THE DIVERGENCE OF LAGRANGE INTERPOLATION IN EQUIDISTANT NODES

It is a classical result of Bernstein that the sequence of Lagrange interpolation polynomials to [x] at equally spaced nodes in [- 1,1 ] diverges everywhere, except at zero and the end-points. In this paper we show that the sequence of Lagrange interpolation polynomials corresponding to the functions which possess better smoothness on equidistant nodes in [- 1,1 ] still diverges every where in the interval except at zero and the end-points.     

One-step estimation for varying coefficient models

A one-step method is proposed to estimate the unknown functions in the varying coefficient models, in which the unknown functions admit different degrees of smoothness. In this method polynomials of different orders are used to approximate unknown functions with different degrees of smoothness. As only one minimization operation is employed, the required computation burden is much less than that required by the existing two-step estimation method. It is shown that the one-step estimators also achieve the optimal convergence rate. Moreover this property is obtained under conditions milder than that imposed in the two-step estimation method. More importantly, as only one minimization operation is employed, the full asymptotic properties, not only the asymptotic bias and variance, but also the asymptotic distributions of the estimators can be derived. The asymptotic distribution results will play a key role for making statistical inference.     

Decomposition and approximation of multivariate functions on the cube

In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jφjψj , where each φj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each φjψj is a product of separated variable type and its smoothness is same as f . Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.     

UNCONSTRAINED AND CONVEX POLYNOMIAL APPROXIMATION IN C[-1,1]

Some estimates for unconstrained and convex polynomial approximation in the uniform metric are obtained. These results are given in terms of the Ditzian-Totik moduli of smoothness≤1 with ψ(x)= 1-x2. The construction of the approximating polynomials does not depend on AAA.     

ON HERMITE MATRIX POLYNOMIALS AND HERMITE MATRIX FUNCTIONS

In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.     

ORTHOGONAL POLYNOMIALS ASSOCIATED WITH THE DIRAC OPERATOR IN EUCLIDEAN SPACE

The author considers the possibility of generalizing the theory of classicalpolynomials to the higher dimensional case.The starting point is the splitting up ofthe second order differential operator of these polynomials into the derivation operator,considered as an operator between Hilbert spaces and its adjoint.In the case of severaldimensions the derivation operator is replaced by the Dirac operator.As however theset of polynomials in the vector variable x is not dense in the Hilbert modulesconsidered,first a decomposition of these modules in terms of spherical monogenicfunctions is proved.Then by applying the theory to each of the constituents,generalizations of the Gegenbauer and the Hermite polynomials are obtained.     

Non-stationary subdivision for exponential polynomials reproduction

In this paper we develop a novel approach to construct non-stationary subdivision schemes with a tension control parameter which can reproduce functions in a finite-dimensional subspace of exponential polynomials. The construction process is mainly implemented by solving linear systems for primal and dual subdivision schemes respectively, which are based on different parameterizations. We give the theoretical basis for the existence, uniqueness, and refinement rules of schemes proposed in this paper. The convergence and smoothness of the schemes are analyzed as well. Moreover, conics reproducing schemes are analyzed based on our theory, and a new idea that the tensor parameter ωk of the schemes can be adjusted for conics generation is proposed.     

QUASIPERIODICITY OF TRANSCENDENTAL MEROMORPHIC FUNCTIONS

This paper is devoted to considering the quasiperiodicity of complex differential polynomials,complex difference polynomials and complex delay-differential polynomials of certain types,and to studying the similarities and differences of quasiperiodicity compared to the corresponding properties of periodicity.     

关于多项式多重卷积利用差分与移位算子的计值法

Here concerned and further investigated is a certain operator method for the computation of convolutions of polynomials. We provide a general formulation of the method with a refinement for certain old results, and also give some new applications to convolved sums involving several noted special polynomials. The advantage of the method using operators is illustrated with concrete examples. Finally, also presented is a brief investigation on convolution polynomials having two types of summations.     

RECURSIVE FORMULA FOR CALCULATING THE CHROMATIC POLYNOMIAL OF A GRAPH BY VERTEX DELETION

A new recursive vertex-deleting formula for the computation of the chromatic polynomial of a graph is obtained in this paper. This algorithm is not only a good tool for further studying chromatic polynomials but also the fastest among all the algorithms for the computation of chromatic polynomials.     

Unconstrained and convex polynomial approximation inC[−1,1]

Some estimates for unconstrained and convex polynomial approximation in the uniform metric are obtained. These results are given in terms of the Ditzian-Totik moduli of smoothness , ≤1 with . The construction of the approximating polynomials does not depend on λ.     

