SWhat changed
A sequence of curves can settle down in average squared error without settling at every point. Fischer proves that, on a finite interval, a sequence that is Cauchy in this mean-square sense has a square-integrable mean-square limit. Riesz shows that every square-summable list of coefficients can be realized against an arbitrary normalized orthogonal sequence; Fischer independently obtains this using completeness. Together these results connect functions with infinite lists of coordinates. If the orthogonal system is complete, the correspondence describes the whole function space, with functions differing only on a set of measure zero treated as the same.
Scientific scope
A sequence of curves can settle down in average squared error without settling at every point. Fischer proves that, on a finite interval, a sequence that is Cauchy in this mean-square sense has a square-integrable mean-square limit. Riesz shows that every square-summable list of coefficients can be realized against an arbitrary normalized orthogonal sequence; Fischer independently obtains this using completeness. Together these results connect functions with infinite lists of coordinates. If the orthogonal system is complete, the correspondence describes the whole function space, with functions differing only on a set of measure zero treated as the same. The original scope is real Lebesgue-square-integrable functions on a finite interval. Mean-square convergence does not imply pointwise or uniform convergence of the entire sequence. For an incomplete orthogonal system, coefficients determine the closed-span representative but leave an orthogonal-complement component undetermined. The authors wrote independent papers; Fischer acknowledges Riesz publication priority. Riesz explicitly gives a proof sketch in places. Earlier Hilbert, Schmidt, Fatou and Lebesgue results retain their own credit.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The original scope is real Lebesgue-square-integrable functions on a finite interval. Mean-square convergence does not imply pointwise or uniform convergence of the entire sequence. For an incomplete orthogonal system, coefficients determine the closed-span representative but leave an orthogonal-complement component undetermined. The authors wrote independent papers; Fischer acknowledges Riesz publication priority. Riesz explicitly gives a proof sketch in places. Earlier Hilbert, Schmidt, Fatou and Lebesgue results retain their own credit.
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.
- Ueber orthogonale Funktionensysteme
- Sur la convergence en moyenne —originalpage1022
- Sur la convergence en moyenne —originalpage1023
- Sur la convergence en moyenne —originalpage1024
- Applications d'un théorème sur la convergence en moyenne —originalpage1148
- Applications d'un théorème sur la convergence en moyenne —originalpage1149
- Applications d'un théorème sur la convergence en moyenne —originalpage1150
- Applications d'un théorème sur la convergence en moyenne —originalpage1151
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