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

Given $\alpha\in(0,1)$ and $c=h+i\beta$, $h,\beta\in\mathbb R$, the function $f_{\alpha,c}\colon\mathbb R\to\mathbb C$ defined as follows is considered: (1) $f_{\alpha,c}$ is Hermitian, i.e., $f_{\alpha,c}(-x)=\overline{f_{\alpha,c}(x)}$, $x\in\mathbb R$; (2) $f_{\alpha,c}(x)=0$ for $x>1$; moreover, on each of the closed intervals $[0,\alpha]$ and $[\alpha,1]$, the function $f_{\alpha,c}$ is linear and satisfies the conditions $f_{\alpha,c}(0)=1$, $f_{\alpha,c}(\alpha)=c$, and $f_{\alpha,c}(1)=0$. It is proved that the complex piecewise linear function $f_{\alpha,c}$ is positive definite on $\mathbb R$ if and only if
$$m(\alpha)\le h\le 1-\alpha\quad \text{and}\quad |\beta|\le\gamma(\alpha,h),$$
where
$$m(\alpha)= \begin{cases} 0{} &\text{if } 1/\alpha\notin\mathbb N, \\ -\alpha{} &\text{if }1/\alpha\in\mathbb N. \end{cases}$$
If $m(\alpha)<h<1-\alpha$ and $\alpha\in\mathbb Q$, then $\gamma(\alpha,h)>0$; otherwise, $\gamma(\alpha,h)=0$. 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.