SELBERG TRACE FORMULA 19
mentioned inthe introduction:
T
(1.18a) Mr(r,;t,T) = - ^/_T0p(r)E(.,£+ir), E(.,Jfir)dr
(if r is cuspidal)
T
(1.18b) = - ^/.
T
R[|E(.,^ir)|
2
, *+ir]dr
(if a = E(.,Jfir), e.g.)
Much ofthe analysis ofthis paper isdevoted toestimating the contribution
of these continuous spectral objects toVeyl laws and tothe error of the
prime geodesic theorem.
Finally, wewould like todiscuss a technical point raised above.
Namely, some ofour applications ofthe trace formula require the useof
non-compactly supported test functions. Most importantly, these test
functions occur when weuse the version (1.11) ofthe formula. Here j 6
CQ(R+),
but f = H" j t
CQ.
Ofcouse, it isquite commomplace touse test
functions ^ t CQ,butthe validity of (1.11) must beestablished for them.
The main point, aswill bediscussed indetail in§ 5,istojustify the trace
formulae for^ e OlR+) bya continuity argument. Namely, weview the trace
formula asan identity between linear functionals onCQ. Ve show that these
functionals arecontinuous ona certain Banach space inwhich CQ isdense.
The test functions ^ which are required inapplications will turn out to lie
in this space. This argument, implicit in [Z2], § 4,will bemade explicit in
§ 5.
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