Best Rational Approximation of Functions with Logarithmic Singularities

Abstract

We consider functions $\omega$ on the unit circle ${\mathbb T}$ with a finite number of logarithmic singularities. We study the approximation of $\omega$ by rational functions and find an asymptotic formula for the distance in the ${{\mathrm{BMO}}}$-norm between $\omega$ and the set of rational functions of degree n as $n\rightarrow \infty$. Our approach relies on the Adamyan–Arov–Krein theorem and on the study of the asymptotic behavior of singular values of Hankel operators.

Appendix A. Besov and Besov–Lorentz spaces

Here for completeness we recall the definitions of the Besov class $B_{p,p}^{1/p}$ and the Besov–Lorentz class $\mathfrak B_{p,\infty }^{1/p}$ on ${\mathbb T}$. The parameter $p>0$ is arbitrary. We refer to the books [16] (see Section 6.4 and Appendix 2) and [24] for more details.

Let $w\in C_0^\infty ({\mathbb R})$ be a function with the properties $w\ge 0$, ${{\mathrm{supp}}}w=[1/2,2]$, and

$$\begin{aligned} \sum _{n=0}^\infty w(t/2^n)=1, \quad \forall t\ge 1. \end{aligned}$$

Let f be a distribution on $L^1 ({\mathbb T})$ with the Fourier coefficients $\widehat{f}(j)$, ${j\in {\mathbb Z}}$. For $n\in {\mathbb Z}$, let us denote by $f_n$ the polynomial

$$\begin{aligned} f_n(\mu )=\sum _{j\in {\mathbb Z}} w(\pm j/2^{|n|})\widehat{f}(j)\mu ^j, \quad \mu \in {\mathbb T}, \quad \pm n >0, \end{aligned}$$

and let $f_0(\mu )=\widehat{f}(1)\mu + \widehat{f}(0) + \widehat{f}(-1)\overline{\mu }$. The Besov class $B_{p,p}^{1/p}$ is defined by the condition

$$\begin{aligned} \sum _{n\in {\mathbb Z}} 2^{|n|}||f_n||_{L^p}^p <\infty . \end{aligned}$$

(A.1)

By definition, $f\in \mathfrak B_{p,\infty }^{1/p}$ if and only if

$$\begin{aligned} \sup _{t>0}t^p \sum _{n\in {\mathbb Z}} 2^{|n|} m (\{\mu \in {\mathbb T}: |f_n(\mu )|>t\})<\infty , \end{aligned}$$

which is the “weak version” of the condition (A.1). We have

$$\begin{aligned} B^{1/ p}_{p,p} \subset \mathfrak {B}^{1/ p}_{p,\infty }\subset B^{1/ q}_{q,q}, \quad \forall q>p. \end{aligned}$$

The Hölder–Zygmund class $\Lambda _\alpha$, $\alpha >0$, is defined in terms of the difference operator

$$\begin{aligned} (\Delta _{\tau }f) (\mu )= f(\tau \mu )- f(\mu ), \quad \tau \in {\mathbb T}. \end{aligned}$$

By definition, $f\in \Lambda _\alpha$ if and only if

$$\begin{aligned} \Vert (\Delta _{\tau }^n f )(\mu )\Vert _{L^\infty }\le C |\tau -1|^\alpha , \end{aligned}$$

where n is an arbitrary integer such that $n>\alpha$. Observe that $\Lambda _\alpha$ coincides with the Hölder class $C^\alpha$ if $\alpha$ is not an integer and $C^\alpha \subset \Lambda _\alpha$ if $\alpha$ is an integer. We also note that $\Lambda _{\alpha }\subset \mathfrak {B}^{\alpha }_{1/\alpha ,\infty }$.