Lp - Logistic discrimination

The paper presents L_p-solution for logistic discriminant function in dichotomous as well as in the polychotomous problem.     

Local Lp-saturation of positive linear convolution operators

The approximation of unbounded functions by positive linear operators under multiplier enlargement is investigated. It is shown that a very wide class of positive linear operators can be used to approximate functions with arbitrary growth on the real line. Estimates are given in terms of the usual quantities which appear in the Shisha-Mond theorem. Examples are provided.     

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.     

Asymptotics for Lp extremal polynomials on the unit circle

Let p > 1, and dμ a positive finite Borel measure on the unit circle Γ: = {z ε C: ¦z¦ = 1}. Define the monic polynomial φ_{n, p}(z)=zⁿ+…εP_n >(the set of polynomials of degree at most n) satisfying

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. Under certain conditions on dμ, the asymptotics of φ_{n, p}(z) for z outside, on, or inside Γ are obtained (cf. Theorems 2.2 and 2.4). Zero distributions of φ_{n, p} are also discussed (cf. Theorems 3.1 and 3.2).     

Degree of Lp-approximation by certain positive convolution operators

The degree of L_p-approximation for a class of positive convolution operators is investigated. Recent results of De Vore, Bojanic, and Shisha for the uniform approximation by these operators and the K-functional of Peetre are employed to obtain the degree of approximation in terms of the integral modulus of smoothness.     

Best Lp approximate solutions of nonlinear integrodifferential equations

In this paper best L_p approximate solutions are shown to exist for a wide class of integrodifferential equations. Using approximation theory techniques, a local existence theorem for solutions is established, and the convergence of the best approximate solutions to a solution is shown.     

Generalized rational approximation in interpolating subspaces of Lp(μ)

We extend the geometric approach of Cheney and Loeb in [2] to the problem of approximation in L_p(μ) by “admissable” generalized rational functions. We obtain a characterization for locally best approximations and find the interpolating condition sufficient for their local unicity. Our results are comparable to those for the linear approximation problem as investigated by Singer and Ault, Deutsch, Morris, and Olson.     

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.     

The convergence of the best discrete linear Lp approximation as p → 1

It is well known that the best discrete linear L_p approximation converges to a special best Chebyshev approximation as p → ∞. In this paper it is shown that the corresponding result for the case p → 1 is also true. Furthermore, the special best L₁ approximation obtained as the limit is characterized as the unique solution of a nonlinear programming problem on the set of all L₁ solutions.     

关于L_p球的几个新不等式和性质(英文)

本文研究了L_p球的相关问题.利用对偶混合体积、球面Radon变换和Fourier变换的方法,获得了关于L_p球的几个新不等式和性质,其中一个不等式与著名的最大切片猜想有关.     

Lp Markov–Bernstein Inequalities on All Arcs of the Circle

Let 0<p<∞ and 0α<β2π. We prove that for n1 and trigonometric polynomials s_n of degree n, we have

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cn^p ∫^β_α |s_n(θ)|^p dθ, where c is independent of α, β, n, s_n. The essential feature is the uniformity in [α,β] of the estimate and the fact that as [α,β] approaches [0,2π], we recover the L_p Markov inequality. The result may be viewed as the complete L_p form of Videnskii's inequalities, improving earlier work of the second author.     

Minimal Lp projections by Fourier, Taylor, and Laurent series

In appropriate function space settings, it is proved that the Fourier, Taylor, and Laurent series projections are minimal in all L_p norms (1 p ∞). This result unifies and extends known results for the Fourier, Taylor, and Laurent projections in L_∞ and for the Fourier projection in L₁. The proof is based on a generalisation of a kernel summation formula due to Berman.     

The L2 norm of the approximation error for Bernstein polynomials

Wassily Hoeffding (J. Approximation Theory 4 (1971), 347–356) obtained a convergence rate for the L₁ norm of the approximation error, using Bernstein polynomials for a wide class of functions. Here, by a different method of proof, a similar result is obtained for the L₂ norm.     

Multivariate Lp-error estimates for positive linear operators via the first-order τ-modulus

Let M(I) {ƒ:ƒ is a real-valued function that is bounded and measurable on an m-dimensional compact interval I} and let L: M(I) → M(I) be a multivariate positive linear operator. The aim of this paper is to give estimates for the approximation error's L_p-norm ƒ − Lƒ_p using the so-called averaged modulus of smoothness or τ-modulus of first order.     

On the stationary Lp-approximation power to derivatives by radial basis function interpolation

In this paper, we consider approximation to derivatives of a function by using radial basis function interpolation. Most of well-known theories for this problem provide error analysis in terms of the so-called native space, say C_φ. However, if a basis function φ is smooth, the space C_φ is extremely small. Thus, the purpose of this study is to extend this result to functions in the homogenous Sobolev space.