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The error of Chebyshev approximations on shrinking domains

Tobias Jawecki

TL;DR

This work establishes precise asymptotic error descriptions for rational Chebyshev approximants on shrinking domains, showing that the pointwise error converges to a Chebyshev polynomial on $K$ multiplied by the Padé-leading coefficient $a_{mn}$ and that the uniform error scales with the Chebyshev constant $t_{m+n+1}$. It reveals that, for sufficiently small domains, Chebyshev approximants behave as interpolatory best approximants with interpolation nodes approaching scaled Chebyshev nodes, and extends these results to complex, real, and unitary exponential settings. The paper also derives explicit error ratios between Chebyshev and Padé approximants and connects the shrinking-domain results to classical Padé convergence results, with concrete implications for the exponential function and related approximations. Overall, the findings provide a cohesive framework linking Padé, Chebyshev, and interpolatory best approximation theories in shrinking-domain regimes, with clear guidance for selecting nodes and understanding asymptotic errors.

Abstract

Previous works show convergence of rational Chebyshev approximants to the Padé approximant as the underlying domain of approximation shrinks to the origin. In the present work, the asymptotic error and interpolation properties of rational Chebyshev approximants are studied in such settings. Namely, the point-wise error of Chebyshev approximants is shown to approach a Chebyshev polynomial multiplied by the asymptotically leading order term of the error of the Padé approximant, and similar results hold true for the uniform error and Chebyshev constants. Moreover, rational Chebyshev approximants are shown to attain interpolation nodes which approach scaled Chebyshev nodes in the limit. Main results are formulated for interpolatory best approximations and apply for complex Chebyshev approximation as well as real Chebyshev approximation to real functions and unitary best approximation to the exponential function.

The error of Chebyshev approximations on shrinking domains

TL;DR

This work establishes precise asymptotic error descriptions for rational Chebyshev approximants on shrinking domains, showing that the pointwise error converges to a Chebyshev polynomial on multiplied by the Padé-leading coefficient and that the uniform error scales with the Chebyshev constant . It reveals that, for sufficiently small domains, Chebyshev approximants behave as interpolatory best approximants with interpolation nodes approaching scaled Chebyshev nodes, and extends these results to complex, real, and unitary exponential settings. The paper also derives explicit error ratios between Chebyshev and Padé approximants and connects the shrinking-domain results to classical Padé convergence results, with concrete implications for the exponential function and related approximations. Overall, the findings provide a cohesive framework linking Padé, Chebyshev, and interpolatory best approximation theories in shrinking-domain regimes, with clear guidance for selecting nodes and understanding asymptotic errors.

Abstract

Previous works show convergence of rational Chebyshev approximants to the Padé approximant as the underlying domain of approximation shrinks to the origin. In the present work, the asymptotic error and interpolation properties of rational Chebyshev approximants are studied in such settings. Namely, the point-wise error of Chebyshev approximants is shown to approach a Chebyshev polynomial multiplied by the asymptotically leading order term of the error of the Padé approximant, and similar results hold true for the uniform error and Chebyshev constants. Moreover, rational Chebyshev approximants are shown to attain interpolation nodes which approach scaled Chebyshev nodes in the limit. Main results are formulated for interpolatory best approximations and apply for complex Chebyshev approximation as well as real Chebyshev approximation to real functions and unitary best approximation to the exponential function.
Paper Structure (17 sections, 10 theorems, 84 equations)

This paper contains 17 sections, 10 theorems, 84 equations.

Table of Contents

  1. Introduction and overview
  2. Rational interpolation
  3. Padé approximation
  4. Continuity of rational interpolants
  5. Error of rational interpolants with scaled nodes
  6. Chebyshev nodes
  7. Interpolatory best approximations
  8. Examples
  9. Real Chebyshev approximation to real functions
  10. Unitary best approximation to the exponential function
  11. Interpolation properties of complex Chebyshev approximation
  12. Error of best approximations
  13. Applications
  14. The error ratio of Chebyshev and Padé approximants
  15. Relation to previous convergence results
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $\{z_{k,\ell}\}_{\ell\in\mathbb{N}}$ denote a given sequence of nodes with $z_{k,\ell}\to z_k$ for $\ell \to \infty$ and $k=0,\ldots,j$. Let $\{g_{\ell}\}_{\ell\in\mathbb{N}}$ denote a sequence of holomorphic functions $g_{\ell}:E\to \mathbb{C}$. Assume $\|g_{\ell} -g \|_{K'}\to 0$ for $\ell\to

Theorems & Definitions (24)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 14 more