system block within the superblock [@wh93].\
The twofold problem introduced above is therefore solved as follows: First the embedding of the system block within the superblock is achieved by reconstructing the target state of the superblock in a basis, in which the basis vectors are given as a tensor product composition of states of the system block and the chosen environment. In this way the system block is described within the bigger superblock. Since we have not truncated the set of the states which belong to the environment, the RG step for the system block is performed by taking all possible boundary conditions within the selected environment into account.\
From the previous discussion it becomes obvious that the coefficients $\gamma_{p,n}$ can only be determined numerically within real applications of this technique. To develop a complete analytic approach, a method for using a target state will be impractical and we can only use the overall picture represented in figure \[superblock\].

A rigorous real-space RG transformation {#rigorous}
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We start this section by giving a very general but well known definition of a RG transformation (RGT). A RGT $R$ is a map defined on a set of physical variables $\{{\sigma_l}\}$ and a 