Necessary and sufficient conditions of nonnegativity for coordinate trigonometrical splines of the second order

Abstract

There are many different ways to define coordinate splines. The splines, which are defined with approximation relations, have the best approximation properties as to order of N-width for the standard compact sets. The suitable choice of the approximation relations gives the maximal smoothness under the condition of minimal support for coordinate splines. On the other hand, the choice of different generating function gives opportunity to receive splines of different types (polynomial, exponential, trigonometrical ones and so on). The paper is discussed the positivity of coordinate trigonometrical splines of the second order with maximal smoothness. Let us consider the real grid {xj}j∈Z, . . . < x−2 < x−1 < x0 < x1 < x2 < . . . By definition we put M def = ∪j∈Z (xj , xj+1), Sj def = [xj−1, xj+2], and consider three-component vectors: generating vector function ϕ(t) and family of vectors aj , defined with formulas ϕ(t) def = (1, sin t, cos t)T , aj def = det(ϕj ,ϕj+1,ϕ ′ j+1)ϕ ′ j − det(ϕ ′ j , ϕj+1,ϕ ′ j+1)ϕj , where ϕj def = ϕ(xj), ϕ ′ j def = ϕ′(xj). The coordinate trigonometric splines ωj ∈ C1 are defined by approximation relations ak−1ωk−1(t) + akωk(t) + ak+1ωk+1(t) ≡ ϕ(t) ∀t ∈ (xk, xk+1), ∀k ∈ Z, ωj(t) ≡ 0 ∀t ∈ M\Sj , ∀j ∈ Z. Theorem. The inequality xj+1 − xj < π ∀j ∈ Z is necessary and sufficient condition for nonnegativity of the functions ωj(t) ∀j ∈ Z. Furthermore, the sufficient conditions of convexity on the intervals (xj−1, xj), (xj+1, xj+2) and concavity on the interval (xj , xj+1) have been obtained in the paper. Refs 16. Figs 2.