Best polynomial approximation in Sobolev-Laguerre and Sobolev-Legendre spaces

We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space W^N,2([0, ∞); e^−x) and the Sobolev-Legendre space W^N,2([−1, 1]) with respect to the Sobolev-Laguerre inner product

Su un problema di tipo nuovo relativo alle equazioni paraboliche d'ordine superiore al secondo

Sunto Si studia il problema della determinazione di una soluzione dell'equazione a_k(x)∂^ku/∂x^k=f(x, y) entro la semistriscia a≤x≤b, y≥0, che assuma assegnati valori per y=0 e per x=a, x₁, x₂, b (a<x₁<x₂<b). Analogamente si studia il problema della determinazione di una soluzione dell' equazione a_k(x)∂^ku/∂x^k+b(x)∂u/∂y=f(x,y), entro la medesima semistriscia, cha assuma assegnati valori per y=0 e per x=a, x₁, x₂, b e la cui ∂/∂y assuma assegnati valori per y=0. A Giovanni Sansone nel suo 70^mo compleanno.     

Ingham divisor problem with square-free numbers

For the number N(x) of solutions to the equation aq − bc = 1 in positive integers a, b, c and square-free numbers q satisfying the condition aq ≤ x the asymptotic formula

On convergence of a generalized Pál interpolation process

Let f∈C _[−1,1] ^″ (r≥1) and R_n(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions R_n(f,α,β,x_k)=f(x_k),R_n ^′(f,α,β,x_k)=f′(x_k)(k=1,2,…,n), where {x_k} are the roots of n-th Jacobi polynomial P_n(α,β,x),α,β>−1 and {x _k ^″ } are the roots of (1−x²)P_n″(α,β,x). In this paper, we prove that holds uniformly on [0,1]. In Memory of Professor M. T. Cheng Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang.     

Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications

We obtain a strengthened version of the Kolmogorov comparison theorem. In particular, this enables us to obtain a strengthened Kolmogorov inequality for functions x L ^x (r), namely,

On estimation of a density and its derivatives

Summary Letf _n ^(p) be a recursive kernel estimate off ^(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of and show that the rate of almost sure convergence of to zero isO(n ^−α), α<(r−p)/(2r+1), iff ^(r),r>p≧0, is a continuousL ₂(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of to zero under different conditions onf. This work was supported in part by the Research Foundation of SUNY.     

On simultaneous approximation to a differentiable function and its derivative by inverse Pal-Type interpolation polynomials

Let ξ_{n −1} < ξ_{n −2} < ξ_{n − 2} < ... < ξ₁ be the zeros of the the (n−1)-th Legendre polynomial P_n−1(x) and −1=x_n<x_n−1<...<x₁=1, the zeros of the polynomial . By the theory of the inverse Pal-Type interpolation, for a function f(x)∈C _[−1,1] ¹ , there exists a unique polynomial R_n(x) of degree 2n−2 (if n is even) satisfying conditions R_n(f, ξ_k) = f (ε_k) (1 ⩽ k ⩽ n −1); R¹ _n(f,x_k)=f¹(x_k)(1≤k≤n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {R_n(f, x)} (n is even) and the main result of this paper is that if f∈C _[1,1] ^r , r≥2, n≥r+2, and n is even then |R¹ _n(f,x)−f¹(x)|=0(1)|W_n(x)|h(x)·n^3−r·E_2n−r−3(f^(r)) holds uniformly for all x∈[−1,1], where .     

Inequalities for derivatives of functions in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

  We obtain a new sharp inequality for the local norms of functions x ∈ L _{∞, ∞} ^r (R), namely,

Uniform boundedness of (<Emphasis Type="Italic">C</Emphasis>, 1) means of Jacobi expansions in weighted sup norms. II (Some necessary estimations)

The paper is concerned with bounds for integrals of the type

On the Non-Commutative Neutrix Product of the Distributions x＋^λ and x^＋μ

Let f and g be distributions and let gn = （g ＊ δn）（x）, where δn （x） is a certain converging to the Dirac delta function. The non-commutative neutrix product fog of f and g to be the limit of the sequence {fgn }, provided its limit h exists in the sense that sequence is defined N-lim n-∞（f（x）g,, （x）, φ（x）〉 = （h（x）, φ（x）},for all functions p in 2. It is proved that （x^λ＋1n^px＋）0（x^μ＋1n^qx＋）=x＋^λμ1n^p＋qx＋,（x^λ-1n^qx-）=x-^λ＋μ1n^p＋qx-,for λ＋μ〈-1; λ,μ, λ＋μ≠-1,-2…and p,q=0,1,2……     

