Univariate cubic Lp splines and shape-preserving, multiscale interpolation by univariate cubic L1 splines Original Research Article

Univariate cubic Lp interpolating splines, 1≤p≤∞, defined by minimizing the Lp norm of the second derivative over a finite-dimensional spline space, are introduced. Cubic L2 splines, which coincide with conventional cubic splines, and cubic L∞ splines do not preserve shape well. In contrast, cubic L1 splines provide C1-smooth, shape-preserving, multiscale interpolation of arbitrary data, including data with abrupt changes in spacing and magnitude, with no need for monotonicity or convexity constraints, node adjustment or other user input. Extensions to higher-degree and higher-dimensional L1 splines are outlined. Cubic L1 splines are particularly useful in modeling terrain, geophysical features, biological objects and financial processes.