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.     

Some notes on Hermite-Fejer type interpolation

Let S_n(f,x) be the Hermite-Fejér type interpolation satisfying S_n(f,x_k)=f(x_k), S′_n(f,x_k)=0, k=1,2,…,n and S_n(f,y_i)=f(y_i), j=1,2,…,m. For m=0, let H_n(f,x)≔S_n(f,x). This paper investigates relationship between S_n(f,x) and H_n(f,x), as well as, the saturation of S_n(f,x).     

The perfectB-splines and a time-optimal control problem

Letx _v =cos (πν/n) (v=0, 1, …,n). It is shown that theB-splineM(x)=M(x; x ₀ ,x ₁ ,…, x _n ) is such thatM _n ⁽ⁿ⁾ (x) has a constant absolute value (=2 ⁿ⁻² (n−1)!) in [−1, 1]. Its integralf ₀(x)=∫ ₋₁ ^x M(t)dt is shown to have an optimal property that allows to solveexplicitly a certain time-optimal control problem.     

A note on convergent polynomials of interpolation

In this note we define a sequence {L_n(f;x)} of interpolatory polynomials based on a system x_n={x_kn, k=1,2,…n} of nodes to be a sequence of QLIP if for every f(x)∈C[−1,1], L_n(f; x) tends uniformly to f(x) and ρ_n=1+o(1) as n→∞, where ρ_n is the ratio of the number of points in x_n, at which L_n(f;x) coincides with f(x), and the degree of L_n(f;x). Two sequences of QLIP are constructed, one of which is based on a Bernstein process and the other the Freud-Sharma's construction.     

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 .     

Embeddings ofl p m intol ∞ n , 1≦p≦2

Isomorphic embeddings ofl _l ^m intol _∞ ⁿ are studied, and ford(n, k)=inf{‖T ‖ ‖T ⁻¹ ‖;T varies over all isomorphic embeddings ofl ₁ ^[klog2n] intol _∞ ⁿ we have that lim _n→∞ d(n, k)=γ(k)⁻¹,k>1, whereγ(k) is the solution of (1+γ)ln(1+γ)+(1 −γ)ln(1 −γ)=k ⁻¹ln4. Here [x] denotes the integer part of the real numberx.     

On some characterization of probability distributions in Hilbert spaces

Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x₁, x₂, …, x_n be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals f_k(t) = E[e^i(t,x _k)], (k=1, 2, …, n): let y₁=x₁ + x_n, y₂=x₂ + x_n, …, y_n−1=x_n−1 + x_n. If the characteristic functional f(t₁, t₂, …, t_n−1) of the random variables (y₁, y₂, …, y_n−1) does not vanish, then the joint distribution of (y₁, y₂, …, y_n−1) determines all the distributions of x₁, x₂, …, x_n up to change of location.     

The unbounded-error communication complexity of symmetric functions

We prove an essentially tight lower bound on the unbounded-error communication complexity of every symmetric function, i.e., f(x,y)=D(|x∧y|), where D: {0,1,…,n}→{0,1} is a given predicate and x,y range over {0,1} ⁿ . Specifically, we show that the communication complexity of f is between Θ(k/log⁵ n) and Θ(k logn), where k is the number of value changes of D in {0,1,…, n}. Prior to this work, the problem was solved only for the parity predicate D (Forster 2001).     

The intersection of the powers of the radical in Noetherian P.I. rings

LetR be a ring and J its radical. DefineJ ₁=∩Jⁿ, J₂=∩J ₁ ⁿ ,…,… J_k=∩J _k−1 ⁿ .... It is shown that in a ringR satisfying a polynomial identity and the ascending chain condition on ideals,J _k =0 for some appropriatek. The work of the first author was supported by an NSF grant at the University of Chicago. The work of the second author was supported by an NSF grant at the University of California, San Diego.     

On a function associated with the Bessel functions

Summary Defining the function Δn, 1,k;x(J) asΔn, 1,k;x(J)=J _n+1(x)−J _n(x)J _n+k+1(x) associated with the Bessel functionJ _n(x), we derive a series of products of Bessel functions for Δ_{n, f, k, x} (J). Whenk=1,k;x (J) becomes Turàn expression for Bessel functions. Some consequences have been pointed out.

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Riassunto Definita la Δ_{n, f, k, x} (J) come Δ_{n, f, k, x}, (J)=J _n+1(x)J _n+k(x)-J _n(_n+k+1)(x) associata alla funzioneJ _n(x) di Bessel, si ricava una serie di prodotti di funzioni di Bessel per Δ_{n, f, k, x}, (J). 3 Quandok=1, Δ_{n, f, k, x}, (J) diventa una espressione di Turàn per le funzioni di 2 Bessel, vengono inoltre indicate alcune altre conseguenze.

