Average B-width and infinite dimensional G-width of some smooth function classes on the line

In this paper, we get the exact values of average σ-B width and infinite dimensional σ-G width of Sobolev class B^r _p(R) in the metric L_p(R) (1≤p≤∞) and obtain the exact (σ∈N) and strong asymptotic (σ>1) results of infinite dimensional σ-G widths of Sobolev-Wiener class W^r _pq (R) in the metric L_q(R) and its dual case W^r _p(R) in the metric L_qp(R) (1≤q≤p≤∞).     

Average B-width and infinite dimensional G-width of some smooth function classes on the line

In this paper, we get the exact values of average σ-B width and infinite dimensional σ-G width of Sobolev class B^r _p(R) in the metric L_p(R) (1≤p≤∞) and obtain the exact (σ∈N) and strong asymptotic (σ>1) results of infinite dimensional σ-G widths of Sobolev-Wiener class W^r _pq (R) in the metric L_q(R) and its dual case W^r _p(R) in the metric L_qp(R) (1≤q≤p≤∞). Supported by the National Natural Science Foundation of China (No. 19671012) and by the Doctoral Programme Foundation of Institution of Higher Education of National Education Committee of China.     

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.     

The Fine Structure of the n-Widths of H-Spaces in Lq(−1, 1)

The n-widths of the unit ball A_p of the Hardy space H^p in L_q( −1, 1) are determined asymptotically. It is shown that for 1 ≤ q < p ≤∞ there exist constants k₁ and k₂ such that [formula]≤ d_n(A_p, L_q(−1, 1)),dⁿ(A_p, L_q(−1, 1)), δ_n(A_p, L_q(−1, 1))[formula]where d_n, dⁿ, and δ_n denote the Kolmogorov, Gel′fand and linear n-widths, respectively. This result is an improvement of estimates previously obtained by Burchard and Höllig and by the author.     

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.     

Linear widths of Hölder-Nikol'skii classes of periodic functions of several variables

We consider the linear widths _N (W _p ^r (Tⁿ), L_q) and _N (H _p ^r (Tⁿ), L_q) of the classesW _p ^r (Tⁿ) andH _p ^r (Tⁿ) of periodic functions of one or several variables in the spaceL _q. For the Sobolev classesW _p ^r (Tⁿ) of functions of one or several variables, we state some well-known results without proof; for the Hölder-Nikol'skii classesH _p ^r (Tⁿ), we state some well-known results, prove some new results, and present some previously unpublished proofs.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 189–199, February, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00237 and by the International Science Foundation under grant No. MP1000.     

The n-width of the unit ball of H

Let E be a compact subset of the open unit disc Δ and let H^q be the Hardy space of analytic functions f on Δ for which stf¦^q has a harmonic majorant. We determine the value of the Kolmogorov, Gel'fand, and linear n-widths in L^p(E, μ) of the restriction to E of the unit ball of H^q when p q or when 1 q < p < ∞ and E is “small”.     

RELATIVE WIDTH OF SMOOTH CLASSES OF MULTIVARIATE PERIODIC FUNCTIONS WITH RESTRICTIONS ON ITERATED LAPLACE DERIVATIVES IN THE L2-METRIC

For two subsets W and V of a Banach space X, let Kn(W, V, X) denote the relative Kolmogorov n-width of W relative to V defined by Kn (W, V, X) := inf sup Ln f∈W g∈V∩Ln inf ‖f-g‖x,where the infimum is taken over all n-dimensional linear subspaces Ln of X. Let W2(△r) denote the class of 2w-periodic functions f with d-variables satisfying ∫[-π,π]d |△rf(x)|2dx ≤ 1,while △r is the r-iterate of Laplace operator △. This article discusses the relative Kolmogorov n-width of W2(△r) relative to W2(△r) in Lq([-r, πr]d) (1 ≤ q ≤∞), and obtain its weak asymptotic result.     

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.     

Global nonparametric estimation of conditional quantile functions and their derivatives

Let (X, Y) be a random vector such that X is d-dimensional, Y is real valued, and θ(X) is the conditional αth quantile of Y given X, where α is a fixed number such that 0 < α < 1. Assume that θ is a smooth function with order of smoothness p > 0, and set r = (p − m)/(2p + d), where m is a nonnegative integer smaller than p. Let T(θ) denote a derivative of θ of order m. It is proved that there exists estimate of T(θ), based on a set of i.i.d. observations (X₁, Y₁), …, (X_n, Y_n), that achieves the optimal nonparametric rate of convergence n^−r in L_q-norms (1 ≤ q < ∞) restricted to compacts under appropriate regularity conditions. Further, it has been shown that there exists estimate of T(θ) that achieves the optimal rate (n/log n)^−r in L_∞-norm restricted to compacts.     

