DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Positive definite function on the real line

Definition

A function $ψ : R \to C$ is positive definite if for every integer $n \geq 1$ , all $t_{1}, \dots, t_{n} \in R$ and all $z_{1}, \dots, z_{n} \in C$ , the number $\sum_{j = 1}^{n} \sum_{k = 1}^{n} z_{j} \overline{z_{k}} ψ (t_{j} - t_{k})$ is real and nonnegative. The frequencies may repeat and coefficients may vanish. Including $n = 0$ would impose only the automatic inequality $0 \geq 0$ . Taking $n = 1$ , $z_{1} = 1$ forces $ψ (0)$ to be real and nonnegative; normalization to $ψ (0) = 1$ is not part of this definition. Continuity is also not imposed. This definition asserts no representation theorem and makes no choice assumption.

Used by

Characteristic functions are positive definite Lemma

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Durrett, Probability: Theory and Examples, fifth edition (standard reference, not scraped)
Norris, Probability and Measure (standard reference, not scraped)