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

• Published in: Mathematical Notes Vol. 103; no. 3-4; pp. 550 - 564
• Main Authors: Zastavnyi, V. P., Manov, A. D.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.03.2018; Springer Nature B.V
• Subjects: Linear functions; Mathematical analysis; Mathematics; Mathematics and Statistics; positive definite function; completelymonotone function; Bernstein’s inequality; piecewise linear function; Bochner–Khinchine theorem
• ISSN: 0001-4346; 1067-9073; 1573-8876
• DOI: 10.1134/S0001434618030227

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 α , c ( − x ) f α , c ( x ) ¯ , x ∈ 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 ), m ( α ) = { 0 i f 1 / α ∉ ℕ , − α i f 1 / α ∈ ℕ . 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.