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APPENDIX B
FOURIER TAUBERIAN THEOREMS
Michael Levitin*
The objective of this appendix is to formulate and prove Fourier Tauberian theo-
rems as theorems of classical analysis without any reference to partial di®erential
equations,spectral theory etc.
The notion of a Tauberian theorem covers a wide range of di®erent mathematical
results,see the history of the subject in [Ga].These results have the following in
common.Suppose that we have some mathematical object with highly irregular
behaviour (say,a discontinuous function or a divergent series) and suppose that we
apply some averaging procedure which makes our object substantially more regular
(say,a transformation which turns our discontinuous function into an in¯nitely
smooth one or makes our divergent series absolutely convergent).A Tauberian
theorem in our understanding is a mathematical result which recovers properties
of the original irregular object from the properties of the averaged object.
Tauberian theorems described in this appendix are associated mainly with the
Fourier transform.We give four main results | Theorems B.2.1,B.3.1,B.4.1 and
B.5.1.TheoremB.2.1 essentially repeats the original Fourier Tauberian Theoremof
rough estimate.Theorem B.4.1 is a re¯ned (with improved remainder estimate)
theoremof the type introduced by Safarov [Sa2]{[Sa6] for studying general (i.e.,not
necessarily polynomial) spectral asymptotics.Finally,Theorem B.5.1 is a special
version of Theorem B.4.1 speci¯cally designed for studying polynomial two-term
spectral asymptotics;a theorem of this sort was implicitly used by Duistermaat
and Guillemin in their pioneering paper [DuiGui].
Hereinafter in this appendix we use the\hat"to denote the Fourier transform
of a function.By a prime we denote the derivative.
B.1.Introductory remarks
It is well known that,under certain conditions,the asymptotic behaviour of a
function at in¯nity is determined by the singularities of its Fourier transform.We
illustrate this fact by the following elementary example.
Example B.1.1.Let
^
f(t) be a complex-valued function on R which is in¯nitely
smooth outside the point t = 0,at which the function
^
f(t) together with all its
derivatives has ¯nite left and right limits.We denote
±
k
=
^
f
(k)
(+0) ¡
^
f
(k)
(¡0);k = 0;1;:::
*supported by a Royal Society grant
Typeset by A
M
S-T
E
X
309
310 APPENDIX B.FOURIER TAUBERIAN THEOREMS
If
^
f(t) and all its derivatives vanish faster than any given negative power of jtj as
t!1 then the inverse Fourier transform f(¸) = F
¡1
t!¸
[
^
asymptotic expansion:
(B.1.1) f(¸) »
1

