Nonlinear Approximation and the Space BV(R)

Given a function f 2 L2(Q), Q := [0; 1) and a real number t > 0, let U(f; t) := infg2BV(Q) kf gk 2 L2(I) + tVQ(g); where the in mum is taken over all functions g 2 BV of bounded variation on I. This and related extremal problems arise in several areas of mathematics such as interpolation of operators and statistical estimation, as well as in digital image processing. Techniques for nding minimizers g for U(f; t) based on variational calculus and nonlinear partial di erential equations have been put forward by several authors ([DMS], [LOR], [MS], [CL]). The main disadvantage of these approaches is that they are numerically intensive. On the other hand, it is well-known that more elementarymethods based on wavelet shrinkage solve related extremal problems, for example, the above problem with BV replaced by the Besov space B 1(L1(I)) (see e.g. [CDLL]). However, since BV has no simple description in terms of wavelet coe cients, it is not clear that minimizers for U(f; t) can be realized in this way. We shall show in this paper that simple methods based on Haar thresholding provide near minimizers for U(f; t). Our analysis of this extremal problem brings forward many interesting relations between Haar decompositions and the space BV.