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.     

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 .     

Convergence of weighted sums of independent random variables

Let {X_k} be a sequence of i.i.d. random variables with d.f. F(x). In the first part of the paper the weak convergence of the d.f.'s F_n(x) of sums is studied, where 0<α≤2, a_nk>0, 1≤k≤m_n, and, as n→∞, bothmax _1≤k≤mn_{a nk}→0 and . It is shown that such convergence, with suitably chosen A_n's and necessarily stable limit laws, holds for all such arrays {α_nk} provided it holds for the special case α_nk=1/n, 1≤k≤n. Necessary and sufficient conditions for such convergence are classical. Conditions are given for the convergence of the moments of the sequence {F_n(x)}, as well as for its convergence in mean. The second part of the paper deals with the almost sure convergence of sums , where a_n≠0, b_n>0, andmax _1≤k≤n a_k/b_n→0. The strong law is said to hold if there are constants A_n for which S_n→0 almost surely. Let N(0)=0 and N(x) equal the number of n≥1 for which b_n/|a_n|<x if x>0. The main result is as follows. If the strong law holds,EN (|X₁|)<∞. If for some 0<p≤2, then the strong law holds with if 1≤p≤2 and A_n=0 if 0<p<1. This extends the results of Heyde and of Jamison, Orey, and Pruitt. The strong law is shown to hold under various conditions imposed on F(x), the coefficients a_n and b_n, and the function N(x). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

A joint norm control Nehari type theorem forN-tuples of Hardy spaces

If N ∈ ℕ, 0 < p ≤ 1, and(X_k) _k=1 ^N are r.i.p-spaces, it is shown that there is C(= C(p, N)) > 0, such that for every ƒ ∈ ∩ _k=1 ^N X_k, there exists with , for every 1 ≤ k ≤ N. Also, if ⊓ is a convex polygon in ℝ², it is proved that the N-tuple (H(X₁),…, H(X_n)) is K_⊓-closed with respect to (X₁,…, X_N) in the sense of Pisier. Everything follows from Theorem 2.1, which is a general analytic partition of unity type result.     

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.     

A generalization of Simpson's Rule

Let and let , where P _c ⁿ denoles the Taylor polynomial to f at c of order n, where n is even. TA and TM are reach generalizations of the Trapezoidal rule and the midpoint rule, respectively, and are each exact for all polynomials of degree ≤n+1. We let L(f)=αTM(f)+(1−α)TA(f), where , to obtain a numerical integration rule L which is exact for all polynomials of degree≤n+3 (see Theorem 1). The case n=0 is just the classical Simpson's rule. We analyze in some detail the case n=2, where our formulae appear to be new. By replacing P _(a+b) ^2/n+1 (x) by the Hermite cubic interpolant at a and b, we obtain some known formulae by a different approach (see [1] and [2]). Finally we discuss some nonlinear numerical integration rules obtained by taking piecewise polynomials of odd degree, each piece being the Taylor polynomial of f at a and b, respectively. Of course all of our formulae can be compounded over subintervals of [a,b].     

Inversion of exponentialk-plane transforms

Approximate and explicit inversion formulas are obtained for a new class of exponential k-plane transforms defined by where x∈ℝⁿ, Θ is a k-frame in ℝⁿ, 1≤k≤n−1, μ∈ℂ^k is an arbitrary complex vector. The case k=1, μ∈ℝ corresponds to the exponential X-ray transform arising in single photon emission tomography. Similar inversion formulas are established for the accompanying transform where V is a real (n×k)-matrix. Two alternative methods, leading to the relevant continuous wavelet transforms, are presented. The first one is based on the use of the generalized Calderón reproducing formula and multidimensional fractional integrals with a Bessel function in the kernel. The second method employs interrelation between P_μ and the associated oscillatory potentials.     

Integration and Lipschitz functions

The aim of the paper is to prove that every f ∈ L ¹([0,1]) is of the form f = , where j _n,k is the characteristic function of the interval [k- 1 / 2 ⁿ , k / 2 ⁿ ) and Σ _n=0^∞Σ _k=1²ⁿ |a _n,k | is arbitrarily close to ||f|| (Theorem 2). It is also shown that if μ is any probabilistic Borel measure on [0,1], then for any ɛ > 0 there exists a sequence (b _n,k ) _n≧0 ^k=1,...,2n of real numbers such that and for each Lipschitz function g: [0,1] → ℝ (Theorem 3).      

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.     

The region of values of a system of functionals in the class of typically real functions

Let T_R be the class of functions that are regular and typically real in the disk E={z:⋱z⋱<1}. For this class, the region of values of the system {f(z₀), f(r)} for z₀ ∈ ℝ, r∈(-1,1) is studied. The sets D_r={f(z₀):f∈T_R, f(r)=a} for −1≤r≤1 and Δ_r={(c₂, c₃): f ∈ T_R, −f(−r)=a} for 0<r≤1 are found, where aε(r(1+r)⁻², r(1−r)⁻²) is an arbitrary fixed number. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 69–79.     