1
X
k=0
±
k
(¡i¸)
k+1
;¸!1:
We shall consider a more complicated situation.First,we allow
^
f to be a
distribution.Second,we allow
^
f to have singularities not only at the origin.Third,
we assume that information on
^
f(t) is given only on a ¯nite time interval rather
than for all t 2 R.Under these assumptions it is impossible to construct a full
asymptotic expansion of the type (B.1.1).Nevertheless,it turns out that for a
monotone function f one can still relate the behaviour of f(¸) at in¯nity to the
singularities of its Fourier transform.This relation is the subject of the Fourier
Tauberian theorems formulated in this appendix.
We shall denote by F
+
the class of real-valued monotone non-decreasing func-
tions N on R such that N(¸) = 0 for l 6 0,and ¸
¡p
N(¸)!0 as ¸!+1,
where p is some positive number depending on the particular function N.The
latter condition can be rewritten as
(B.1.2) N(¸) = o(¸
p
);¸!+1:
Let us ¯x a real-valued function ½ on R satisfying the following ¯ve conditions:
(1) ½ 2 S(R);
(2) ½(¸) > 0 for all ¸ 2 R;
(3) ^½(0) =
R
½(¸) d¸ = 1;
(4) supp ^½ is compact;
(5) the function ½(¸) is even.
Such a function ½ exists (see,for instance,[HÄo3,vol.3,Sect.17.5]).Note that
under conditions (1){(5) the Fourier transform ^½ is a real even function.
We shall denote ½
T
(¸):= T½(T¸),^½
T
(t):= ^½(t=T),where T is a positive
parameter.Obviously,if the function ½ satis¯es the ¯ve conditions stated above
then ½
T
satis¯es these conditions as well.
Throughout this appendix º will denote some real number,not necessarily pos-
itive.All subsequent asymptotics in this appendix are written with respect to
¸!+1.Of course,asymptotics with respect to ¸!¡1 are of no interest be-
cause for all N 2 F
+
we have N(¸) ´ 0 and (N¤½)(¸) = O(j¸j
¡1
) as ¸!¡1.
B.2.Basic theorem
In this section we will be using the notation C for various positive constants
which may depend only on the choice of the function ½ and on the number º but
are independent of the function N.
Theorem B.2.1.If N 2 F
+
,and
(B.2.1) (N
0
¤ ½)(¸) 6 ¸
º
;8¸ > 1;
then
(B.2.2) jN(¸) ¡ (N ¤ ½)(¸)j 6 C¸
º
;8¸ > 1:
By an elementary rescaling argument Theorem B.2.1 immediately implies
B.2.BASIC THEOREM 311
Corollary B.2.2.If N 2 F
+
and (N
0
¤ ½)(¸) = O(¸
º
) then
(B.2.3) N(¸) = (N ¤ ½)(¸) + O(¸
º
):
Moreover,if the estimate (N
0
¤ ½)(¸) = O(¸
º
) holds uniformly on some subset of
F
+
,then (B.2.3) is also uniform.
Clearly,formula (B.2.3) can be rewritten as
N(¸) = F
¡1
t!¸
[^½(t)
^
N(t)] +O(¸
º
);
which means that,as promised,we have established the relation between the be-
haviour of N(¸) at in¯nity and the behaviour of
^
N(t) in the neighbourhood of
the origin.
The proof of Theorem B.2.1 is based on the following technical lemma.
Lemma B.2.3.Under the conditions of Theorem B.2.1
(B.2.4) jN(¸ +s) ¡N(¸)j 6 C(1 +jsj)
1+jºj
¸
º
;8¸ > 1;
uniformly over all s 2 R.
Proof.It is su±cient to prove (B.2.4) in the case when ¸ and s are integers.
Indeed,the general case is reduced to this one by perturbing ¸ and s to one of
their two nearest integers and using the monotonicity of N.Note also that the
case s = 0 is trivial.Thus,in order to prove the lemma it is su±cient to establish
the following two estimates:
(B.2.5) N(¸ +s) ¡N(¸) 6 Cs
1+jºj
¸
º
;8¸;s 2 N;
(B.2.6) N(¸) ¡N(¸ ¡s) 6 Cs
1+jºj
¸
º
;8¸;s 2 N:
We have
(B.2.7) (N
0
¤ ½)(¸) =
Z
½(¸ ¡¹) dN(¹) > C
¸+1
Z
¸¡1
dN(¹)
= C
¡
(N(¸ +1) ¡N(¸)) +(N(¸) ¡N(¸ ¡1)
¢
(here we have used the fact that ½(¤) > C > 0 for ¤ 2 [¡1;1] ).Formulae (B.2.1),
(B.2.7) imply
(B.2.8) N(¸ +1) ¡N(¸) 6 C¸
º
;8¸ 2 N;
(B.2.9) N(¸) ¡N(¸ ¡1) 6 C¸
º
;8¸ 2 N:
312 APPENDIX B.FOURIER TAUBERIAN THEOREMS
Adding up the inequalities (B.2.8) we get
N(¸ +s) ¡N(¸) =
s
X
k=1
(N(¸ +k) ¡N(¸ +k ¡1))
6 C
s
X
k=1
(¸ +k ¡1)
º
6
½
Cs(¸ +s ¡1)
º
;if º > 0;
Cs¸
º
;if º < 0:
Since in the case º > 0 we have (¸ +s ¡1)
º
6 s
º
¸
º
,the above formula implies
(B.2.5).
Let now prove (B.2.6).It is su±cient to prove (B.2.6) for s 6 ¸ because for
