SWhat 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.
Scientific 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.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. 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.
Contributors
- The named authors in the research record · SCIENCE: Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record identities are not inferred.
- Über den Approximationssatz von Weierstraß, Schwarz Festschrift pp303–312
- Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen, Mathematische Annalen77 pp482–496
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