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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).
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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.
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People and systems
S. Bochner
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English catalogue narrative
Bochner: continuous positive-definite functions on Euclidean space are Fourier transforms of finite positive measures
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
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.
Reading status
This is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.
People and systems
S. Bochner
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.