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    "summary": "Ordinary polynomials use all whole-number powers. What happens if most powers are unavailable? For increasing positive real exponents p_n, Müntz shows that finite combinations of 1 and x raised to those powers approximate every continuous function on [0,1] exactly when the sum of 1/p_n diverges. Szász later simplifies the proof and extends its scope to complex exponents. When their real parts stay above a fixed positive constant, he obtains an exact corresponding criterion. The contribution identifies a boundary between complete and incomplete approximation systems, building on Weierstrass and Bernstein's question.",
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        "description": "Density of finite combinations does not mean a unique series expansion or a prescribed approximation speed. The constant 1 is needed for functions nonzero at the origin. Szász's unrestricted positive-real-part complex case has different necessary and sufficient conditions, so it is not presented as a universal equivalence. Later full results for exponents accumulating at zero are outside this original scope.",
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          "citation": "Über den Approximationssatz von Weierstraß, Schwarz Festschrift pp303–312",
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          "citation": "Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen, Mathematische Annalen77 pp482–496",
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      "statement": "# Müntz and Szász: when a sparse collection of powers still approximates every curve\n\n## Can a toolbox work when most tools are missing?\n\nA polynomial combines powers such as one, x, x squared and x cubed. Weierstrass tells us that enough such terms can approximate any continuous curve on a closed interval. Müntz asks what happens if only selected powers are available. The answer is not simply that an infinite toolbox is enough. Some infinite choices can approximate every curve and others cannot; a reciprocal-series condition separates the original cases.\n\n## Dense means arbitrarily accurate finite combinations\n\nFor increasing positive real exponents p_n, include the constant one and powers x raised to p_n. On [0,1], their finite linear combinations are uniformly dense exactly when the sum of 1/p_n diverges. Uniformly dense means that every continuous target and positive error allowance has a suitable finite combination. It does not promise a unique infinite expansion, a fixed number of terms or one approximation rate for every target.\n\n## Even powers can work on one side of zero\n\nChoose exponents two, four, six and so on. Their reciprocal sum is half the harmonic series and diverges, so the constant and these even powers suffice on [0,1]. This can sound surprising if the target is x. The interval matters: on [-1,1], every combination of even powers takes the same value at minus one and plus one, whereas x takes opposite values there. At least one endpoint error must then be at least one.\n\n## Do not erase the conditions in the extension\n\nSzász simplifies the proof and treats complex exponents with positive real parts. His general necessary and sufficient conditions are different; they coincide under an additional positive lower bound on the real parts. Exponents approaching zero require further care. Also, without the constant term, every available function vanishes at zero, so a target that does not vanish there cannot be approximated uniformly. These are structural restrictions, not technical decoration.\n\n## Make a comparison table your agent can check\n\nAsk your agent to compare exponents n, two n and n squared. The first two reciprocal series diverge; the last converges. Require a reason for each, rather than judging from a finite decimal sum. It can use 1/n squared less than 1/[n(n minus one)] for n at least two to obtain a telescoping bound. The output should state the interval and whether the constant term is present before making any density claim.\n\n## Scientific scope\n\nDensity of finite combinations does not mean a unique series expansion or a prescribed approximation speed. The constant 1 is needed for functions nonzero at the origin. Szász's unrestricted positive-real-part complex case has different necessary and sufficient conditions, so it is not presented as a universal equivalence. Later full results for exponents accumulating at zero are outside this original scope.\n\n## Sources for this explanation\n\n- [Über den Approximationssatz von Weierstraß, Schwarz Festschrift pp303–312](https://history-of-approximation-theory.com/fpapers/muntz.pdf): Müntz1914,printed303–312,especially theorem304and proof306–310.\n- [Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen, Mathematische Annalen77 pp482–496](https://history-of-approximation-theory.com/fpapers/szasz1916.pdf): Szász1916,printed484,486and491–493;different complex necessary/sufficient conditions and bounded-real-part equivalence.\n",
      "summary": "Ordinary polynomials use all whole-number powers. What happens if most powers are unavailable? For increasing positive real exponents p_n, Müntz shows that finite combinations of 1 and x raised to those powers approximate every continuous function on [0,1] exactly when the sum of 1/p_n diverges. Szász later simplifies the proof and extends its scope to complex exponents. When their real parts stay above a fixed positive constant, he obtains an exact corresponding criterion. The contribution identifies a boundary between complete and incomplete approximation systems, building on Weierstrass and Bernstein's question.",
