Cesàro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures

The convergence in L²( ) of the even approximants of the Wall continued fractions is extended to the Cesàro–Nevai class CN, which is defined as the class of probability measures σ with lim_n→∞ ∑ⁿ⁻¹_k=0 |a_k|=0, {a_n}_n0 being the Geronimus parameters of σ. We show that CN contains universal measures, that is, probability measures for which the sequence {|_n|² dσ}_n0 is dense in the set of all probability measures equipped with the weak-* topology. We also consider the “opposite” Szeg class which consists of measures with ∑^∞_n=0 (1−|a_n|²)^1/2<∞ and describe it in terms of Hessenberg matrices.     

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(f θ)¦^p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.     

On Positive and Copositive Polynomial and Spline Approximation inLp[−1, 1], 0<p<∞

For a functionfL_p[−1, 1], 0<p<∞, with finitely many sign changes, we construct a sequence of polynomialsP_nΠ_nwhich are copositive withfand such that f−P_npCω(f, (n+1)⁻¹)_p, whereω(f, t)_pdenotes the Ditzian–Totik modulus of continuity inL_pmetric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, thenωcannot be replaced byω²if 1<p<∞. In fact, we show that even for positive approximation and all 0<p<∞ the same conclusion is true. Also, some results for (co)positive spline approximation, exact in the same sense, are obtained.     

Completeness of Trigonometric System with Integer Indices {e; x }

Necessary and sufficient conditions are given which ensure the completeness of the trigonometric systems with integer indices; {e^inx; x }^∞_n=−∞ or {e^inx; x }^∞_n=1 in L^α(μ,  ), α1. If there exists a support Λ of the measure μ which is a wandering set, that is, Λ+2kπ, k=0, ±1, ±2, … are mutually disjoint for different k's, then the linear span of our trigonometric system {e^inx; x }^∞_n=−∞ is dense in L^α(μ,  ) α1. The converse statement is also true.     

Best polynomial and durrmeyer approximation in Lp(S)

On a simplex SR^d, the best polynomial approximation is E_n()_Lp(S)=Inf{P_n−_Lp(S): P_n of total degree n}. The Durrmeyer modification, M_n, of the Bernstein operator is a bounded operator on L_p(S) and has many “nice” properties, most notably commutativity and self-adjointness. In this paper, relations between M_n−z.dfnc;_Lp(S) and E_[√n]()_Lp(S) will be given by weak inequalities will imply, for 0<α<1 and 1≤p≤∞, E_n()_Lp(S)=O(n^-2α)M_n−z.dfnc;_Lp(S)=O(n^-α). We also see how the fact that P(D)εL_p(S) for the appropriate P(D) affects directional smoothness.     

Note on multidimensional empirical processes for φ-mixing random vectors

In 1974, Sen proved weak convergence of the empirical processes (in the J₁-topology on D^p[0, 1]) for a stationary φ-mixing sequence of stochastic p( 1)-vectors. In this note, we show that Sen's theorem on weak convergence of the multidimensional empirical process for a stationary φ-mixing sequence of stochastic vectors remains true under a less restrictive condition on the mixing constants {φ_n}, i.e., φ_n = O(n^−1−δ) for some δ > 0.     

L approximation with trigonometric n-nomials

We compare the degree of approximation to L²(−π, π) by nth degree trigonometric polynomials, with the degree of approximation by trigonometric n-nomials, which are linear combinations, with constant (complex) coefficients, of any 2n + 1 members of the sequence {exp (ikx)}, − ∞ < k < ∞.     

Asymptotic Behaviour of Best lp-Approximations from Affi ne Subspaces

In this paper we consider the problem of best approximation in ℓ_pⁿ, 1<p∞. If h_p, 1<p<∞, denotes the best ℓ_p-approximation of the element h ⁿ from a proper affine subspace K of ⁿ, hK, then lim_p→∞h_p=h_∞^*, where h_∞^* is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to prove that for all r there are α_j ⁿ, 1jr, such that

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, with γ_p^{(r) n} and γ_p^(r)= (p^−r−1).     

On an infinite series of Abel occurring in the theory of interpolation

The purpose of this paper is to show that for a certain class of functions f which are analytic in the complex plane possibly minus (−∞, −1], the Abel series f(0) + Σ_{n = 1}^∞ f⁽ⁿ⁾(nβ) z(z − nβ)^{n − 1}/n! is convergent for all β>0. Its sum is an entire function of exponential type and can be evaluated in terms of f. Furthermore, it is shown that the Abel series of f for small β>0 approximates f uniformly in half-planes of the form Re(z) − 1 + δ, δ>0. At the end of the paper some special cases are discussed.     

