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        "description": "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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      "statement": "# Bochner: continuous positive-definite functions on Euclidean space are Fourier transforms of finite positive measures\n\nFor 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).\n\n## Scientific scope\n\nNo 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.\n\n## Reading status\n\nThis 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.\n\n## People and systems\n\nS. Bochner\n\nSource-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.\n\n\n\n## Original sources\n\n- [Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse (MathAnn 108, 378–410; full-volume container)](https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0108/PPN235181684_0108.pdf)\n",
      "summary": "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).",
      "title": "Bochner: continuous positive-definite functions on Euclidean space are Fourier transforms of finite positive measures"
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    "title": "Bochner: continuous positive-definite functions on Euclidean space are Fourier transforms of finite positive measures",
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    "what_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).",
    "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.",
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