotangent bundles

We prove that the problem of finding all Mf _m -natural operators C: Q ? QT ^r * lifting classical linear connections ? on m-manifolds M into classical linear connections C _M (?) on the r-th order cotangent bundle T ^r *M = J ^r (M, ?)₀ of M can be reduced to the well known one of describing all M f _m -natural operators D: Q ? ? ^p T ? ? ^q T* sending classical linear connections ? on m-manifolds M into tensor fields D _M (?) of type (p, q) on M.     

Some new results on weak compactness

We consider the scattering problem in Lsu2$\text{Rsu3}$ for Schrödinger operators with momentum-dependent interactions, i.e., for the pair of Hamiltonians H₀ = ? Δ and H = H₀ + T, where T is a pseudodifferential operator. The existence of the wave operators is proved by estimating the integrals appearing in the CookHack convergence criterion, and their asymptotic completeness is established under more restrictive conditions on the symbol of T by using a local trace criterion.     

Means and Følner Condition on Polynomial Hypergroups

We investigate summing sequences in the context of polynomial hypergroups. It will be shown that the summing sequence ${(S_{n})_{n\in\mathbb{N}_{0}}}We investigate summing sequences in the context of polynomial hypergroups. It will be shown that the summing sequence (S_n)_{n ? \mathbbN0}{(S_{n})_{n\in\mathbb{N}_{0}}}, where S _n = {0, 1, . . . , n}, is equivalent to the growth condition (H) of the Haar weights h(n). This condition is also sufficient for the existence of means which satisfy a strong form of translation invariance. Furthermore we give exact representations of the unique translation invariant mean on the space of weakly almost periodic sequences for a large class of polynomial hypergroups.     

Krengel-Lin decomposition for probability measures on hypergroups

A Markov operator P on a σ-finite measure space (X,Σ,m) with invariant measure m is said to have Krengel-Lin decomposition if L²(X)=E₀⊕L²(X,Σ_d) where E₀={f∈L²(X)∣‖Pⁿ(f)‖→0} and Σ_d is the deterministic σ-field of P. We consider convolution operators and we show that a measure λ on a hypergroup has Krengel-Lin decomposition if and only if the sequence converges to an idempotent or λ is scattered. We verify this condition for probabilities on Tortrat groups, on commutative hypergroups and on central hypergroups. We give a counter-example to show that the decomposition is not true for measures on discrete hypergroups.     

Average Error Bounds of Best Approximation of Continuous Functions on the Wiener Space

In this paper, we study the approximation of the identity operator and the integral operator T_m by Jackson operators, discrete Jackson operators, and spline operators, respectively, on the Wiener space and obtain average error estimation.     

A moment theorem for completely hyperexpansive operators

Given a family $ \{ A_m^x \} _{\mathop {m \in \mathbb{Z}_ + ^d }\limits_{x \in X} } $ (X is a non-empty set) of bounded linear operators between the complex inner product space $ \mathcal{D} $ and the complex Hilbert space ? we characterize the existence of completely hyperexpansive d-tuples T = (T ₁, … , T _d ) on ? such that A _m ^x = T ^m A ₀ ^x for all m ? ? ₊ ^d and x ? X.     

Blowing up of a finite difference solution tou t = uxx + u2

This paper studies a finite difference approximation to the similinear heat equation (1) with special emphasis on the case when the exact solution blows up with the blowing-up timeT _∞. The key results will be given in Propositions 1 and 2. Proposition 1 states the local convergence, i.e., the convergence of the proposed finite difference solution to the exact solution in any fixed time interval 0 ?t ? T, whereT < T _∞. Proposition 2 states the convergence of the numerical blowing-up time to the exact oneT _∞.     

A Generalization of the Heine�CStieltjes Theorem

The Heine?CStieltjes theorem describes the polynomial solutions, (v,f) such that T(f)=vf, to specific second-order differential operators, T, with polynomial coefficients. We extend the theorem to concern all (nondegenerate) differential operators preserving the property of having only real zeros, thus solving a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question of how to generalize their results to higher degrees. Many of the results are new even for the classical case.     

Boundary conditions for quasi-differential operators in L p spaces

For a very general class of ordinary quasi-differential expressions M with matrix-valued coefficients and for p, q ∈ [1, ∞) or p, q ∈ (1, ∞] all operators T of a subspace of LP into Lq which satisfy T _M ⁰ ? T ? Tm and are Fredholm operators with index zero are characterized by suitable boundary conditions, where Tm and T _M ⁰ are the maximal and minimal operators associated to M. This generalizes a result of Evans and Ibrahim for p = q = 2.     

Wavelet shrinkage estimators of Calderón-Zygmund operators with odd kernels

Wavelet shrinkage is a strategy to obtain a nonlinear approximation to a given function f and is widely used in data compression,signal processing and statistics,etc.For Calder′on-Zygmund operators T,it is interesting to construct estimator of T f,based on wavelet shrinkage estimator of f.With the help of a representation of operators on wavelets,due to Beylkin et al.,an estimator of T f is presented in this paper.The almost everywhere convergence and norm convergence of the proposed estimators are established.     

Some properties of bivariate Bernstein-Schoenberg operators

The bivariate Bernstein-Schoenberg operatorV _T of degreem, introduced in [5], is a spline approximation operator that generalizes the Bernstein polynomial operatorB _m . It is shown here that for a convex functionf,f≤V _T (f)≤B _m (f). This result is then used to show that for a twice differentiable functiong, the asymptotic error limm(V _T (g)-g) depends only on the asymptotic error for quadratic polynomials. The latter is evaluated explicitly in the special circumstances thatV _T is, in a sense, asymptotically close toB _m .     

