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    "title": "Bernstein's binomial-expectation polynomial approximation method",
    "summary": "For every continuous F on[0,1], the explicit polynomials B_nF(x)=sum_{k=0}^n F(k/n) binom(n,k)x^k(1-x)^(n-k) converge uniformly to F. The original derives this construction as the expected payout from n Bernoulli trials, using continuity and a small tail probability.",
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        "title": "Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités",
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      "what_changed": "For every continuous F on[0,1], the explicit polynomials B_nF(x)=sum_{k=0}^n F(k/n) binom(n,k)x^k(1-x)^(n-k) converge uniformly to F. The original derives this construction as the expected payout from n Bernoulli trials, using continuity and a small tail probability.",
      "scope": "For every continuous F on[0,1], the explicit polynomials B_nF(x)=sum_{k=0}^n F(k/n) binom(n,k)x^k(1-x)^(n-k) converge uniformly to F. The original derives this construction as the expected payout from n Bernoulli trials, using continuity and a small tail probability. This is the explicit sampling/positive-polynomial method, not a second discovery of Weierstrass density. No modern optimal convergence rate, all-function interpolation property, arbitrary interval without rescaling or new probabilistic limit theorem is claimed. An explicit method and proof contribution, not rediscovery of the earlier Weierstrass existence theorem. No optimal modern rate is assigned. Archive bibliographic explanation resolves1912 separate issue versus1913 bound-volume heading; neither creates a second achievement.",
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        "description": "This is the explicit sampling/positive-polynomial method, not a second discovery of Weierstrass density. No modern optimal convergence rate, all-function interpolation property, arbitrary interval without rescaling or new probabilistic limit theorem is claimed. An explicit method and proof contribution, not rediscovery of the earlier Weierstrass existence theorem. No optimal modern rate is assigned. Archive bibliographic explanation resolves1912 separate issue versus1913 bound-volume heading; neither creates a second achievement.",
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      "statement": "# Bernstein's binomial-expectation polynomial approximation method\n\n## Can a coin help draw a curve?\n\nSuppose you know the values of a continuous curve at evenly spaced points between zero and one. You want a simple polynomial that approaches the entire curve, rather than merely matching a few values. Bernstein turned this approximation problem into an imagined betting game. The coin is a mathematical model, not a physical experiment reported in the paper. Its probability changes with the position where the curve is being approximated.\n\n## Average the possible payouts\n\nAt position x, imagine n independent trials whose success probability is x. If k trials succeed, pay the value of the curve at k divided by n. Multiply each possible payout by its binomial probability, then add. The result is the Bernstein polynomial. Every probability is nonnegative and the probabilities sum to one, so the polynomial is a weighted average of sampled curve values, not an instruction to force the approximation through every sample.\n\n## Try a curve you can calculate\n\nUse the teaching curve f(x) equals x squared and just two trials. At x equals one-half, the possible sample locations are zero, one-half and one. Their probabilities are one-quarter, one-half and one-quarter. Their squared values are zero, one-quarter and one. The expected payout is zero plus one-eighth plus one-quarter, giving three-eighths. The true curve value is one-quarter. This small approximation is not exact, and increasing the number of trials is part of the method.\n\n## Why the approximation improves\n\nFor many trials, the success fraction tends to stay close to x. Continuity makes curve values at nearby sample locations close to the value at x. The remaining distant samples can have different values, but their total probability becomes small, while continuity on the compact interval keeps the payouts bounded. Combining the nearby contribution with the small tail controls the error across the interval. This is a constructive proof route to an already known approximation theorem.\n\n## A small project with an honest limit\n\nAsk your agent to plot the squared curve and its Bernstein approximations for several trial counts. Require it to calculate the midpoint example above before trusting the plot. Then try a continuous curve with a sharp corner and describe what changes. Do not conclude that an arbitrary polynomial interpolation rule must converge: these polynomials average sampled values instead of matching every value exactly. Keep the 1912 separate issue and the 1913 bound-volume date as one historical paper.\n\n## Scientific scope\n\nThis is the explicit sampling/positive-polynomial method, not a second discovery of Weierstrass density. No modern optimal convergence rate, all-function interpolation property, arbitrary interval without rescaling or new probabilistic limit theorem is claimed. An explicit method and proof contribution, not rediscovery of the earlier Weierstrass existence theorem. No optimal modern rate is assigned. Archive bibliographic explanation resolves1912 separate issue versus1913 bound-volume heading; neither creates a second achievement.\n\n## Sources for this explanation\n\n- [Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités](https://history-of-approximation-theory.com/fpapers/bern1.pdf): Original two-page note, formula(1), continuity/tail estimates(2)–(5), final sampling remarks; archive catalogue explains1912 separate issue versus1913 bound volume.\n",
