Well, the modern viewpoint relays on the interpretation of "fourier transform" (in any generalized fashion you like to define "fourier transform") in representation-theoretic language. As a consequence, there are several approaches today to get the trace formula (either more analytic by green functions or the more general representation theoretic manner).

A nice introductory account can be found here by Marklof - http://arxiv.org/pdf/math/0407288v2.pdf Another representation-free approach is done in Iwaniec's "spectral methods of automorphic forms".

The theorem that Hejhal mentioned is very well known for general (compact, closed) manifolds (follows from Poincare inequality), but in the (cocompact) homogeneous case, one can overcome many analytical complications by just mimicking the proof of the Peter-Weyl theorem in representation theory of compact groups (Hilbert-Schmidt operators and so on). In particular, no Sobolev computations whatsoever, that shows one simple example of the advantages of using representation theory.

A more advanced approach (which uses some representation theory) is found in Knapp's article - http://sporadic.stanford.edu/bump/match/trace.pdf Probably a good introduction to this article is Bump's book about automorphic representations (chapters 1-2 I guess, you only need the real part for this article).

Well, the modern viewpoint relays on the interpretation of "fourier transform" (in any generalized fashion you like to define "fourier transform") in representation-theoretic language. As a consequence, there are several approaches today to get the trace formula (either more analytic by Green's functions or the more general representation theoretic manner).

A nice introductory account can be found here by Marklof - http://arxiv.org/pdf/math/0407288v2.pdf Another representation-free approach is done in Iwaniec's "spectral methods of automorphic forms".

The theorem that Hejhal mentioned is very well known for general (compact, closed) manifolds (follows from Poincare inequality), but in the (cocompact) homogeneous case, one can overcome many analytical complications by just mimicking the proof of the Peter-Weyl theorem in representation theory of compact groups (Hilbert-Schmidt operators and so on). In particular, no Sobolev computations whatsoever, that shows one simple example of the advantages of using representation theory.

A more advanced approach (which uses some representation theory) is found in Knapp's article - http://sporadic.stanford.edu/bump/match/trace.pdf Probably a good introduction to this article is Bump's book about automorphic representations (chapters 1-2 I guess, you only need the real part for this article).