Matching Polynomials And Duality

Let G be a simple graph on n vertices. An r-matching in G is a set of r independent edges. The number of r-matchings in G will be denoted by p(G, r). We set p(G, 0) = 1 and define the matching polynomial of G by and the signless matching polynomial of G by .It is classical that the matching polynomials of a graph G determine the matching polynomials of its complement . We make this statement more explicit by proving new duality theorems by the generating function method for set functions. In particular, we show that the matching functions and are, up to a sign, real Fourier transforms of each other.Moreover, we generalize Foatas combinatorial proof of the Mehler formula for Hermite polynomials to matching polynomials. This provides a new short proof of the classical fact that all zeros of µ(G, x) are real. The same statement is also proved for a common generalization of the matching polynomial and the rook polynomial.     

On the riemann zeta-function and the divisor problem

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If with , then we obtain

Composition operators on the Lipschitz space in polydiscs

Let Un be the unit polydisc of Cn and φ= (φ1,...,φn? a holomorphic self-map of Un. Let 0≤α< 1. This paper shows that the composition operator Cφ, is bounded on the Lipschitz space Lipa(Un) if and only if there exists M > 0 such thatfor z∈Un. Moreover Cφ is compact on Lipa(Un) if and only if Cφ is bounded on Lipa(Un) and for every ε > 0, there exists a δ > 0 such that whenever dist(φ(z),σUn) <δ     

Global Existence of Positive Periodic Solutions for a Distributed Delay Competition Model

By using the continuation theorem of Mawhin's coincidence degree theory, a sufficient condition is derived for the existence of positive periodic solutions for a distributed delay competition modelwhere ri and r2 are continuous w-periodic functions in R+=[0,∞) with ,ai,ci(i =1,2) are positive continuous w-periodic functions in R+=[0,∞),bi (i = 1,2) is nonnegative continuous w-periodic function in R+=[0,∞), w and T are positive constants. Ki,Lt ∈ C([-T,0], (01 88)) and Ki(s)ds = 1,ds - 1. i = 1,2. Some known results are improved and extended.     

Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold

Let M ⁿ be an n-dimensional compact C ^∞-differentiable manifold, n ≥ 2, and let S be a C ¹-differential system on M ⁿ . The system induces a one-parameter C ¹ transformation group φ _t (−∞ < t < ∞) over M ⁿ and, thus, naturally induces a one-parameter transformation group of the tangent bundle of M ⁿ . The aim of this paper, in essence, is to study certain ergodic properties of this latter transformation group. Among various results established in the paper, we mention here only the following, which might describe quite well the nature of our study. (A) Let M be the set of regular points in M ⁿ of the differential system S. With respect to a given C ^∞ Riemannian metric of M ⁿ , we consider the bundle of all (n−2) spheres Q _x ⁿ⁻², x∈M, where Q _x ⁿ⁻² for each x consists of all unit tangent vectors of M ⁿ orthogonal to the trajectory through x. Then, the differential system S gives rise naturally to a one-parameter transformation group ψ _t ^# (−∞<t<∞) of . For an l-frame α = (u ₁, u ₂,⋯, u _l ) of M ⁿ at a point x in M, 1 ≥ l ≥ n−1, each u _i being in , we shall denote the volume of the parallelotope in the tangent space of M ⁿ at x with edges u ₁, u ₂,⋯, u _l by υ(α), and let . This is a continuous real function of t. Let

Generalizations of Non-Commutative Neutrix Convolution Products of Functions

The non-commutative neutrix convolution product of the functions x ^r cos_–(x) and x ^s cos₊(x) is evaluated. Further similar non-commutative neutrix convolution products are evaluated and deduced.     

Higher integrability for minimizers of variational integrals with nonstandard growth

We consider a (possibly) vector-valued function u: Ω→R ^N, Ω⊂R ⁿ, minimizing the integral , whereD _iu=∂u/∂x _i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D ₁u,…,D_n−1u∈L^q, under suitable assumptions ona _i(x).

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Sunto Consideriamo una funzione u: Ω→R ^N, Ω⊂R ⁿ che minimizzi l'integrale , doveD _iu=∂u/∂x_i, o un funzionale con un comportamento simile; sotto opportune ipotesi sua _i(x), dimostriamo la seguente maggiore integrabilità perDu:D ₁u,…,D_n−1uεL^q.

     

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.     

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,