### $\mathcal{O}$-regularly varying functions in approximation theory.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Refining the idea used in [24] and employing very careful computation, the present paper shows that for 0 < p ≤ ∞ and k ≥ 1, there exists a function $f\in {C}_{[-1,1]}^{k}$, with ${f}^{\left(k\right)}\left(x\right)\ge 0$ for x ∈ [0,1] and ${f}^{\left(k\right)}\left(x\right)\le 0$ for x ∈ [-1,0], such that lim supn→∞ (en(k)(f)p) / (ωk+2+[1/p](f,n-1)p) = + ∞ where ${e}_{n}^{\left(k\right)}{\left(f\right)}_{p}$ is the best approximation of degree n to f in ${L}^{p}$ by polynomials which are comonotone with f, that is, polynomials P so that ${P}^{\left(k\right)}\left(x\right){f}^{\left(k\right)}\left(x\right)\ge 0$ for all x ∈ [-1,1]. This theorem, which is a particular case of a more general one, gives a complete solution...

The paper gives such an iterative method for special Chebyshev approxiamtions that its order of convergence is $\ge 2$. Somewhat comparable results are found in [1] and [2], based on another idea.

This is mainly a review paper, concerned with some applications of the concept of Nonlinear Approximation to adaptive convex minimization. At first, we recall the basic ideas and we compare linear to nonlinear approximation for three relevant families of bases used in practice: Fourier bases, finite element bases, wavelet bases. Next, we show how nonlinear approximation can be used to design rigorously justified and optimally efficient adaptive methods to solve abstract minimization problems in...

For sequences of rational functions, analytic in some domain, a theorem of Montel’s type is proved. As an application, sequences of rational functions of the best ${L}_{p}$-approximation with an unbounded number of finite poles are considered.