Estimates of Freud-Christoffel functions for some weights with the whole real line as support

Upper and lower bounds for generalized Christoffel functions, called Freud-Christoffel functions, are obtained. These have the form λ_n,p(W,j,x) = infPW_Lp(R)/|P^(j)(X)| where the infimum is taken over all polynomials P(x) of degree at most n − 1. The upper and lower bounds for λ_n,p(W,j,x) are obtained for all 0 < p ∞ and J = 0, 1, 2, 3,… for weights W(x) = exp(−Q(x)), where, among other things, Q(x) is bounded in [− A, A], and Q″ is continuous in β(−A, A) for some A > 0. For p = ∞, the lower bounds give a simple proof of local and global Markov-Bernstein inequalities. For p = 2, the results remove some restrictions on Q in Freud's work. The weights considered include W(x) = exp(− ¦x¦^α/2), α > 0, and W(x) = exp(− exp(¦x¦)), > 0.     

Onα-Symmetric Multivariate Characteristic Functions

Ann-dimensional random vector is said to have anα-symmetric distribution,α>0, if its characteristic function is of the form((|u₁|^α+…+|u_n|^α)^1/α). We study the classesΦ_n(α) of all admissible functions: [0, ∞)→ . It is known that members ofΦ_n(2) andΦ_n(1) are scale mixtures of certain primitivesΩ_nandω_n, respectively, and we show thatω_nis obtained fromΩ_2n−1byn−1 successive integrations. Consequently, curious relations between 1- and 2- (or spherically) symmetric distributions arise. An analogue of Askey's criterion gives a partial solution to a question of D. St. P. Richards: If(0)=1,is continuous, lim_t→∞ (t)=0, and⁽²ⁿ⁻²⁾(t) is convex, thenΦ_n(1). The paper closes with various criteria for the unimodality of anα-symmetric distribution.     

Asymptotic expansions in the conditional central limit theorem

Let X_n, n , be i.i.d. with mean 0, variance 1, and E(¦X_n¦^r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√n Σ_{i = 1}ⁿ X_i t¦B) can be approximated by a modified Edgeworth expansion up to order o(1/n^{(r − 2)/2)}), if the distances of the set B from the σ-fields σ(X₁, …, X_n) are of order O(1/n^{(r − 2)/2)}(lg n)^β), where β < −(r − 2)/2 for r and β < −r/2 for r . An example shows that if we replace β < −(r − 2)/2 by β = −(r − 2)/2 for r (β < −r/2 by β = −r/2 for r ) we can only obtain the approximation order O(1/n^{(r − 2)/2)}) for r (O(lg lgn/n^{(r − 2)/2)}) for r ).     

On the Strong Law of Large Numbers and the Law of the Logarithm for Weighted Sums of Independent Random Variables with Multidimensional Indices

Let (X, X ; ^d} be a field of independent identically distributed real random variables, 0 < p < 2, and {a _, ; ( , ) ^d × ^d, ≤ } a triangular array of real numbers, where ^d is the d-dimensional lattice. Under the minimal condition that sup _, |a _, | < ∞, we show that | |^{− 1/p} ∑ _≤ a _, X → 0 a.s. as | | → ∞ if and only if E(|X|^p(L|X|)^{d − 1}) < ∞ provided d ≥ 2. In the above, if 1 ≤ p < 2, the random variables are needed to be centered at the mean. By establishing a certain law of the logarithm, we show that the Law of the Iterated Logarithm fails for the weighted sums ∑ _≤ a _, X under the conditions that EX = 0, EX² < ∞, and E(X²(L|X|)^{d − 1}/L₂|X|) < ∞ for almost all bounded families {a _, ; ( , ) ^d × ^d, ≤ of numbers.     

Tractability of Multivariate Integration for Weighted Korobov Classes

We study the worst-case error of multivariate integration in weighted Korobov classes of periodic functions of d coordinates. This class is defined in terms of weights γ_j which moderate the behavior of functions with respect to successive coordinates. We study two classes of quadrature rules. They are quasi-Monte Carlo rules which use n function values and in which all quadrature weights are 1/n and rules for which all quadrature weights are non-negative. Tractability for these two classes of quadrature rules means that the minimal number of function values needed to guarantee error in the worst-case setting is bounded by a polynomial in d and ⁻¹. Strong tractability means that the bound does not depend on d and depends polynomially on ⁻¹. We prove that strong tractability holds iff ∑^∞_j=1 γ_j<∞, and tractability holds iff lim sup_d→∞ ∑^d_j=1 γ_j/log d<∞. Furthermore, strong tractability or tractability results are achieved by the relatively small class of lattice rules. We also prove that if ∑^∞_j=1 γ^1/α_j<∞, where α measures the decay of Fourier coefficients in the weighted Korobov class, then for d1, n prime and δ>0 there exist lattice rules that satisfy an error bound independent of d and of order n^−α/2+δ. This is almost the best possible result, since the order n^−α/2 cannot be improved upon even for d=1. A corresponding result is deduced for weighted non-periodic Sobolev spaces: if ∑^∞_j=1 γ^1/2_j<∞, then for d1, n prime and δ>0 there exist shifted lattice rules that satisfy an error bound independent of d and of order n^−1+δ. We also check how the randomized error of the (classical) Monte Carlo algorithm depends on d for weighted Korobov classes. It turns out that Monte Carlo is strongly tractable iff ∑^∞_j=1 log γ_j<∞ and tractable iff lim sup_d→∞ ∑^d_j=1 log γ_j/log d<∞. Hence, in particular, for γ_j=1 we have the usual Korobov space in which integration is intractable for the two classes of quadrature rules in the worst-case setting, whereas Monte Carlo is strongly tractable in the randomized setting.     