     

Averaging distances in real quasihypermetric Banach spaces of finite dimension

The average distance theorem of Gross implies that for each realN-dimensional Banach space (N≥2) there is a unique positive real numberr(E) with the following property: For each positive integern and for all (not necessarily distinct)x ₁,x ₂, …,x _n inE with ‖x ₁‖=‖x ₂‖=…=‖x _n‖=1, there exists anx inE with ‖x‖=1 such that The main result of this paper shows, thatr(E)≤2−1/N for each realN-dimensional Banach spaceE (N≥2) with the so-called quasihypermetric property (which is equivalent toE isL ¹-embeddable). Moreover, equality holds if and only ifE is isometrically isomorphic to ℝ ^N equipped with the usual 1-norm.     

Polynomials of best approximation inC[−1,1]

Equivalences between the condition |P _n ^(k) (x)|≦K(n ⁻¹√1−x ²+1/n ²) ^k n ^-a, whereP _n(x) is the bestn-th degree polynomial approximation tof(x), and the Peetre interpolation space betweenC[−1,1] and the space (1−x ²) ^k f ^(2k)(x)∈C[−1,1] is established. A similar result is shown forE _n(f)= ‖f−P _n‖ _C[−1,1]. Rates other thann ^-a are also discussed. Supported by NSERC grant A4816 of Canada.     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

A result on the composition of distributions

LetF be a distribution and letf be a locally summable function. The distributionF(f) is defined as the neutrix limit of the sequenceF _n (f), whereF _n (x) = F(x) * δ _n (x) andδ _n (x) is a certain sequence of infinitely differentiable functions converging to the Dirac delta-functionδ(x). The distribution (x^r)^−s is valuated forr, s = 1,2, ….     

The additive subgroup generated by a polynomial

SupposeR is a prime ring with the centerZ and the extended centroidC. Letp(x ₁, …,x _n) be a polynomial overC in noncommuting variablesx ₁, …,x _n. LetI be a nonzero ideal ofR andA be the additive subgroup ofRC generated by {p(a ₁, …,a _n):a ₁, …,a _n ∈I}. Then eitherp(x ₁, …,x _n) is central valued orA contains a noncentral Lie ideal ofR except in the only one case whereR is the ring of all 2 × 2 matrices over GF(2), the integers mod 2.     

On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces

We prove that ifC is a bounded closed convex subset of a uniformly convex Banach space,T:C→C is a nonlinear contraction, andS _n =(I+T+…+T ⁿ⁻¹ )/n, then lim _n ‖S _n (x)−TS _n (x)‖=0 uniformly inx inC. T also satisfies an inequality analogous to Zarantonello’s Hilbert space inequality. which permits the study of the structure of the weak ω-limit set of an orbit. These results are valid forB-convex spaces if some additional condition is imposed on the mapping. Partially supported by NSF Grant MCS-7802305A01.     

The Johnson–Lindenstrauss Lemma Almost Characterizes Hilbert Space, But Not Quite

Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer n and any x ₁,…,x _n ∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that ‖x _i −x _j ‖≤‖L(x _i )−L(x _j )‖≤O(1)⋅‖x _i −x _j ‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion 2^2O(log*n)2^{2^{O(\log^{*}n)}} . On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-dimensional subspace E _n ⊆Y whose Euclidean distortion is at least 2^Ω(α(n)), where α is the inverse Ackermann function.     

Regularity of (0,1, …,r−2, r)* interpolation on the unit circle

The object of this paper is to show regularity of (0,1, …, r−2, r)^* interpolation on the set obtained by projecting vertically the zeros of (1−x²)p _n ^(λ) (x) (λ≥1/2) onto the unit circle, where P _n ^(λ) (x) stands for the nth ultraspherical polynomial.     

Finiteness conditions in Krull subrings of a ring of polynomials

LetR be a Krull subring of a ring of polynomialsk[x ₁, …, x_n] over a fieldk. We prove that ifR is generated by monomials overk thenr is affine. We also construct an example of a non-affine Krull ringR, such thatk[x, xy]⊂R⊂k[x, y], and a non-Noetherian Krull ringS, such thatk[x, xy, z]⊂S⊂k[x, y, z].     

Some properties of multivariate extreme value distributions and multivariate tail equivalence

Summary Denote byH ak-dimensional extreme value distribution with marginal distributionH _i (x)=Λ(x)=exp(−e ^−x ),x∈R ¹. Then it is proved thatH(x)=Λ(x ₁)...Λ(x _k ) for anyx=(x ₁, ...,x _k ) ∈R ^k , if and only if the equation holds forx=(0,...,0). Next some multivariate extensions of the results by Resnick (1971,J. Appl. Probab.,8, 136–156) on tail equivalence and asymptotic distributions of extremes are established.