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      "statement": "# Jackson: a quantitative speed for approximating Lipschitz functions by polynomials\n\n## How many polynomial terms buy a useful approximation?\n\nKnowing that a curve can be approximated eventually is different from knowing how the error falls as the polynomial degree increases. Jackson adds a quantitative answer for Lipschitz functions. A Lipschitz condition limits how much the function can change when its input moves: the output difference is at most a fixed constant times the input difference. It allows corners, but rules out arbitrarily steep local changes.\n\n## The degree budget must stay in the calculation\n\nJackson constructs degree-at-most-n polynomial approximants whose largest error is bounded by K/n, with K depending on the function and interval. The route first smooths the function using a kernel, a weighted averaging rule. It then replaces the kernel with a polynomial and tracks both approximation error and resulting degree. A small smoothing error alone would be insufficient if the required polynomial degree grew without control.\n\n## A rate is not a published numerical constant\n\nIn an invented classroom example, suppose a valid bound has K equal to two. Degree twenty would then guarantee error at most 0.1, and degree two hundred at most 0.01. These numbers illustrate the formula; they are not constants measured or certified for a particular curve in the dissertation. A real use must establish its own K and account for the interval. A bound is also different from the actual best error achieved.\n\n## A corner is allowed, an infinite slope needs care\n\nThe absolute-value function is Lipschitz: changing x by h changes its absolute value by no more than the absolute value of h. Near zero, the square-root function does not satisfy one fixed Lipschitz bound, because square root of h divided by h grows without bound as h shrinks. That does not mean square root cannot be approximated. It means this particular theorem cannot be applied without checking its assumptions.\n\n## Inspect a statement your agent can reuse\n\nAsk your agent to identify the function, interval, Lipschitz constant, degree limit and meaning of the error before using any rate. It can compare a fitted polynomial with a sample grid, but grid error is not automatically a certified uniform bound. The linked historical scan omits printed page 19 while preserving the theorem statement. The source limitation and the distinction between a published theorem and a fresh proof reconstruction remain visible.\n\n## Scientific scope\n\nThe constant depends on the function and interval; the theorem does not give the same rate for all continuous functions, a sharp best constant or convergence of ordinary Fourier partial sums. The linked archival scan repeats printed page 18 and omits page 19. Its theorem statement is available, but the scan has this gap in the construction. Other dissertation extensions are outside this contribution.\n\n## Sources for this explanation\n\n- [Über die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung](https://history-of-approximation-theory.com/fpapers/jackson1911.pdf): 1911thesis,printed16–22,SatzI and degree-controlled construction;linkedscan repeats18and omits19.\n",
      "summary": "Weierstrass guarantees that a continuous curve can be approximated by polynomials, but does not give one useful speed for all continuous curves. Jackson adds a regularity condition: the change in the function must be bounded by a constant times the change in its input. For such a Lipschitz function on a compact interval, he constructs polynomials of degree at most n whose largest error is bounded by a constant divided by n. His route first smooths the curve with a fourth-power sinc kernel, then replaces that kernel by a polynomial while tracking error and degree. This supplies a quantitative approximation capability.",
      "title": "Jackson: a quantitative speed for approximating Lipschitz functions by polynomials"
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    "what_changed": "Weierstrass guarantees that a continuous curve can be approximated by polynomials, but does not give one useful speed for all continuous curves. Jackson adds a regularity condition: the change in the function must be bounded by a constant times the change in its input. For such a Lipschitz function on a compact interval, he constructs polynomials of degree at most n whose largest error is bounded by a constant divided by n. His route first smooths the curve with a fourth-power sinc kernel, then replaces that kernel by a polynomial while tracking error and degree. This supplies a quantitative approximation capability.",
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        "title": "How many polynomial terms buy a useful approximation?",
        "text": "Knowing that a curve can be approximated eventually is different from knowing how the error falls as the polynomial degree increases. Jackson adds a quantitative answer for Lipschitz functions. A Lipschitz condition limits how much the function can change when its input moves: the output difference is at most a fixed constant times the input difference. It allows corners, but rules out arbitrarily steep local changes."
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        "text": "Jackson constructs degree-at-most-n polynomial approximants whose largest error is bounded by K/n, with K depending on the function and interval. The route first smooths the function using a kernel, a weighted averaging rule. It then replaces the kernel with a polynomial and tracks both approximation error and resulting degree. A small smoothing error alone would be insufficient if the required polynomial degree grew without control."
      },
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        "title": "A rate is not a published numerical constant",
        "text": "In an invented classroom example, suppose a valid bound has K equal to two. Degree twenty would then guarantee error at most 0.1, and degree two hundred at most 0.01. These numbers illustrate the formula; they are not constants measured or certified for a particular curve in the dissertation. A real use must establish its own K and account for the interval. A bound is also different from the actual best error achieved."
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        "title": "A corner is allowed, an infinite slope needs care",
        "text": "The absolute-value function is Lipschitz: changing x by h changes its absolute value by no more than the absolute value of h. Near zero, the square-root function does not satisfy one fixed Lipschitz bound, because square root of h divided by h grows without bound as h shrinks. That does not mean square root cannot be approximated. It means this particular theorem cannot be applied without checking its assumptions."
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