Approximation ofk-Monotone Functions

It is shown that an algebraic polynomial of degree k−1 which interpolates ak-monotone functionfatkpoints, sufficiently approximates it, even if the points of interpolation are close to each other. It is well known that this result is not true in general for non-k-monotone functions. As an application, we prove a (positive) result on simultaneous approximation of ak-monotone function and its derivatives inL_p, 0<p<1, metric, and also show that the rate of the best algebraic approximation ofk-monotone functions (with bounded (k−2)nd derivatives inL_p, 1<p<∞, iso(n^−k/p).     

Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case

In a sequence ofn independent random variables the pdf changes fromf(x, 0) tof(x, 0 + δv_n⁻¹) after the firstnλ variables. The problem is to estimateλ (0, 1 ), where 0 and δ are unknownd-dim parameters andv_n → ∞ slower thann^1/2. Let_n denote the maximum likelihood estimator (mle) ofλ. Analyzing the local behavior of the likelihood function near the true parameter values it is shown under regularity conditions that ifn_n²(− λ) is bounded in probability asn → ∞, then it converges in law to the timeT_(δjδ)^1/2 at which a two-sided Brownian motion (B.M.) with drift1/2(δ′Jδ)^1/2ton(−∞, ∞) attains its a.s. unique minimum, whereJ denotes the Fisher-information matrix. This generalizes the result for small change in mean of univariate normal random variables obtained by Bhattacharya and Brockwell (1976,Z. Warsch. Verw. Gebiete37, 51–75) who also derived the distribution ofT_μ forμ > 0. For the general case an alternative estimator is constructed by a three-step procedure which is shown to have the above asymptotic distribution. In the important case of multiparameter exponential families, the construction of this estimator is considerably simplified.     

Average CaseL∞-Approximation in the Presence of Gaussian Noise

We consider the average caseL_∞-approximation of functions fromC^r([0, 1]) with respect to ther-fold Wiener measure. An approximation is based onnfunction evaluations in the presence of Gaussian noise with varianceσ²>0. We show that the n th minimal average error is of ordern^{−(2r+1)/(4r+4)} ln^1/2 n, and that it can be attained either by the piecewise polynomial approximation using repetitive observations, or by the smoothing spline approximation using non-repetitive observations. This completes the already known results forL_q-approximation withq<∞ andσ0, and forL_∞-approximation withσ=0.     

On piecewise-polynomial approximation of functions with a bounded fractional derivative in an Lp-norm

We study the error in approximating functions with a bounded (r + α)th derivative in an L_p-norm. Here r is a nonnegative integer, α ε [0, 1), and ƒ^{(r + α)} is the classical fractional derivative, i.e., ƒ^{(r + α)}(y) = ∝₀¹, ^α d(ƒ^(r)(t)). We prove that, for any such function ƒ, there exists a piecewise-polynomial of degree s that interpolates ƒ at n equally spaced points and that approximates ƒ with an error (in sup-norm) ƒ^{(r + α)}_p O(n^{−(r+α−1/p}). We also prove that no algorithm based on n function and/or derivative values of ƒ has the error equal ƒ^{(r + α)}_p O(n^{−(r+α−1/p}) for any ƒ. This implies the optimality of piecewise-polynomial interpolation. These two results generalize well-known results on approximating functions with bounded rth derivative (α = 0). We stress that the piecewise-polynomial approximation does not depend on α nor on p. It does not depend on the exact value of r as well; what matters is an upper bound s on r, s r. Hence, even without knowing the actual regularity (r, α, and p) of ƒ, we can approximate the function ƒ with an error equal (modulo a constant) to the minimal worst case error when the regularity were known.     

On the completeness of the system of function {e sin λnx}n = 1

In this paper, the biorthogonal system corresponding to the system {e^−αnx sin nx}_{n = 1}^∞ is represented in an appropriate form so that it is possible to obtain sufficiently good estimates of its norm. Then, by the stability of a completeness property we prove that the system of functions {e^−αλ_nx sin λ_nx}_{n = 1}^∞ is complete.     

On deviations between empirical and quantile processes for mixing random variables

Let {X_n} be a strictly stationary φ-mixing process with Σ_j=1^∞ φ^1/2(j) < ∞. It is shown in the paper that if X₁ is uniformly distributed on the unit interval, then, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = (O(n^−3/4(log n)^1/2(log log n)^1/4) a.s., where F_n and F_n⁻¹(t) denote the sample distribution function and tth sample quantile, respectively. In case {X_n} is strong mixing with exponentially decaying mixing coefficients, it is shown that, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)^1/2(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)(log log n)^1/4) a.s. The results are further extended to general distributions, including some nonregular cases, when the underlying distribution function is not differentiable. The results for φ-mixing processes give the sharpest possible orders in view of the corresponding results of Kiefer for independent random variables.