One interesting article

Recently I've read the article Phys. Rev. B 84, 075145 (2011) devoted to study of Matsubara Green functions of quantum impurity models on the basis of Legendre polynomials. It was shown that Green functions for models of some class have coefficients of decomposition into Legandre polynomial decreasing much faster than coefficients of Fourier series for Green functions.

Some thoughts rose from this article:

1) Is this property true for every model or only for some class of models? If second, what is that class?
2) If it is general property of Green functions then it must be inferred by some general consideration that hasn't been done.

3) Does this property depend on dimension and type of lattice of the model?

4) Observed property of Green functions restricts the class of functions to which the Green functions belong. Is it possible to formulate this restriction strictly mathematically? How could it be presented?