Positive Definiteness of Complex Piecewise Linear Functions and Some of Its Applications

Abstract

Given α ∈ (0, 1) and c = h + iβ, h, β ∈ R, the function f_α,c: R → C defined as follows is considered: (1) f_α,c is Hermitian, i.e., \({f_{\alpha ,c}}\left( { - x} \right)\overline {{f_{\alpha ,c}}\left( x \right)} ,x \in \mathbb{R};\), x ∈ R; (2) f_α,c(x) = 0 for x > 1; moreover, on each of the closed intervals [0, α] and [α, 1], the function f_α,c is linear and satisfies the conditions f_α,c(0) = 1, f_α,c(α) = c, and f_α,c(1) = 0. It is proved that the complex piecewise linear function f_α,c is positive definite on R if and only if m(α) ≤ h ≤ 1 − α and |β| ≤ γ(α, h), \(where m\left( \alpha \right) = \left\{ \begin{gathered}
0if1/\alpha \notin \mathbb{N}, \hfill \\
- \alpha if1/\alpha \in \mathbb{N}. \hfill \\
\end{gathered} \right.\) If m(α) ≤ h ≤ 1 − α and α ∈ Q, then γ(α, h) > 0; otherwise, γ(α, h) = 0. This result is used to obtain a criterion for the complete monotonicity of functions of a special form and prove a sharp inequality for trigonometric polynomials.