Weak infinitesimal Hilbert’s 16th problem

The following weak infinitestimal Hilbert’s 16th problem is solved. Given a real polynomial H in two variables, denote by M(H, m) the maximal number possessing the following property: for any generic set {γ _i } of at most M(H,m) compact connected components of the level lines H = c _i of the polynomial H, there exists a form θ = P dx + Q dy with polynomials P and Q of degrees no greater than m such that the integral ∫ _H=c θ has nonmultiple zeros on the connected components {γ _i }. An upper bound for the number M(H,m) in terms of the degree n of the polynomial H is found; this estimate is sharp for almost every polynomial H of degree n. A multidimensional version of this result is proved. The relation between the weak infinitesimal Hilbert’s 16th problem and the following question is discussed: How many limit cycles can a polynomial vector field of degree n have if it is close to a Hamiltonian vector field?     

Asymptotics of the “minimumL 1-norm” estimates in nonparametric regression models

Consider the nonparametric regression modelY=g ₀(T)+u, whereY is real-valued,u is a random error,T ranges over a nondegenerate compact interval, say [0,1], andg ₀(·) is an unknown regression function, which ism(m≥0) times continuously differentiable and itsmth derivative,g ₀ ^(m) , satisfies a Hölder condition of order γ(m+γ>1/2). A piecewise polynomialL ₁-norm estimator ofg ₀ is proposed. Under some regularity conditions including that the random errors are independent but not necessarily have a common distribution, it is proved that the rates of convergence of the piecewise polynomialL ₁-norm estimator are $o(n^{ - \frac{{m + \gamma - 1/2 - \delta }}{{2\left( {m + \gamma } \right) + 1}}} )$ almost surely and $o(n^{ - \frac{{m + \gamma - \delta }}{{2\left( {m + \gamma } \right) + 1}}} )$ in probability, which can arbitrarily approach the optimal rates of convergence for nonparametric regression, where σ is any number in (0, min((m+γ?1/2)/3,γ)).     

Estimation effective de la perte de positivité dans la régularisation des courants

Let (X,ω) be a compact complex Hermitian manifold, and let T?γ be a d-closed (1,1) almost positive current on X. A variant of Demailly's regularization-of-currents theorem states that T is the weak limit of a sequence of (1,1)-currents T_m with analytic singularities of coefficient 1/m, lying in the same cohomology class as T, whose Lelong numbers converge to those of T, and with a loss of positivity decaying to zero. We prove that if the (1,1)-form γ is assumed to be closed and C^∞, the regularizing currents T_m can be chosen such that $\text{T}{}_{\mathrm{m}}\text{?γ?}\text{C}\text{m}$ for a constant C>0 independent of m. To cite this article: D. Popovici, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Continuity of iteration and approximation of iterative roots

Motivated by computing iterative roots for general continuous functions, in this paper we prove the continuity of the iteration operators T_n, defined by T_nf=fⁿ. We apply the continuity and introduce the concept of continuity degree to answer positively the approximation question: If lim_m→∞F_m=F, can we find an iterative root f_m of F_m of order n for each m∈N such that the sequence (f_m) tends to the iterative root of F of order n associated with a given initial function? We not only give the construction of such an approximating sequence (f_m) but also illustrate the approximation of iterative roots with an example. Some remarks are presented in order to compare our approximation with the Hyers-Ulam stability.     

On the convergence of interpolating periodic spline functions of high degree

Let the spline functionS _m of degree 2m?1 and period 1 be the unique solution of the interpolation problem in § 1. An interesting question was posed by Schoenberg [1], p. 125: What happens toS _m if we letm→∞? In this paper, we prove that the spline functionsS _m and their derivatives converge form→∞ to a well determined trigonometric polynomial and its derivatives. Estimates for the rate of convergence are given.     

Polynomials and Hankel matrices

Compatibility of a Hankel n × n matrix H and a polynomial f of degree m, m ? n, is defined. If m = n, compatibility means that HC^T_f=C_fH where C_f is t companion matrix of f. With a suitable generalization of C_f, this theorem is generalized to the case that m < n.     

Stability theory for absolutely continuous parts of commutative m-tuples of self-adjoint operators in complex Hilbert spaces

By the method of direction wave operators, we prove that absolutely continuous parts of commutative m-tuples of self-adjoint operators in complex Hilbert spaces are invariant up to unitary equivalence under perturbations of class B_pfor p < m, m ? 2.     

Intersection properties of finite sets

Let X₁, X₂, …, X_m be finite sets. The present paper is concerned with the m² ? m intersection numbers |X_i ∩ X_j| (i ≠ j). We prove several theorems on families of sets with the same prescribed intersection numbers. We state here one of our conclusions that requires no further terminology. Let T₁, T₂, …, T_m be finite sets and let m ? 3. We assume that each of the elements in the set union T₁ ∪ T₂ ∪ … ∪ T_m occurs in at least two of the subsets T₁, T₂, …, T_m. We further assume that every pair of sets T_i and T_j (i ≠ j) intersect in at most one element and that for every such pair of sets there exists exactly one set T_k (k ≠ i, k ≠ j) such that T_k intersects both T_i and T_j. Then it follows that the integer m = 2m′ + 1 is odd and apart from the labeling of sets and elements there exist exactly m′ + 1 such families of sets. The unique family with the minimal number of elements is {1}, {2}, …, {m′}, {1}, {2}, …, {m′}, {1, 2, …, m′}.