s > ¸ we have N(¸) ¡N(¸¡s) = N(¸) ¡N(¸¡¸).So further on s 6 ¸.Adding
up the inequalities (B.2.9) we get
N(¸) ¡N(¸ ¡s) =
s
X
k=1
(N(¸ ¡k +1) ¡N(¸ ¡k))
6 C
s
X
k=1
(¸ ¡k +1)
º
6
½
Cs¸
º
;if º > 0;
Cs(¸ ¡s +1)
º
;if º < 0:
Since in the case º < 0 we have (¸ ¡s +1)
º
6 s
jºj
¸
º
,the above formula implies
(B.2.6).¤
Proof of Theorem B.2.1.Using (B.2.4) we obtain
j(N ¤ ½)(¸) ¡N(¸)j =
¯
¯
¯
¯
Z
(N(¸ ¡¹) ¡N(¸)) ½(¹) d¹
¯
¯
¯
¯
6
Z
jN(¸ ¡¹) ¡N(¸)j ½(¹) d¹ 6 C¸
º
Z
(1 +j¹j)
1+jºj
½(¹) d¹ = C¸
º
for all ¸ > 1.¤
The following is a weighted version of Theorem B.2.1.
Theorem B.2.4.Let N 2 F
+
and
(B.2.10) (N
0
¤ ½
T
)(¸) 6 d¸
º
;8¸ > T
¡1
;
where d > 0,T > 0 are parameters.Then
(B.2.11) jN(¸) ¡ (N ¤ ½
T
)(¸)j 6 CdT
¡1
¸
º
;8¸ > T
¡1
:
Here the constant C is the same as in Theorem B.2.1.
Proof.Set
e
¸:= T¸,
e
N(
e
¸):= d
¡1
T
1+º
N(
e
¸=T).Then (B.2.10) is equivalent
to
(
e
N
0
¤ ½)(
e
¸) 6
e
¸
º
;8
e
¸ > 1:
Therefore by Theorem B.2.1
j
e
N(
e
¸) ¡ (
e
N ¤ ½)(
e
¸)j 6 C
e
¸
º
;8
e
¸ > 1:
The latter is equivalent to (B.2.11).¤
B.4.GENERAL REFINED THEOREM 313
B.3.Rough estimate for the nonzero singularities
In this section we give a simple result which shows that the singularities of the
distribution F
¸!t
[N
0
(¸)] at di®erent t are not totally independent.Namely,we
show that the singularities at t 6
= 0 cannot be stronger than the singularity at
t = 0.
Theorem B.3.1.If N 2 F
+
and
(B.3.1) (N
0
¤ ½)(¸) = O(¸
º
);
then for any function ° such that ^° 2 C
1
0
(R) we have
(B.3.2) (N
0
¤ °)(¸) = O(¸
º
);
and moreover
(B.3.3) limsup
¸!+1
j(N
0
¤ °)(¸)j
¸
º
6 C
½;°
limsup
¸!+1
(N
0
¤ ½)(¸)
¸
º
;
where C
½;°
> 0 is a constant independent of the choice of the function N and of
the number º.
Proof.Let us choose a number ± > 0 such that ^½(t) 6
= 0 on [¡±;±].Further
on we assume without loss of generality that diam(supp ^°) 6 2±.Indeed,this
can always be achieved by representing the original function ° as a ¯nite sum of
functions possessing this property.
Let us choose a ¿ 2 R such that supp ^° 2 [¿ ¡±;¿ +±] and denote
®(¸) = e
i¿¸
½(¸);¯(¸) = F
¡1
t!¸
·
^°(t)
^®(t)
¸
:
Then
(B.3.4) (N
0
¤ °)(¸) = ((N
0
¤ ®) ¤ ¯)(¸):
As ½ is a nonnegative function and dN is a nonnegative measure we have
(B.3.5) j(N
0
¤ ®)(¸)j 6 (N
0
¤ ½)(¸):
Formulae (B.3.4),(B.3.5),(B.3.1) imply (B.3.3) with C
½;°
=
R
j¯(¹)j d¹:¤
B.4.General re¯ned theorem
In this section we denote by C positive constants (maybe,di®erent) which
may depend only on the functions N
j
,½ and on the number º,but not on the
parameters s,T and".If a constant depends on some of the parameters s,T
or"this will be indicated by respective subscripts.
314 APPENDIX B.FOURIER TAUBERIAN THEOREMS
Theorem B.4.1.Let N
j
2 F
+
,(N
0
j
¤ ½)(¸) = O(¸
º
),j = 1;2,
(B.4.1) (N
2
¤ ½)(¸) = (N
1
¤ ½)(¸) + o(¸
º
)
and
(B.4.2) (N
0
2
¤ °)(¸) = (N
0
1
¤ °)(¸) + o(¸
º
)
for any function ° such that ^° 2 C
1
0
(R),supp ^° ½ (0;+1).Then
N
1
(¸ ¡") ¡ o(¸
º
) 6 N
2
(¸) 6 N
1
(¸ +") + o(¸
º
);8"> 0;
or equivalently
(B.4.3) N
1
(¸ ¡o(1)) ¡ o(¸
º
) 6 N
2
(¸) 6 N
1
(¸ +o(1)) + o(¸
º
):
Formula (B.4.3) means that there exists a (positive) function f
such that f(¸)!0 as ¸!+1,and
N
1
(¸ ¡f(¸)) ¡ ¸
º
f(¸) 6 N
2
(¸) 6 N
1
(¸ +f(¸)) + ¸
º
f(¸):
Formula (B.4.3) di®ers fromstandard asymptotic formulae (like (B.2.3) above or
(B.5.2) below) in the sense that we are comparing the graphs of the functions N
1
(¸)
and N
2
(¸) not only in the\vertical"direction,but in the\horizontal"direction
as well.This is natural when one compares the graphs of two counting functions:
trying to increase accuracy we inevitably have to start comparing the graphs in
both directions due to the possible presence of discontinuities.This phenomenon is
well known in probability theory,where graphs of monotone functions are compared
using the so-called L¶evy metric,[GnKo,Chapter 2,Section 9].The basic idea in the
de¯nition of the classical L¶evy metric is to measure the distance between the graphs
in the direction forming an angle of

