Nonlinear Approximation and the Space

Given a function f 2 L2(Q), Q := [0, 1)2 and a real number t > 0, let U( f , t) := infg2BV (Q) kf gkL2(I) + t VQ (g), where the infimum 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 finding minimizers g for U( f , t) based on variational calculus and nonlinear partial differential equations have been put forward by several authors [DMS], [RO], [MS], [CL]. The main disadvantage of these approaches is that they are numerically intensive. On the other hand, it is well known that more elementary methods based on wavelet shrinkage solve related extremal problems, for example, the above problem with BV replaced by the Besov space B1(L1(I)) (see e.g. [CDLL]). However, since BV has no simple description in terms of wavelet coefficients, 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.