Smooth approximation of Lipschitz functions on Riemannian manifolds

We show that for every Lipschitz function f defined on a separable Riemannian manifold M (possibly of infinite dimension), for every continuous , and for every positive number r>0, there exists a C^∞ smooth Lipschitz function such that |f(p)−g(p)|?ε(p) for every p∈M and Lip(g)?Lip(f)+r. Consequently, every separable Riemannian manifold is uniformly bumpable. We also present some applications of this result, such as a general version for separable Riemannian manifolds of Deville-Godefroy-Zizler's smooth variational principle.     

Algebraic-polynomial approximation of functions satisfying a Lipschitz condition

For functionsf(x) ε KH^(α) [satisfying the Lipschitz condition of order α (0 < α < 1) with constant K on [?1, 1], the existence is proved of a sequence P_n (f; x) of algebraic polynomials of degree n = 1, 2,..., such that $$|f(x) - P_{n - 1} (f;x)| \leqslant \mathop {\sup }\limits_{f \in KH^{(\alpha )} } E_n (f)[(1 - x^2 )^{a/2} + o(1)],$$ when n → ∞, uniformly for x ε [?1, 1], where E_n(f) is the best approximation off(x) by polynomials of degree not higher than n.     

New constructions for local approximation of Lipschitz functions.II

In this paper, some properties of the set-valued mapping _Dαf(.)$D_{\alpha }f(.)$ connected with the new approximation method of a function f(.)$f(.)$ defined in the first part of the article are given. Continuity and Lipschitz properties of _Dαf(.)$D_{\alpha }f(.)$ are formulated. A continuous extension of the Clarke subdifferential of any function represented as a difference of two convex functions is given. For the convex case, the set-valued mapping _Dαf(.)$D_{\alpha }f(.)$ is similar to the ε$\epsilon$-subdifferential mapping.     

Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group

We characterize intrinsic Lipschitz functions as maps which can be approximated by a sequence of smooth maps, with pointwise convergent intrinsic gradient. We also provide an estimate of the Lipschitz constant of an intrinsic Lipschitz function in terms of the $L^{\infty }$ -norm of its intrinsic gradient.     

A simple proof of the approximation by real analytic Lipschitz functions

A theorem in Azagra et al. (preprint) [1] asserts that on a real separable Banach space with separating polynomial every Lipschitz function can be uniformly approximated by real analytic Lipschitz function with a control over the Lipschitz constant. We give a simple proof of this theorem.     

Complexity of approximate realizations of Lipschitz functions by schemes in continuous bases

We show that any function satisfying the Lipschitz condition on a given closed interval can be approximately computed by a scheme (nonbranching program) in the basis composed of functions $$x - y, \left| x \right|, x*y = \min (\max (x,0),1)\min (\max (y,0),1),$$ and all constants from the closed interval [0, 1]; here the complexity of the scheme is $O\left( {{1 \mathord{\left/ {\vphantom {1 {\sqrt \varepsilon }}} \right. \kern-0em} {\sqrt \varepsilon }}} \right)$ , where ? is the accuracy of the approximation. This estimate of complexity, is in general, order-sharp.     

Complexity of approximation of functions of few variables in high dimensions

In DeVore et al. (2011) [7] we considered smooth functions on [0,1]^N which depend on a much smaller number of variables ? or continuous functions which can be approximated by such functions. We were interested in approximating those functions when we can calculate point values at points of our choice. The number of points we needed for non-adaptive algorithms was higher than that in the adaptive case. In this paper we improve on DeVore et al. (2011) [7] and show that in the non-adaptive case one can use the same number of points (up to a multiplicative constant depending on ?) that we need in the adaptive case.     

An approximation property of harmonic functions in Lipschitz domains and some of its consequences

An extension of an inequality of J. B. Garnett (1979), with improvements by B. E. J. Dahlberg (1980), on an approximation property of harmonic functions is proved. The weighted inequality proved here was suggested by the work of J. Pipher (1993) and it implies an extension of a result of S. Y. A. Chang, J. Wilson and T. Wolff (1985) and C. Sweezy (1991) on exponential square integrability of the boundary values of solutions to second-order linear differential equations in divergence form. This implies a solution of a problem left open by R. Bañuelos and C. N. Moore (1989) on sharp estimates for the area integral of harmonic functions in Lipschitz domains.

     

Complexity of approximation problems

We consider some aspects of optimal encoding and renewal related to the problem of complexity of the ε-definition of functions posed by Kolmogorov in 1962. We present some estimates for the ε-complexity of the problem of renewal of functions in the uniform metric and Hausdorff metric.     

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).      

The pointwise complete asymptotic expansion for the approximation of Lipschitz functions by Hermite-Fejer interpolation polynomials

The pointwise complete asymptotic expansion is derived for the approximation of Lipschitz functions by Hermite-Fejér interpolation polynomials based on the Chebyshev polynomials of the first kind.     

Parabolic Lipschitz truncation and caloric approximation

We develop an improved version of the parabolic Lipschitz truncation, which allows qualitative control of the distributional time derivative and the preservation of zero boundary values. As a consequence, we establish a new caloric approximation lemma. We show that almost p-caloric functions are close to p-caloric functions. The distance is measured in terms of spatial gradients as well as almost uniformly in time. Both results are extended to the setting of Orlicz growth.