Tamtam Proceedings - lamsin

356 Mraoui et al.in the standard way by thresholding out small coefficients of detail functions g t r,s, and weshow that this process leads to the omission of some Hermite data, without affecting thesmoothness of the Hermite interpolant and sacrificing the quality of the approximation.2. Numerical examplesIn this section we give three numerical examples. In the first one, we give the smoothingof a surface. The second is reserved for the decomposition of an Hermite interpolant,and in the last one we give the compression of some Hermite data.2.1. Smoothing a surfaceLet Ω = [0, 2] × [0, 2]. In this example, we consider the piecewise function f definedon Ω by⎧− ln(1 + (x − 1) ⎪⎨2 (y − 1) 2 ) if (x, y) ∈ [0, 1] × [0, 1],sin(2πx) ln(1 + (y − 1f(x, y) =2 )) if (x, y) ∈ [1, 2] × [0, 1],sin(2π(x − 1)(y − 1))/3 if (x, y) ∈ [0, 1] × [1, 2],⎪⎩sin(πx) sin(πy) if (x, y) ∈ [1, 2] × [1, 2].It is obvious that f is only of class C 0,0 on Ω, see the graph of f in Figure 1. Ouraim is to smooth f in order to make it of class C 1,2 on Ω. For this, we use the smoothingalgorithm. We study this example using two different partitions of Ω. we consider the∆ n,m = ∆ n × ∆ m is defined from ∆ n = (0, 0.5, 0.7, 1, 1.3, 1.5, 2) and ∆ m =(0, 0.5, 0.7, 1, 1.1, 1.5, 2). In this case, the function f 1,2 given in Figure 2 is close tof. The corresponding graphs are given in the left column of Figure 2 for the functionsf 1,2 and in the right column for the error functions f − f 1,2 .Figure 1.TAMTAM –Tunis– 2005