Necessary and sufficient nonnegativity conditions for second-order coordinate trigonometric splines

Abstract

Necessary and sufficient nonnegativity conditions for continuous differentiable coordinate trigonometric splines of the second order are obtained; the convexity and concavity intervals of these splines are determined. The method of investigation consists in recognizing concavity in intervals adjacent to the endpoints of the support of a coordinate spline under consideration and applying arguments related to the number of zeros of the solution of the corresponding boundary value problem for a second-order differential equation.