      "summary": "For every continuous F on[0,1], the explicit polynomials B_nF(x)=sum_{k=0}^n F(k/n) binom(n,k)x^k(1-x)^(n-k) converge uniformly to F. The original derives this construction as the expected payout from n Bernoulli trials, using continuity and a small tail probability.",
      "title": "Bernstein's binomial-expectation polynomial approximation method"
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    "what_changed": "For every continuous F on[0,1], the explicit polynomials B_nF(x)=sum_{k=0}^n F(k/n) binom(n,k)x^k(1-x)^(n-k) converge uniformly to F. The original derives this construction as the expected payout from n Bernoulli trials, using continuity and a small tail probability.",
    "scope": "For every continuous F on[0,1], the explicit polynomials B_nF(x)=sum_{k=0}^n F(k/n) binom(n,k)x^k(1-x)^(n-k) converge uniformly to F. The original derives this construction as the expected payout from n Bernoulli trials, using continuity and a small tail probability. This is the explicit sampling/positive-polynomial method, not a second discovery of Weierstrass density. No modern optimal convergence rate, all-function interpolation property, arbitrary interval without rescaling or new probabilistic limit theorem is claimed. An explicit method and proof contribution, not rediscovery of the earlier Weierstrass existence theorem. No optimal modern rate is assigned. Archive bibliographic explanation resolves1912 separate issue versus1913 bound-volume heading; neither creates a second achievement.",
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        "url": "https://history-of-approximation-theory.com/fpapers/bern1.pdf",
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      {
        "title": "Can a coin help draw a curve?",
        "text": "Suppose you know the values of a continuous curve at evenly spaced points between zero and one. You want a simple polynomial that approaches the entire curve, rather than merely matching a few values. Bernstein turned this approximation problem into an imagined betting game. The coin is a mathematical model, not a physical experiment reported in the paper. Its probability changes with the position where the curve is being approximated."
      },
      {
        "title": "Average the possible payouts",
        "text": "At position x, imagine n independent trials whose success probability is x. If k trials succeed, pay the value of the curve at k divided by n. Multiply each possible payout by its binomial probability, then add. The result is the Bernstein polynomial. Every probability is nonnegative and the probabilities sum to one, so the polynomial is a weighted average of sampled curve values, not an instruction to force the approximation through every sample."
      },
      {
        "title": "Try a curve you can calculate",
        "text": "Use the teaching curve f(x) equals x squared and just two trials. At x equals one-half, the possible sample locations are zero, one-half and one. Their probabilities are one-quarter, one-half and one-quarter. Their squared values are zero, one-quarter and one. The expected payout is zero plus one-eighth plus one-quarter, giving three-eighths. The true curve value is one-quarter. This small approximation is not exact, and increasing the number of trials is part of the method."
      },
      {
        "title": "Why the approximation improves",
        "text": "For many trials, the success fraction tends to stay close to x. Continuity makes curve values at nearby sample locations close to the value at x. The remaining distant samples can have different values, but their total probability becomes small, while continuity on the compact interval keeps the payouts bounded. Combining the nearby contribution with the small tail controls the error across the interval. This is a constructive proof route to an already known approximation theorem."
      },
      {
        "title": "A small project with an honest limit",
        "text": "Ask your agent to plot the squared curve and its Bernstein approximations for several trial counts. Require it to calculate the midpoint example above before trusting the plot. Then try a continuous curve with a sharp corner and describe what changes. Do not conclude that an arbitrary polynomial interpolation rule must converge: these polynomials average sampled values instead of matching every value exactly. Keep the 1912 separate issue and the 1913 bound-volume date as one historical paper."
      }
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