4
with the positive ¸-semiaxis.TheoremB.4.1
can be reformulated in terms of a weighted L¶evy metric,in which the angle is a
function of ¸.
We shall prove Theorem B.4.1 in several steps.First,we prove
Lemma B.4.2.Let N 2 F
+
and (N
0
¤ ½)(¸) = O(¸
º
).Then for all s > 0,
T > 1,¸ > 1 we have the uniform estimate
(B.4.4) (N¤½
T
)(¸¡s)¡C(1+sT)
¡1
¸
º
6 N(¸) 6 (N¤½
T
)(¸+s)+C(1+sT)
¡1
¸
º
:
Proof.For the sake of brevity,we shall prove only the left inequality (B.4.4);
the right one is proved in a similar way.
We have
(N ¤ ½
T
)(¸ ¡s) ¡N(¸) =
Z
¡
N(¸ ¡s ¡T
¡1
¿) ¡N(¸)
¢
½(¿) d¿
6
¡sT
Z
¡1
¡
N(¸ ¡s ¡T
¡1
¿) ¡N(¸)
¢
½(¿) d¿
B.4.GENERAL REFINED THEOREM 315
(here we used the monotonicity of N ).Estimating the integrand by Lemma B.2.3
we obtain
(B.4.5) (N ¤ ½
T
)(¸ ¡s) ¡N(¸) 6 C¸
º
+1
Z
sT
(1 ¡s +T
¡1
¿)
1+jºj
½(¿) d¿
for ¸ > 1.But
(1 ¡s +T
¡1
¿)
1+jºj
6 (1 +¿)
1+jºj
;8¿ > sT
(here we used the inequalities s > 0,T > 1),and
½(¿) 6 C(1 +¿)
¡3¡jºj
;8¿ > 0;
so
+1
Z
sT
(1 ¡s +T
¡1
¿)
1+jºj
½(¿) d¿ = C
+1
Z
sT
(1 +¿)
¡2
d¿ = C(1 +sT)
¡1
:
Substituting the latter into (B.4.5) we arrive at (B.4.4).¤
Remark B.4.3.Obviously,if the estimate (N
0
¤½)(¸) = O(¸
º
) holds uniformly
on some subset of F
+
,then the constant C in (B.4.4) is independent of the par-
ticular function N from this subset (i.e.,(B.4.4) is also uniform).
Lemma B.4.2 implies
Lemma B.4.4.Let N
j
2 F
+
,(N
0
j
¤ ½)(¸) = O(¸
º
),j = 1;2,and
(B.4.6) (N
2
¤ ½
T
)(¸) = (N
1
¤ ½
T
)(¸) + o(¸
º
);8T > 1:
Then (B.4.3) holds.
Proof.Note that both functions N
2
and N
1
satisfy the conditions of Lemma
B.4.2,and therefore estimates (B.4.4) hold.With account of (B.4.6) the inequalities
(B.4.4) can be rewritten as
(B.4.7) (N
k
¤ ½
T
)(¸ ¡s) ¡C(1 +sT)
¡1
¸
º
¡o((¸ ¡s)
º
)
6 N
j
(¸) 6
(N
k
¤ ½
T
)(¸ +s) +C(1 +sT)
¡1
¸
º
+o((¸ +s)
º
);
where j;k = 1;2,j 6
= k.Formulae (B.4.7) imply
(B.4.8) N
1
(¸ ¡2s) ¡C(1 +sT)
¡1

