A quasi-interpolation scheme for periodic data based on multiquadric trigonometric B-splines

Multiquadric (MQ) quasi-interpolation has been studied extensively in the literature. However, the MQ quasi-interpolant is not well defined for periodic data, since its kernel (the MQ function) itself is not periodic. Note that in many applications, the data may arise from a closed curve (surface) and thus possess some kind of periodicity, for example, analysis of geodetic and meteorological data, constructing active contours, estimating the region of attraction of dynamical systems, and so on. Therefore, it is meaningful to construct a quasi-interpolant (whose kernel itself is periodic) for periodic data. Combining the construction techniques of the MQ quasi-interpolant and the trigonometric B-spline quasi-interpolant, the paper constructs such a quasi-interpolant. The quasi-interpolant couples together the periodicity of the trigonometric B-spline quasi-interpolant and the smoothness of the MQ quasi-interpolant. Moreover, both the quasi-interpolant and its derivatives are periodic. The quasi-interpolant covers the trigonometric B-spline quasi-interpolant as a special case. In addition, the error estimate shows that a proper shape parameter can be chosen such that the quasi-interpolant provides the same approximation order as a trigonometric B-spline quasi-interpolant for a periodic function. Furthermore, the quasi-interpolant gives better approximations to high-order derivatives than the trigonometric B-spline quasi-interpolant, as illustrated by numerical examples presented at the end of the paper.