Hermitean Clifford-Hermite Polynomials

Orthogonal Clifford analysis in flat m–dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator , where ( ) forms an orthogonal basis for the quadratic space underlying the construction of the Clifford algebra . When allowing for complex constants and taking the dimension to be even: m = 2n, the same set of generators produces the complex Clifford algebra , which we equip with a Hermitean Clifford conjugation and a Hermitean inner product. Hermitean Clifford analysis then focusses on the simultaneous null solutions of two mutually conjugate Hermitean Dirac operators, naturally arising in the present context and being invariant under the action of a realization of the unitary group U (n). In this so–called Hermitean setting Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Dirac operators mentioned. Due to the specific features of the Hermitean setting, four different types of polynomials are obtained, two types of even degree and two types of odd degree. We investigate their properties: recurrence relations, structure, explicit form and orthogonality w.r.t. a deliberately chosen weight function. They also give rise to the definition of the Hermitean Clifford–Hermite functions, and may be used to develop a Hermitean continuous wavelet transform, see [4].     

Graded monads and rings of polynomials

Models for free graded monads over the category of sets are constructed. Certain rings of generalized noncommutative polynomials, generated by an operation of arbitrary arity, are implemented as subrings of classical rings of noncommutative polynomials. It is shown that natural homomorphisms from rings of generalized polynomials to rings of the usual commutative polynomials are not inclusions as a rule. For instance, the natural homomorphism , where t is a binary variable, is not an inclusion even if t is subject to the alternating condition. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 349, 2007, pp. 174–210.     

Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation

Some Limit Theorems for a Particle System of Single Point Catalytic Branching Random Walks

We study the scaling limit for a catalytic branching particle system whose particles perform random walks on Z and can branch at 0 only. Varying the initial （finite） number of particles, we get for this system different limiting distributions. To be more specific, suppose that initially there are n^β particles and consider the scaled process Zt^n（·） = Znt（√n·）, where Zt is the measure-valued process 1 and to a representing the original particle system. We prove that Ztn converges to 0 when β 〈1/4 and to a nondegenerate discrete distribution when β=1/4.In addition,if 1/4〈β〈1/2 then n-^（2β-1/2）Zt^n converges to a random limit,while if β 〉21then n^-βZtn converges to a deterministic limit.     

On the Littlewood-Paley theorem for arbitrary intervals

We extend the results of Rubio de Francia and Bourgain by showing that, for arbitrary mutually disjoint intervals Δ_k ⊂ ℤ₊, arbitrary p ∈, (0, 2], and arbitrary trigonometric polynomials f _k with supp , we have

Generalized interval exchanges and the 2–3 conjecture

We introduce the notion of a generalized interval exchange induced by a measurable k-partition of [0,1). can be viewed as the corresponding restriction of a nondecreasing function on ℝ with . A is called λ-dense if λ(A _i ∩(a, b))>0 for each i and any 0≤ a< b≤1. We show that the 2–3 Furstenberg conjecture is invalid if and only if there are 2 and 3 λ-dense partitions A and B of [0,1), such that . We give necessary and sufficient conditions for this equality to hold. We show that for each integer m≥2, such that 3∤2m+1, there exist 2 and 3 non λ-dense partitions A and B of [0,1), corresponding to the interval exchanges on 2m intervals, for which and commute.     

On Sikkema-Kantorovič polynomials of order K

Let a_n≥0 and F(u)∈C [0,1], Sikkema constructed polynomials: , ifα _n ≡0, then B_n (0, F, x) are Bernstein polynomials. Let , we constructe new polynomials in this paper: Q _n ^(k) (α _n ,f(t))=d ^k /dx ^k B _n+k (α _n ,F _k (u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα _n ≡0, k=1, then Q_n ⁽¹⁾ (0, f(t), x) are Kantorovič polynomials P_n(f). Ifα _n =0, k=2, then Q_n ⁽²⁾, (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is: Theorem 2. Let 1≤p≤∞, in order that for every f∈L^P [0, 1], , it is sufficient and necessary that , § 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define^{[1–2],[8–10]}: . As usual, for the space L^p [a,b](1≤p<∞), we have and L[a, b]=l¹[a, b]. Letα _n ⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials ^{[3] [4]}. The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports.     

Herz-type Triebel-Lizorkin Spaces, Ⅰ

Let s ∈R,0〈β≤∞, 0〈 q, p〈 ∞ and-n/q〈α. In this paper the authors introduce the Herz-type Triebel-Lizorkin spaces,Kq^α,pFβ^s（R^n）andKq^α,pFβ^s（R^n）which are the generalizations of the well-known Herz-type spaces and the inhomogeneous Triebel-Lizorkin spaces, Some properties on these Herz-type Triebel Lizorkin spaces are also given.     

Isometries and direct decompositions of pseudo MV-algebras

In the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) there exists an internal direct decomposition of with commutative such that and for each x ∈ A. On the other hand, if is an internal direct decomposition of a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) with commutative, then the mapping g: A → A defined by is an isometry in and .