I have a question about the restriction and the interpolation operators of a Multigrid algorithm.
Let those be given:
The full weighting restriction stencil (in 2D): $\frac{1}{16} \left[ \begin{array}{ccc} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end{array} \right] $
The bilinear interpolation: $\frac{1}{4} \left. \right] \begin{array}{ccc} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end{array} \left[ \right. $
It is always said, that one is the transpose of the other. For the matrix that is pretty obvious, but how come the prefactors differ.
The reason why I ask is the following: I have given the cubic interpolation stencil. And from this, I'd like to know the corresponding restriction operator. The matrix is easy to get, but I'm not sure about the coefficient.
Thanks