Self-intersections of random walks on lattices

Let {X _n ^d }_n≥0be a uniform symmetric random walk on Z^d, and Π^(d) (a,b)={X _n ^d ∈ Z^d : a ≤ n ≤ b}. Suppose f(n) is an integer-valued function on n and increases to infinity as n↑∞, and let

Moments of randomly stopped sums-revisited

Conditions are obtained for (*)E|S _T |^γ<∞, γ>2 whereT is a stopping time and {S _n=∑ ₁ ⁿ ,X _j ℱ _n ,n⩾1} is a martingale and these ensure when (**)X _n ,n≥1 are independent, mean zero random variables that (*) holds wheneverET ^γ/2<∞, sup _n≥1 E|X _n |^γ<∞. This, in turn, is applied to obtain conditions for the validity ofE|S _k,T |^γ<∞ and of the second moment equationES _k,T ² =σ ² EΣ _j=k ^T S _k−1,j−1 ² where and {X _n , n≥1} satisfies (**) and ,n≥1. The latter is utilized to elicit information about a moment of a stopping rule. It is also shown for i.i.d. {X _n , n≥1} withEX=0,EX ²=1 that the a.s. limit set of {(n log logn)^−k/2 S _k,n ,n≥k} is [0,2 ^k/2/k!] or [−2 ^k/2/k!] according ask is even or odd and this can readily be reformulated in terms of the corresponding (degenerate kernel)U-statistic .     

Addendum To Schrijver's Work On Minimum Permanents

Let denote the set of n×n binary matrices which have each row and column sum equal to k. For 2≤k≤n→∞ we show that is asymptotically equal to (k−1)^k−1k^2−k. This confirms Conjecture 23 in Minc's catalogue of open problems. * Written while the author was employed by the Department of Computer Science at the Australian National University.     

Über lineare Komplexe von Räumen höchster Dimension einer nicht entarteten Quadrik

Sunto Conoscendo il modello minimo per la totalità degli S _k ^I d'una schiera giacenti su di una quadrica Q_2k di S_2k+1 si definisce il ? complesso lineare di S _k ^I ? come imagine d'una sezione iperpiana di . Il problema della classificazione projettiva di tali complessi lineari è posto e risolto per k≤5.      

A Ramsey-Sperner theorem

Letn≥k≥1 be integers and letf(n, k) be the smallest integer for which the following holds: If ℱ is a family of subsets of ann-setX with |ℱ|<f(n,k) then for everyk-coloring ofX there existA B ∈ ℱ,A∈B, A⊂B such thatB-A is monochromatic. Here it is proven that for a fixedk there exist constantsc _k andd _k such that and ask→∞. The proofs of both the lower and the upper bounds use probabilistic methods.     

Some inequalities about mixed volumes

We prove inequalities about the quermassintegralsV _k (K) of a convex bodyK in ℝ ⁿ (here,V _k (K) is the mixed volumeV((K, k), (B _n ,n − k)) whereB _n is the Euclidean unit ball). (i) The inequality

The value regions of initial coefficients in a certain class of meromorphic functions

Let M_k,λ(0≤λ≤1, k≥2) be the class of functions f(z)=1/z+a_o+a₁z+... that are regular and locally univalent for 0<⩛z⩛<1 and satisfy the condition where J_λ(z)=λ(1+zf″(z)/f'(z))+(1-λ)zf'(z)/f(z). In the class M_k,λ we consider sorne coefficient problems and problems concerning distortion theorems. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 91–96. Translated by N. Yu. Netsvetaev.     

Precise asymptotics in self-normalized sums of iterated logarithm for multidimensionally indexed random variables

In the case of Zd (d ≥ 2)-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k ∈ Zd } i.i.d. random variables with mean 0, Sn = ∑k≤nXk and Vn2 = ∑j≤nX2j, the precise asymptotics for ∑n1/|n|(log|n|)dP(|Sn/vn|≥ ε√loglog|n|) and ∑n(logn|)δ/|n|(log|n|)d-1 P(|Sn/Vn| ≥ ε√log n), as ε ↘ 0, is established.     

Structural formulas and value regions of functionals in certain classes of regular functions

We study the structural properties of the class M_k,λ,b(k≥2, 0≤λ≤1, b∈ℂ\{0}) of functions f(z)=z+ ... which are regular in |z|<1 and satisfy the conditions f(z)f′(z)z⁻¹≠0 and , where J(z)=λ(1+b⁻¹zf″(z)/f′(z)+(1−λ)(b⁻¹zf′(z)/f(z)+1−b⁻¹). The value regions of some functionals on this class are found. The case λ=1 was considered in our previous paper. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 55–60. Translated by O. A. Ivanov.     

Exchangeable Gibbs partitions and Stirling triangles

For two collections of nonnegative and suitably normalized weights W = (W_j) and V = (V_n,k), a probability distribution on the set of partitions of the set {1, …, n} is defined by assigning to a generic partition {A_j, j ≤ k} the probability V_n,k , where |A_j| is the number of elements of A_j. We impose constraints on the weights by assuming that the resulting random partitions Π _n of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights W must be of a very special form depending on a single parameter α ∈ [− ∞, 1]. The case α = 1 is trivial, and for each value of α ≠ = 1 the set of possible V-weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalized Stirling triangle. In particular, we show that the boundary is discrete for − ∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by (α,θ), while for 0 < α < 1 the extremes are obtained by conditioning an (α,θ)-partition on the asymptotics of the number of blocks of Π_n as n tends to infinity. Bibliography: 29 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 83–102.