A univariate quasi-multiquadric interpolation with better smoothness

In this paper, we propose a multilevel univariate quasi-interpolation scheme usingmultiquadric basis. It is practical as it does not require derivative values of the function being interpolated. It has a higher degree of smoothness than the original level-0 formula as it allows a shape parameter c=O(h). Our level-1 quasi-interpolation costs O(nlogn) flops to set up. It preserves strict convexity and monotonicity. When c=O(h), we prove the proposed scheme converges with a rate of O(h2.5logh).Furthermore, if both |f″(a)| and |f″| are relatively small compared with ∥f″∥ _∞, the convergence rate will increase. We verify numerically that c = h is a good shape parameter to use for our method, hence we need not find the optimal parameter. For all test functions, both convergence speed and error are optimized for c between 0.5h and 1.5h. Our method can be generalized to a multilevel scheme; we include the numerical results for the level-2 scheme. The shape parameter of the level-2 scheme can be chosen between 2h to 3h.