Abstract EN:

In 2000, on computational experiments, Lavery noticed that L1 cubic interpolating C1 splines have the property of well preserving the shape of data with abrupt changes in magnitude and spacing. His methods and his results constitute the basis of the present thesis. In the first part of the thesis, we show that in the case of a Heaviside function, if we take at least three points to the left of the discontinuity and at least three points to the right of the discontinuity, then there is a unique L1 cubic interpolating C1 spline which entirely coincides with the Heaviside function except in the interval containing the discontinuity. At the extremities of this interval, the tangents of the spline are the straight lines of the graph of the Heaviside function. We notice three consequences : there is no Gibbs’ phenomenon for the L1 cubic interpolating C1 spline, in the Lp-norm the spline converges at rate O(h1/p) to the Heaviside function, and for any fixed >0, the spline converges uniformly to the Heaviside function outside any interval of length , centered at the discontinuity. In the second part of the thesis, we define parametric polynomial Lp interpolating splines. After giving some important properties of these splines, we focus on the case p=1 for C1 cubics, G1 cubics and C2 quintics. Since the L1-norm treats each coordinate independently, the L1 splines are not invariant by rotation. To obtain invariance by rotation, we propose to make a local change of coordinates on each segment. In this way, we achieve the desired rotational invariance, and we obtain an additional shape parameter