Optimality Conditions for Rational Minimax Approximations: Bridging Ruttan's Criteria to Dual-Based Methods
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911431753990144 |
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| author | Zhang, Lei-Hong |
| author_facet | Zhang, Lei-Hong |
| contents | This paper presents a theoretical discussion on Ruttan's optimality conditions for rational minimax approximations in discrete and continuum settings, integrating analytical foundations with computational practice. We develop extended second-order optimality criteria for the discrete case, demonstrating that Ruttan's sufficient condition for global solutions [Ruttan, {Constr. Approx.}, 1 (1985), 287-296] becomes necessary when the number of extreme points is minimal. Our analysis further uncovers fundamental relationships between these conditions and the dual-based {d-Lawson} method [L.-H. Zhang et al., {Math. Comp.}, 94 (2025), 2457-2494], proving that strong duality in {d-Lawson} ensures simultaneous satisfaction of both Ruttan's and Kolmogorov's criteria. Additionally, we show that minimax approximants on a continuum satisfying Ruttan's sufficient global optimality can be captured through discrete minimax approximations at properly chosen boundary points, thereby enabling efficient computation of minimax approximants on a continuum using discrete methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07862 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimality Conditions for Rational Minimax Approximations: Bridging Ruttan's Criteria to Dual-Based Methods Zhang, Lei-Hong Numerical Analysis 41A52, 41A50, 65D15, 49K35, 41A20 This paper presents a theoretical discussion on Ruttan's optimality conditions for rational minimax approximations in discrete and continuum settings, integrating analytical foundations with computational practice. We develop extended second-order optimality criteria for the discrete case, demonstrating that Ruttan's sufficient condition for global solutions [Ruttan, {Constr. Approx.}, 1 (1985), 287-296] becomes necessary when the number of extreme points is minimal. Our analysis further uncovers fundamental relationships between these conditions and the dual-based {d-Lawson} method [L.-H. Zhang et al., {Math. Comp.}, 94 (2025), 2457-2494], proving that strong duality in {d-Lawson} ensures simultaneous satisfaction of both Ruttan's and Kolmogorov's criteria. Additionally, we show that minimax approximants on a continuum satisfying Ruttan's sufficient global optimality can be captured through discrete minimax approximations at properly chosen boundary points, thereby enabling efficient computation of minimax approximants on a continuum using discrete methods. |
| title | Optimality Conditions for Rational Minimax Approximations: Bridging Ruttan's Criteria to Dual-Based Methods |
| topic | Numerical Analysis 41A52, 41A50, 65D15, 49K35, 41A20 |
| url | https://arxiv.org/abs/2602.07862 |