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The harmonic analysis of automorphic functions has been extensively
studied; references to the pertinent parts of this theory are contained
in  our monograph. We recall that the Poincar\'e plane $Pi$, that is the 
upper half plane
$$w=x+iy,  y > 0,\leqno l.l)$$
serves as a model for a non-Euclidean geometry in which the motions are
given by the group $G$ of fractional linear transformations:
$$ w → {aw+b\over cw+d}\leqno l.2)$$
where $a,b,c,d$ are real and
$$ad-bc=l;\leqno l.3)$$
$G$ is isomorphic with $SL(2,R)/\pm I$.  The Riemannian metric
$${dx↑2+dy↑2\over y↑2}\leqno l.4)$$
is invariant under this group of motions. The invariant $L↓2$ form is
$$\int\int u↑2 {dx\,dy\over y↑2}.\leqno l.5)$$
The invariant Dirichlet form is
$$\int\!\int(u↓x↑2+u↓y↑2)dx\,dy;\eqno 1.6)$$
The corresponding Laplace-Beltrami operator
$$L↓o=y↑2\Delta=y↑2(∂↑2↓x+∂↓y↑2)\eqno l.7)$$
is then clearly invariant.It turns out that the operator $L$ defined as
$$L=L↓o+1/4,\eqno 1.7)\prim$$
also invariant, has more useful analytic properties, as will be seen in
what follows.

A subgroup $\Gama$ of $G$ is called discrete if the identity is not a limit
point of $\Gama$. A fundamental dmain $F$ for a discrete subgroup $\Gama$is
a subdomain of 

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