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.

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Acknowledgments

The authors are grateful to the Departments of Mathematics of King’s College London and of the University of Rennes 1 (France) for the financial support. The second author (D.Y.) acknowledges also the support and hospitality of the Isaac Newton Institute for Mathematical Sciences (Cambridge University, UK), where a part of this work was done during the program Periodic and Ergodic Spectral Problems.

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Correspondence to Alexander Pushnitski.

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Communicated by Pencho Petrushev.

Appendix A. Besov and Besov–Lorentz spaces

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 \(\Lambda _{\alpha }\subset \mathfrak {B}^{\alpha }_{1/\alpha ,\infty }\).

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Pushnitski, A., Yafaev, D. Best Rational Approximation of Functions with Logarithmic Singularities. Constr Approx 46, 243–269 (2017). https://doi.org/10.1007/s00365-016-9347-1

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Keywords

  • Hankel operators
  • Asymptotics of singular values
  • Logarithmic singularities
  • Rational approximation

Mathematics Subject Classification

  • 41A20
  • 47B06
  • 47B35