SWhat changed
For finite-dimensional Euclidean R^k, Bochner proves that a bounded continuous complex function is positive definite exactly when it has a Fourier–Stieltjes representation f(x)=∫exp(i x·α)dV(α) with a nonnegative bounded distribution function. Positive definiteness is the nonnegativity of every finite quadratic form, not strict positivity. His uniqueness theorem identifies the distribution up to the original continuity-interval equivalence, corresponding to a unique finite positive Borel measure; its total mass is f(0).
Scientific scope
For finite-dimensional Euclidean R^k, Bochner proves that a bounded continuous complex function is positive definite exactly when it has a Fourier–Stieltjes representation f(x)=∫exp(i x·α)dV(α) with a nonnegative bounded distribution function. Positive definiteness is the nonnegativity of every finite quadratic form, not strict positivity. His uniqueness theorem identifies the distribution up to the original continuity-interval equivalence, corresponding to a unique finite positive Borel measure; its total mass is f(0). No normalization f(0)=1 is assumed; probability measures are the normalized case. Continuity is part of the original positive-definite function class and is essential in the limit theorem used in the proof. The result concerns R^k, not arbitrary nonabelian groups, vector-valued integration, or a claim that every kernel is a Fourier transform without translation invariance. The introduction explicitly credits Bochner's 1932 book for the one-variable criterion; this 1933 paper develops the multidimensional form and supporting Stieltjes/limit theory. The later sections on generalized Wiener spectra have a distinct scientific 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. No normalization f(0)=1 is assumed; probability measures are the normalized case. Continuity is part of the original positive-definite function class and is essential in the limit theorem used in the proof. The result concerns R^k, not arbitrary nonabelian groups, vector-valued integration, or a claim that every kernel is a Fourier transform without translation invariance. The introduction explicitly credits Bochner's 1932 book for the one-variable criterion; this 1933 paper develops the multidimensional form and supporting Stieltjes/limit theory. The later sections on generalized Wiener spectra have a distinct scientific scope.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
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