  In this paper, the following type Tikhonov regularization problem will be
systematically studied: [(u_t,v_t):=\argmin_{u+v=f} {|v|_X+t|u|_Y},] where $Y$
is a smooth space such as a $\BV$ space or a Sobolev space and $X$ is the pace
in which we measure distortion. Examples of the above problem occur in
denoising in image processing, in numerically treating inverse problems, and in
the sparse recovery problem of compressed sensing. It is also at the heart of
interpolation of linear operators by the real method of interpolation. We shall
characterize of the minimizing pair $(u_t,v_t)$ for
$(X,Y)=(L_2(\Omega),\BV(\Omega))$ as a primary example and generalize Yves
Meyer's result in [11] and Antonin Chambolle's result in [6]. After that, the
following multiscale decomposition scheme will be studied:
[u_{k+1}:=\argmin_{u\in \BV(\Omega)\cap L_2(\Omega)}
{1/2|f-u|^2_{L_2}+t_{k}|u-u_k|_{\BV}},] where $u_0=0$ and $\Omega$ is a bounded
Lipschitz domain in $\R^d$. This method was introduced by Eitan Tadmor et al.
and we will improve the $L_2$ convergence result in \cite{Tadmor}. Other pairs
such as $(X,Y)=(L_p,W^{1}(L_\tau))$ and $(X,Y)=(\ell_2,\ell_p)$ will also be
mentioned. In the end, the numerical implementation for
$(X,Y)=(L_2(\Omega),\BV(\Omega))$ and the corresponding convergence results
will be given.
