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        "description": "The conclusion is uniform approximation on a fixed finite real interval, with polynomial choice depending on the function, interval and requested error. It does not make every continuous function analytic, give one fixed polynomial valid to arbitrary accuracy, guarantee uniform approximation on the entire real line or establish the later abstract Stone–Weierstrass theorem. Selected German reprint pages were inspected; full proof certification is not awarded.",
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          "citation": "Karl Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen reeller Argumente, collected-works reprint of communications of 9 and 30 July 1885, pp.1–37",
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      "statement": "# Weierstrass: uniform polynomial approximation on finite real intervals\n\nWeierstrass showed how continuous real functions can be approximated uniformly on a fixed finite interval by polynomials. An integral smoothing construction and truncation of an analytic expansion connect continuous functions with simpler algebraic expressions.\n\nThe approximation error is controlled over the interval, rather than separately at each point. After smoothing a continuous function, a finite part of an analytic expansion provides a polynomial with the required accuracy. The paper also connects the construction with polynomial series converging uniformly on finite intervals; that is the same original contribution family. Later abstract approximation and practical error-rate algorithms retain separate credit.\n\n## Scientific scope\n\nThe conclusion is uniform approximation on a fixed finite real interval, with polynomial choice depending on the function, interval and requested error. It does not make every continuous function analytic, give one fixed polynomial valid to arbitrary accuracy, guarantee uniform approximation on the entire real line or establish the later abstract Stone–Weierstrass theorem. Selected German reprint pages were inspected; full proof certification is not awarded.\n\n## People and systems\n\nKarl Weierstrass\n\n\n\n## Original sources\n\n- [Karl Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen reeller Argumente, collected-works reprint of communications of 9 and 30 July 1885, pp.1–37](https://history-of-approximation-theory.com/fpapers/wei5.pdf)\n",
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