º
+(¸ ¡s)
º
) ¡o((¸ ¡s)
º
)
6 N
2
(¸) 6
N
1
(¸ +2s) +C(1 +sT)
¡1

º
+(¸ +s)
º
) +o((¸ +s)
º
):
316 APPENDIX B.FOURIER TAUBERIAN THEOREMS
Formula (B.4.8) holds for any ¯xed s > 0 and T > 1.So we can set s = T
¡1=2
.
Then (B.4.8) takes the form
(B.4.9) N
1
(¸ ¡2T
¡1=2
) ¡CT
¡1=2
³
¸
º
+(¸ ¡T
¡1=2
)
º
´
¡o
³
(¸ ¡T
¡1=2
)
º
´
6 N
2
(¸) 6
N
1
(¸ +2T
¡1=2
) +CT
¡1=2
³
¸
º
+(¸ +T
¡1=2
)
º
´
+o
³
(¸ +T
¡1=2
)
º
´
:
It remains only to set T = T(¸),where T(¸) is an increasing function which
tends to +1 as ¸!+1.The function T = T(¸) can be chosen to increase so
slowly that for both o-terms appearing in (B.4.9) we have o
¡
(¸ ¨T
¡1=2
(¸))
º
¢
=
o(¸
º
);this remark is necessary because our o-terms (originating from (B.4.6))
might depend on T.
Formula (B.4.9) with the substitution T = T(¸) implies (B.4.3).¤
Proof of Theorem B.4.1.In view of Lemma B.4.4,it is su±cient to show
that (B.4.1) and (B.4.2) imply (B.4.6).Let us split ^½
T
into the sum of functions

T;1
;^½
T;2
2 C
1
0
(R) such that ^½(t) > C > 0 for t 2 supp ^½
T;1
,and ^½
T;2
(t) vanishes
in a neighbourhood of t = 0.Then (B.4.2) implies (B.4.6) for ½
T;2
because
(N
j
¤ ½
T;2
)(¸) = F
¡1
t!¸
h
(it)
^
N
j
(t) (it)
¡1

T;2
(t)
i
= (N
0
j
¤ e½
T;2
)(¸)
where j = 1;2,and e½
T;2
(¸) = F
¡1
t!¸
£
(it)
¡1

T;2
(t)
¤
.Obviously,
(B.4.10) N
j
¤ ½
T;1
= N
j
¤ ½ ¤ ¯;¯(¸) = F
¡1
t!¸
·

T;1
(t)
^½(t)
¸
;j = 1;2:
Since the function ¯ is rapidly decreasing,(B.4.1) and (B.4.10) imply (B.4.6) for
½
T;1

