On a Weighted Chaikin-type Subdivision Scheme for Non-Smooth Data

In this paper, we introduce a nonlinear subdivision scheme designed to converge to piecewise smooth functions. The scheme is formulated as a convex nonlinear combination of Chaikin-type linear schemes. In regions of regularity, it attains a high level of smoothness and accuracy, comparable to that of the C2(R) fourth-order linear algorithm that it approximates. Near discontinuities, the scheme successfully avoids Gibbs-type oscillations and does not introduce any artificial intermediate points within the intervals containing the discontinuities. Furthermore, the proposed scheme adaptively responds to discontinuities while utilizing the maximum number of available points within smooth regions, thus ensuring the highest attainable order of accuracy in every case. A set of numerical experiments is provided, illustrating and supporting the theoretical analysis with respect to convergence, stability, accuracy, and regularity across both smooth regions and regions with discontinuities. To the best of our knowledge, this represents the first algorithm capable of simultaneously exhibiting all these desirable properties.