      "title": "Müntz and Szász: when a sparse collection of powers still approximates every curve"
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    "title": "Müntz and Szász: when a sparse collection of powers still approximates every curve",
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    "what_changed": "Ordinary polynomials use all whole-number powers. What happens if most powers are unavailable? For increasing positive real exponents p_n, Müntz shows that finite combinations of 1 and x raised to those powers approximate every continuous function on [0,1] exactly when the sum of 1/p_n diverges. Szász later simplifies the proof and extends its scope to complex exponents. When their real parts stay above a fixed positive constant, he obtains an exact corresponding criterion. The contribution identifies a boundary between complete and incomplete approximation systems, building on Weierstrass and Bernstein's question.",
    "scope": "Ordinary polynomials use all whole-number powers. What happens if most powers are unavailable? For increasing positive real exponents p_n, Müntz shows that finite combinations of 1 and x raised to those powers approximate every continuous function on [0,1] exactly when the sum of 1/p_n diverges. Szász later simplifies the proof and extends its scope to complex exponents. When their real parts stay above a fixed positive constant, he obtains an exact corresponding criterion. The contribution identifies a boundary between complete and incomplete approximation systems, building on Weierstrass and Bernstein's question. Density of finite combinations does not mean a unique series expansion or a prescribed approximation speed. The constant 1 is needed for functions nonzero at the origin. Szász's unrestricted positive-real-part complex case has different necessary and sufficient conditions, so it is not presented as a universal equivalence. Later full results for exponents accumulating at zero are outside this original scope.",
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        "title": "Can a toolbox work when most tools are missing?",
        "text": "A polynomial combines powers such as one, x, x squared and x cubed. Weierstrass tells us that enough such terms can approximate any continuous curve on a closed interval. Müntz asks what happens if only selected powers are available. The answer is not simply that an infinite toolbox is enough. Some infinite choices can approximate every curve and others cannot; a reciprocal-series condition separates the original cases."
      },
      {
        "title": "Dense means arbitrarily accurate finite combinations",
        "text": "For increasing positive real exponents p_n, include the constant one and powers x raised to p_n. On [0,1], their finite linear combinations are uniformly dense exactly when the sum of 1/p_n diverges. Uniformly dense means that every continuous target and positive error allowance has a suitable finite combination. It does not promise a unique infinite expansion, a fixed number of terms or one approximation rate for every target."
      },
      {
        "title": "Even powers can work on one side of zero",
        "text": "Choose exponents two, four, six and so on. Their reciprocal sum is half the harmonic series and diverges, so the constant and these even powers suffice on [0,1]. This can sound surprising if the target is x. The interval matters: on [-1,1], every combination of even powers takes the same value at minus one and plus one, whereas x takes opposite values there. At least one endpoint error must then be at least one."
      },
      {
        "title": "Do not erase the conditions in the extension",
        "text": "Szász simplifies the proof and treats complex exponents with positive real parts. His general necessary and sufficient conditions are different; they coincide under an additional positive lower bound on the real parts. Exponents approaching zero require further care. Also, without the constant term, every available function vanishes at zero, so a target that does not vanish there cannot be approximated uniformly. These are structural restrictions, not technical decoration."
      },
      {
        "title": "Make a comparison table your agent can check",
        "text": "Ask your agent to compare exponents n, two n and n squared. The first two reciprocal series diverge; the last converges. Require a reason for each, rather than judging from a finite decimal sum. It can use 1/n squared less than 1/[n(n minus one)] for n at least two to obtain a telescoping bound. The output should state the interval and whether the constant term is present before making any density claim."
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        "title": "Über den Approximationssatz von Weierstraß, Schwarz Festschrift pp303–312",
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        "title": "Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen, Mathematische Annalen77 pp482–496",
        "url": "https://history-of-approximation-theory.com/fpapers/szasz1916.pdf",
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