B.5.Special version of the general re¯ned theorem
We formulate below a special version of the general re¯ned theorem speci¯cally
oriented towards the case when N has a polynomial two-term asymptotics,i.e.,
when the nonzero singularities of the Fourier transform of N
0
are weaker than the
singularity at the origin.
Theorem B.5.1.Let N 2 F
+
,(N
0
¤ ½)(¸) = O(¸
º
),and
(B.5.1) (N
0
¤ °)(¸) = o(¸
º
);
for any function ° such that ^° 2 C
1
0
(R),supp ^° ½ (0;+1).Then
(B.5.2) N(¸) = (N ¤ ½)(¸) + o(¸
º
):
Remark B.5.2.As will be clear from the proof,if the conditions of Theorem
B.5.1 are ful¯lled uniformly on some subset of F
+
then (B.5.2) also holds uniformly.
B.5.SPECIAL VERSION OF THE GENERAL REFINED THEOREM 317
Theorem B.5.1 has a simpler formulation than Theorem B.4.1,and it can be
viewed as a natural extension of Corollary B.2.2.However,Theorem B.5.1 requires
more restrictive conditions on the functions involved than TheoremB.4.1.Basically,
the conditions of TheoremB.5.1 ensure that the (discontinuous) monotone function
N does not have very big\jumps",and this is why we are able to compare the
graphs in the\vertical"direction only.
Proof of Theorem B.5.1.According to the Lagrange formula we have for
some µ
¨
(¸) 2 (0;1)
(B.5.3) (N ¤ ½)(¸ ¨o(1)) ¡ (N ¤ ½)(¸)
= o(1) (N
0
¤ ½) (¸ ¨µ
¨
(¸) o(1)) = o(1) O(¸
º
) = o(¸
º
):
In view of Lemma B.4.2 and (B.5.3),it is su±cient to prove that
(B.5.4) (N ¤ ½)(¸) ¡ (N ¤ ½
T
)(¸) = (N ¤ (½ ¡½
T
)) (¸) = o(¸
º
);8T > 1:
Indeed,then (B.5.2) is obtained from(B.4.4) by substitution s = f
1
(¸),T = f
2
(¸),
where f
1
and f
2
are arbitrary functions such that
f
1
(¸)!0;f
2
(¸)!+1;f
1
(¸) f
2
(¸)!+1;¸!+1:
Since ^½(0) = 1 and ^½
0
(0) = 0,the function ^½(t) ¡ ^½
T
(t) has a second order
zero at t = 0.Therefore (B.5.4) is a consequence of the following lemma,which
completes the proof of Theorem B.5.1.
Lemma B.5.3.Let N 2 F
+
satisfy the conditions of Theorem B.5.1.Assume
that ^½
0
2 C
1
0
(R) and ^½
0
(t) = t
2
^®(t),where ^® 2 C
1
0
(R).Then
(B.5.5) (N ¤ ½
0
)(¸) = o(¸
º
):
Proof.We have
(B.5.6) (N ¤ ½
0
)(¸) = F
¡1
t!¸
[
^
N(t) t
2
^®(t)] = (N
0
¤ ¯)(¸);
where ¯ = F
¡1
t!¸
[¡it ^®(t)].
Let
^
f 2 C
1
0
(R) and
^
f(t) = 1 in a neighbourhood of origin.Denote
^
f
±
(t) =
^
f(t=±) and
e
f
±
(¸) = F
¡1
t!¸
[¡it
^
f
±
(t)].Then
e
f
±
(¸) = ±
2
F
¡1
t!±¸
[¡it
^
f(t)] and
Z
j
e
f
±
(¸)j d¸ = ± C
f
;
where C
f
=
R
jF
¡1
t!¸
[¡it
^
f(t)] j d¸.
Clearly,
(B.5.8) ¯(¸) = F
¡1
t!¸
[¡it (1 ¡
^
f
±
(t)) ^®(t)] + F
¡1
t!¸
[¡it
^
f
±
(t) ^®(t)]:
In view of (B.5.1),the contribution to (B.5.6) of ¯rst term in the right-hand side of
(B.5.8) is o(¸
n
).The contribution to (B.5.6) of the second term in the right-hand
side of (B.5.8) is (N
0
¤ ® ¤ f
±
).By Theorem B.3.1 (N
0
¤ ®)(¸) is estimated by
C(¸
º
+1),so
j (N
0
¤ ® ¤ f
±
)(¸) j 6 ± CC
f

º
+1);
Thus,
j (N ¤ ½
0
)(¸) j 6 ± CC
f

º
+1) + o(¸
n
):
Since ± can be chosen arbitrarily small,this implies (B.5.5).¤