I know non-negative definiteness for a matrix, but I am reading some notes that use it for a function $f \colon \mathbb{Z} \rightarrow \mathbb{R}$.
I am not sure since it doesnt' say so explictily, but I inferred that the definition is something along the lines of
$$\sum_{i, j = 1}^N a_i a_j f(t_i - t_j) \ge 0, \ \forall t_i, t_j \in Z, a_i, a_j \in \mathbb{R}$$ where $a_i, a_j$ are real, $N$ is some integer, and $t_i, t_j$ are integers as well.
My questions are:
Is this definition accurate?
Why is it not the same as the definition for a matrix? (In particular, the $a_i a_j$ are not $a_i^2$).
- What special properties do such functions have? Or, to make it less general: how do the properties of such functions relate to the corresponding properties that the nonnegative definite matrices have.