An introduction to a hybrid trigonometric box spline surface producing subdivision scheme (Q6145563)

In this paper, the authors construct new bivariate hybrid trigonometric box splines by directional convolution products of three univariate hybrid trigonometric B-splines. Note that here a hybrid trigonometric B-spline \({\mathbf T}_{0,\ell}(x;h)\) with step size \(h \in (0,\,\pi]\) is a linear combination of \(\sin(2x)\), \(\cos(2x)\), \(x^{\ell -3}\), \(x^{\ell-4}\), \(\ldots\), \(x\), 1 with \(\ell \ge 3\). Then for \(\ell,\,m,\, n \ge 3\), the hybrid trigonometric box spline \({\mathbf T}_{\ell,m,n}(x,y;h)\) is defined as a directional convolution product of \({\mathbf T}_{0,\ell}\), \({\mathbf T}_{0,m}\), and \({\mathbf T}_{0,n}\). Such a hybrid trigonometric box spline has compact support, is non-negative, and is symmetric with respect to the center of its support. The special hybrid trigonometric box spline \({\mathbf T}_{3,3,3}\) is studied in detail. The bivariate Fourier transform of \({\mathbf T}_{3,3,3}\) is equal to the product of an exponential function and nine cardinal sine functions. From the two-scale relation of the Fourier transform of \({\mathbf T}_{3,3,3}\), the authors derive the subdivision mask for a non-stationary subdivision scheme. They investigate the convergence and smoothness of the subdivision scheme in regular and irregular regions of a mesh. This subdivision scheme yields a \(C^4\)-continuous limit surface in the regular part of the mesh, if \(h \in (0,\,\frac{\pi}{3}]\). Further they analyze the exponential polynomial generation and reproduction property of this non-stationary subdivision scheme.