More on the order of prolongations and restrictions

For the analysis of multi-grid methods applied to finite difference discretizations, two definitions of orders of intergrid transfer operators are being used. The order is either defined in terms of the symbol of the transfer operator, i.e., its Fourier transform, or as the order of polynomials being preserved. In [J. Comput. Appl. Math. 32 (3) (1990) 423–429], Hemker showed that the second definition is stronger than the first one. In this note, we show that both definitions are even equivalent.