A supplementary note on the completeness of the Rossby normal modes in a rectangular basin

In previous studies of the completeness of Rossby normal modes in a closed basin, proofs have been obtained by rather technical methods based on the Green's function for the Laplacian operator or based on the theory of compact self-adjoint operators. This short note (1) gives a much simpler proof by the use of only the norm relations between the sine functions and the Rossby normal modes and (2) clarifies the relations between the two different sets of eigenfunctions and between the ordinary norm and the Dirichlet norm, though the validity of the present method is restricted to a special one-dimensional case. The expansion theorem is also derived from the completeness theorem with the aid of transformation laws, where the assumption about the smoothness of the expanded function is weaker than in the elementary proof of Masuda (1987a). Both the set of Rossby normal modes and that of sine functions are shown to be complete for the Dirichlet norm as well.