A Non-stationary Combined Ternary 5-point Subdivision Scheme with $C^{4}$ Continuity

Abstract

In this paper, a family of non-stationary combined ternary $5$-point subdivision schemes with multiple variable parameters is proposed. The construction of the scheme is based on the generalized ternary subdivision scheme of order $4$, which is built upon refinement of a family of generalized B-splines, using the variable displacements. For such a non-stationary scheme, we study its smoothness and get that it can generate $C^{2}$ interpolating limit curves and $C^{4}$ approximating limit curves. Besides, we investigate the exponential polynomial generation/reproduction property and approximation order. It can generate/reproduce certain exponential polynomials with suitable choices of the variable parameters, and reach approximation order $5$.