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.     

Tests for the mean direction of the Langevin distribution with large concentration parameter

In this paper we study the asymptotic behaviors of the likelihood ratio criterion (T_L^(s)), Watson statistic (T_W^(s)) and Rao statistic (T_R^(s)) for testing H_0s: μ (a given subspace) against H_1s: μ , based on a sample of size n from a p-variate Langevin distribution M_p(μ, κ) when κ is large. For the case when κ is known, asymptotic expansions of the null and nonnull distributions of these statistics are obtained. It is shown that the powers of these statistics are coincident up to the order κ⁻¹. For the case when κ is unknown, it is shown that T_R^(s) T_L^(s) T_W^(s) in their powers up to the order κ⁻¹.     

Product properties of Hilbert transforms

Let fεL^p(R), gεL^q(R) with 1<p<∞, 1<q<∞ and let Hf, Hg be their respective Hilbert transforms. We give a simple proof of the identity Hf · Hg − f · G = H(f · Hg + g · Hf) a.e. and of its inverse in the case (1/p) + (1/q) 1 which includes the cases already considered by Cossar and Tricomi. We next derive applications, especially to boundary values of analytic functions.     

Centres of Convex Sets inLMetrics

It is shown that for each convex bodyARⁿthere exists a naturally defined family _AC(Sⁿ⁻¹) such that for everyg _A, and every convex functionf: R→Rthe mappingy∫_Sⁿ⁻¹ f(g(x)−y, x) dσ(x) has a minimizer which belongs toA. As an application, approximation of convex bodies by balls with respect toL^pmetrics is discussed.     

W(R)-splines

In [3] Golomb describes, for 1 < p < ∞, the H^r,p(R)-extremal extension F_* of a function ƒ:E → R (i.e., the H^r,p-spline with knots in E) and studies the cone H_*E^r,p of all such splines. We study the problem of determining when F_* is in W^r,p ≡ H^r,p ∩ L^p. If F_* ε W^r,p, then F_* is called a W^r,p-spline, and we denote by W_*E^r,p the cone of all such splines. If E is quasiuniform, then F_* ε W^r,p if and only if {ƒ(t_i)}_tiεE ε l^p. The cone W_*E^r,p with E quasiuniform is shown to be homeomorphic to l^p. Similarly, H_*E^r,p is homeomorphic to h^r,p. Approximation properties of the W^r,p-splines are studied and error bounds in terms of the mesh size ¦ E ¦ are calculated. Restricting ourselves to the case p = 2 and to quasiuniform partitions E, the second integral relation is proved and better error bounds in terms of ¦ E ¦ are derived.     

Kolmogorov and Linear Widths of Weighted Sobolev-Type Classes on a Finite Interval, II

Let I be a finite interval, r and ρ(t)=dist{t, ∂I}, tI. Denote by Δ^s₊W^r_p, α, 0α<∞, the class of functions x on I with the seminorm x^(r)ρ^α_Lp1 for which Δ^s_τx, τ>0, is nonnegative on I. We obtain two-sided estimates of the Kolmogorov widths d_n(Δ^s₊W^r_p, α)_Lq and of the linear widths d_n(Δ^s₊W^r_p, α)^lin_Lq, s=0, 1, …, r+1.     

On Lp-Generalization of a Theorem of Adamyan, Arov, and Kreın

The paper deals with problems relating to the theory of Hankel operators. Let G be a bounded simple connected domain with the boundary Γ consisting of a closed analytic Jordan curve. Denote by _n,p(G), 1p<∞, the class of all meromorphic functions on G that can be represented in the form h=β/α, where β belongs to the Smirnov class E_p(G), α is a polynomial degree at most n, α0. We obtain estimates of s-numbers of the Hankel operator A_f constructed from fL_p(Γ), 1p<∞, in terms of the best approximation Δ_n,p of f in the space L_p(Γ) by functions belonging to the class _n,p(G).     

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).     

Mappings of finite distortion: the degree of regularity

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRⁿ, n2, and such that exp(βK(x))L_loc¹(Ω), β>0. We show that there exist two universal constants c₁(n),c₂(n) with the following property: Let f be a mapping in W_loc^1,1(Ω,Rⁿ) with |Df(x)|ⁿK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L¹ log^−c₁(n)βL. Then automatically J(x,f) is locally in L¹ log^c₂(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings.