Fourier interpolation on the real line

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by  Danylo Radchenko

FOURIER INTERPOLATION ON THE REAL LINE
DANYLO RADCHENKO AND MARYNA VIAZOVSKA
Abstract. We use weakly holomorphic modular forms for the Hecke theta group to
construct an explicit interpolation formula for Schwartz functions on the real line. The
formula expresses the value of a function at any given point in terms of the values of
√ √ √
the function and its Fourier transform on the set {0,± 1,± 2,± 3,...}.
7 1. Introduction
1
0 Let f: R → R be an integrable function and let f(cid:98)be the Fourier transform of f:
2
(cid:90) ∞
n f(cid:98)(ξ) = f(x)e−2πiξxdx.
a
J −∞

1 The classical Whittaker-Shannon interpolation formula (see [16], [12]) states that if the

] Fourier transform f(cid:98)is supported in [−w/2,w/2], then
T
(cid:88)
N f(x) = f(n/w)sinc(wx−n),
h. n∈Z
t
a where sinc(x) = sin(πx)/(πx) is the cardinal sine function. In other words, the func-
m tions s (x) = sinc(wx − n) form an interpolation basis on the set 1Z for the space
n w
[
of functions whose Fourier transform is supported in [−w/2,w/2] (the so-called Paley-

1 Wiener space PW ). For a nice overview of history of the Whittaker-Shannon formula,
w
v
its generalizations and other related results, see [10]. A generalization of the formula that
5
6 is useful for certain extremal problems in Fourier analysis is described in [13]
2 Note that it is not possible to apply the Whittaker-Shannon formula directly to func-
0
0 tions whose Fourier transform f(cid:98) has unbounded support, say, to f(x) = exp(−πx2).
. The main goal of this paper is to prove an interpolation formula that can be applied to
1
0 arbitrary Schwartz functions on the real line.
7
1 Theorem 1. There exists a collection of even Schwartz functions a : R → R with the
: n
v property that for any even Schwartz function f: R → R and any x ∈ R we have
i
X
(cid:88)∞ √ (cid:88)∞ √
ar (1) f(x) = an(x)f( n)+ (cid:98)an(x)f(cid:98)( n),
n=0 n=0
where the right-hand side converges absolutely.
As immediate corollary of Theorem 1, we get the following.
Corollary 1. Let f: R → R be an even Schwartz function that satisfies
√ √
f( n) = f(cid:98)( n) = 0, n ∈ Z .
≥0
Then f vanishes identically.
Denote by s the vector space of all rapidly decaying sequences of real numbers, i.e.,
sequences(x ) suchthatforallk > 0wehavenkx → 0,n → ∞. IfwedenotebyS
n n≥0 n even
1
2 RADCHENKO AND VIAZOVSKA
the space of even Schwartz functions on R (see Section 6), then there is a well-defined
map Ψ: S → s⊕s given by
even
√ √
Ψ(f) = (f( n)) ⊕(f(cid:98)( n)) .
n≥0 n≥0
Together with Theorem 1 the following result gives a complete description of what values
√
an even Schwartz function and its Fourier transform can take at ± n for n ≥ 0.
Theorem 2. The map Ψ is an isomorphism of the space of even Schwartz functions onto
the vector space kerL ⊂ s⊕s, where L: s⊕s → R is the linear functional
(cid:88) (cid:88)
L((x ) ,(y ) ) = x − y .
n n≥0 n n≥0 n2 n2
n∈Z n∈Z
In the proof of Theorem 1 we will give an explicit construction of the interpolating
basis {a (x)} . For instance, the Fourier invariant part of a will be given by
n n≥0 n
(cid:90) 1
a (x)+a (x) = g(z)eiπx2zdz,
n (cid:98)n
−1
where g is a certain weakly holomorphic modular form of weight 3/2, and the integral is
over a semicircle in the upper half-plane. The anti-invariant part a (x)−a (x) will be
n (cid:98)n
defined by a similar expression. For an explicit example, we define a (x) by
0
1 (cid:90) 1
a (x) = θ3(z)eiπx2zdz,
0
4
−1
where θ(z) is the classical theta series
(cid:88)
(2) θ(z) = eiπn2z.
n∈Z
The modular transformation property of g is chosen in such a way that it complements
the action of the Fourier transform on Gaussian functions:
1
e (ξ) = √ e (ξ),
(cid:98)z −1/z
−iz
where e (x) = eiπzx2, and the square root is chosen to be positive when z lies on the
z
imaginary axis (this comment also applies whenever expression (−iz)α occurs throughout
the paper). For instance, using the identity
(cid:16) 1(cid:17) √
θ − = −izθ(z)
z
and applying the change of variable z (cid:55)→ −1/z in the integral that defines a (x) we see
0
that a = a . The general definition of a needs some preparation, and will be given in
(cid:98)0 0 n
Section 4. The plots of the first three functions are shown in Figure 1.
An analogue of Theorem 1 holds also for odd Schwartz functions, but we postpone
its formulation until Section 7. It is possible to combine the two results into a general
interpolation theorem, but it is more convenient to work with the two cases separately.
Remark. Another way to interpret equation (1) is to think of it as a “deformation”
of the classical Poisson summation formula
(cid:88) (cid:88)
(3) f(n) = f(cid:98)(n),
n∈Z n∈Z
which will be a special case of (1) for x = 0 (more precisely, −a (0) = a (0) = 1
n2 (cid:99)n2
for n ≥ 1, a (0) = a (0) = 1/2, and all other values are zero).
0 (cid:98)0
Our general approach fits into the framework of Eichler cohomology (see [5]; some
relevant results can also be found in [6] and [7]), but for the most part we avoid using its
FOURIER INTERPOLATION ON THE REAL LINE 3
1 1
2a
(cid:98)0
a
(cid:98)1
a
(cid:98)2
√ √ √ √
0 1 2 3 0 1 2 3
2a
0
a
1
a
2
Figure 1. Plots of a and a for n = 0,1,2.
n (cid:98)n
general results and terminology. In our case we obtain estimates with explicit constants
by simpler methods, and this also allows us to keep the proofs relatively self-contained.
Let us also note that functions with properties similar to that of a have recently been
n
used in [14] and [2] to solve the sphere packing problem in dimensions 8 and 24.
The paper is organized as follows. In Section 2 we recall some known facts about
modular forms for the Hecke theta group Γ . In Section 3 we compute explicit basis of
θ
a certain space of weakly holomorphic modular forms of weight 3/2 for the group Γ .
θ
Then, in Section 4 we use these modular forms to construct an interpolation basis for the
even Schwartz functions and prove some of its properties. In the next section we prove an
estimate on the growth of this sequence functions, this is by far the most technical part
of the paper. In Section 6 we prove the main result for even functions, and in Section 7
we define the interpolation basis and formulate corresponding statements for the odd
functions.
Acknowledgements. The authors would like to thank Max Planck Institute for Math-
ematics, Bonn for hospitality and support while this paper was being written. The first
author would also like to thank the Absus Salam International Center for Theoretical
Physics, Trieste for the financial support.
The authors are grateful to Don Zagier, Emanuel Carneiro, and Andrew Bakan for
many helpful remarks and comments.
2. Hecke theta group
In this section we set up notation and collect facts about the Hecke theta group and
related modular forms. Most of the material from this section can be found, in much
greater detail, in [9]. For a motivated general introduction to the theory of modular
forms, see [18].
2.1. Upper half-plane and the action of SL (R). Denote by H the complex upper
2
half-plane {z ∈ C : Im(z) > 0}. The group SL (R) of 2×2 matrices with real coefficients
2
4 RADCHENKO AND VIAZOVSKA
and determinant 1 acts on the upper half-plane on the left by Moebius transformations
(cid:18) (cid:19)
az +b a b
γz = , γ = ∈ SL (R).
cz +d c d 2
The kernel of this action coincides with the center {±I} of SL (R) and thus we can work
2
with the action of PSL (R) = SL (R)/{±I} instead.
2 2
We will use special notation for the following elements of SL (Z) (or, by abuse of
2
notation, of PSL (Z)):
2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
1 0 1 1 0 −1
I = , T = , S = .
0 1 0 1 1 0
Recall that Γ(2) ⊂ SL (Z) is defined as
2
(cid:26) (cid:12) (cid:18) (cid:19) (cid:27)
Γ(2) = A ∈ SL (Z) (cid:12)(cid:12) A ≡ 1 0 (mod 2) ,
2 (cid:12) 0 1
and Γ is the subgroup of SL (Z) generated by S and T2, or, equivalently,
θ 2
(cid:26) (cid:12) (cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
Γ = A ∈ SL (Z) (cid:12)(cid:12) A ≡ 1 0 or 0 1 (mod 2) .
θ 2 (cid:12) 0 1 1 0
Note the obvious inclusions SL (Z) ⊃ Γ ⊃ Γ(2). The group Γ(2) has three cusps 0, 1,
2 θ
and ∞, while the group Γ has only two cusps: 1 and ∞. The standard fundamental
θ
domain for Γ is
θ
(4) D = {τ ∈ H : |τ| > 1, Re(τ) ∈ (−1,1)}.
Finally, we are going to use the “θ-automorphy factor” on the group Γ , which we
θ
define for all z ∈ H and γ ∈ Γ by
θ
θ(z)
(5) j (z,γ) = .
θ
θ(γz)
From the definition it immediately follows that j (z,γ γ ) = j (z,γ )j (γ z,γ ), so j is
θ 1 2 θ 2 θ 2 1 θ
indeed an automorphy factor on Γ . We have j (z,T2) = 1 and j (z,S) = (−iz)−1/2, and
θ θ θ
in general we have j (z,(a b)) = ζ · (cz + d)−1/2 for some suitable 8-th root of unity ζ
θ c d
(an explicit expression for ζ can be found in [9, Th. 7.1]). Using this automorphy factor
we define the following slash operator in weight k/2 (that acts on holomorphic functions
defined on the upper half-plane H)
(cid:16)az +b(cid:17)
(cid:0) (cid:1)
(6) f| A (z) = j (z,A)kf ,
k/2 θ
cz +d
where A = (a b) ∈ Γ . More generally, for ε ∈ {−,+} define a slash operator |ε by
c d θ k/2
(7) f|ε A = χ (A)f| A,
k/2 ε k/2
where χ : Γ → {±1} is the homomorphism defined by χ (S) = ε and χ (T2) = 1. The
ε θ ε ε
slash operator defines a group action, that is, f|AB = (f|A)|B. Another fact that we
will use is that for all (a b) ∈ SL (R) we have
c d 2
(cid:16)aτ +b(cid:17) Im(τ)
(8) Im = .
cτ +d |cτ +d|2
For any real number a we will denote by qa the analytic function
qa = qa(z) = exp(2πiaz).
Any T-periodic holomorphic function on H admits an expansion in powers of q1/T (in
general as a Laurent series, but in our case such expansions will have only finitely many
FOURIER INTERPOLATION ON THE REAL LINE 5
negative powers). We will be using subscripts to indicate the main variable of q, i.e., qa
τ
is the same as qa(τ); by default the variable of qa is z.
2.2. Modular forms for the group Γ . We begin by defining the classical Jacobi theta
θ
series (the so-called Thetanullwerte):
(cid:88) η(2z)2
Θ (z) = q1n2 = 2 ,
2 2
η(z)
n∈Z+1
2
(cid:88) η(z)5
Θ (z) = q1n2 = (= θ(z)),
3 2 η(z/2)2η(2z)2
n∈Z
(cid:88) η(z/2)2
Θ (z) = (−1)nq1n2 = ,
4 2
η(z)
n∈Z
where η(z) = q1/24(cid:81) (1 − qn) is the Dedekind eta function. The functions Θ4, Θ4,
n≥1 2 3
and Θ4 generate the ring of holomorphic modular forms on Γ(2) and satisfy the Jacobi
4
identity
(9) Θ4 = Θ4 +Θ4.
3 2 4
The q-expansions of these forms at the cusp i∞ are as follows:
Θ4(z) = 16q1/2 +64q3/2 +96q5/2 +O(q3),
2
Θ4(z) = 1+8q1/2 +24q +32q3/2 +24q2 +48q5/2 +O(q3),
3
Θ4(z) = 1−8q1/2 +24q −32q3/2 +24q2 −48q5/2 +O(q3).
4
Under the action of SL (Z) the theta functions transform as follows. Under the action
2
of S we have
(−iz)−1/2Θ (−1/z) = −Θ (z),
2 4
(10) (−iz)−1/2Θ (−1/z) = −Θ (z),
3 3
(−iz)−1/2Θ (−1/z) = −Θ (z),
4 2
and under the action of T we have
Θ (z +1) = eiπ/4Θ (z),
2 2
(11) Θ (z +1) = Θ (z),
3 4
Θ (z +1) = Θ (z)
4 3
Together with the q-series for Θ , Θ , and Θ , these transformations allow to compute
2 3 4
the q-series expansion of any expression in theta functions at any of the three cusps
of Γ(2).
Using these theta functions we can define the classical modular lambda invariant
Θ4(z)
λ(z) = 2 = 16q1/2 −128q +704q3/2 +... ,
Θ4(z)
3
which is a Hauptmodul for Γ(2). In particular, we have
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:16)az +b(cid:17) a b 1 0
λ = λ(z), ≡ (mod 2),
cz +d c d 0 1
and any function with these transformation properties and with appropriate behavior at
the cusps can be expressed as a rational function of λ. From (9) – (11) we see that under
6 RADCHENKO AND VIAZOVSKA
the action of PSL (Z) the function λ(z) transforms as follows:
2
(cid:16) 1(cid:17)
λ − = 1−λ(z),
z
(12)
λ(z)
λ(z +1) = .
λ(z)−1
Since Θ , Θ , and Θ do not vanish in H (by the product expression using η(z)), we get
3 2 4
the well-known fact that λ(z) omits the values 0 and 1.
Using λ(z), define a Hauptmodul J for the group Γ
θ
1 Θ4(z)Θ4(z)
(13) J(z) = λ(z)(1−λ(z)) = 2 4 = q1/2 −24q +300q3/2 +... .
16 16Θ8(z)
3
Note that J(z) = η(z/2)24η(2z)24η(z)−48, hence it does not have zeros in H. This function
satisfies the transformation laws
(cid:16) 1(cid:17)
J − = J(z),
z
J(z +2) = J(z),
and it maps the fundamental domain D conformally onto the cut plane C(cid:114)[1/64,+∞).
Finally, note that 1/J vanishes at the cusp 1, since
1
(14) = −4096q −98304q2 +O(q3).
J(1−1/z)
2.3. Asymptotic notation. We freely use the standard big O notation. In addition,
we also use Vinogradov’s “(cid:28)” sign
f (cid:28) g ⇔ f = O (g).
ε,δ,... ε,δ,...
Notationally, we prefer to use “O” for sequences and additive remainders, while for most
inequalities with implied constants we use “(cid:28)”.
3. Weakly holomorphic modular forms on Γ of weight 3/2
θ
We begin by constructing a basis for a certain space of weakly holomorphic modular
forms of weight 3/2. Namely, let {g+(z)} and {g−(z)} be two collections of holo-
n n≥0 n n≥1
morphic functions on the upper half-plane H that satisfy the transformation properties
gε(z +2) = gε(z),
n n
(15)
(−iz)−3/2gε(−1/z) = εgε(z),
n n
as well as the following behavior at the cusps
g+(z) = q−n/2 +O(q1/2), z → i∞,
n
(16) g−(z) = q−n/2 +O(1), z → i∞,
n
gε(1+i/t) → 0, t → ∞.
n
The reason behind these conditions will be made clear in the next section. We make the
following ansatz:
g+(z) = θ3(z)P+(J−1(z)),
n n
(17)
g−(z) = θ3(z)(1−2λ(z))P−(J−1(z)),
n n
where P± ∈ Q[x] are monic polynomials of degree n and P−(0) = 0. The polynomials P±
n n n
are uniquely determined by the first two conditions in (16), since J−1 has q-expansion
starting with q−1/2 + 24 + O(q1/2), and thus the coefficients of P± can be found by
n
FOURIER INTERPOLATION ON THE REAL LINE 7
inverting an upper-triangular matrix. The transformation properties (15) follow from the
properties of J(z) and λ(z). The first few of these functions are
g+ = θ3, g− = θ3 ·(1−2λ)·(J−1),
0 1
g+ = θ3 ·(J−1 −30), g− = θ3 ·(1−2λ)·(J−2 −22J−1),
1 2
g+ = θ3 ·(J−2 −54J−1 +192), g− = θ3 ·(1−2λ)·(J−3 −46J−2 +252J−1).
2 3
The polynomials P± are analogues of Faber polynomials, see [3]. The main difference
n
here is that the group Γ has two cusps, so some additional normalization is needed. In
θ
the next theorem we give closed form expressions for generating functions of {g±}.
n
Theorem 3. The generating functions for {g+(z)} and {g−(z)} are given by
n n≥0 n n≥1
(cid:88)∞ θ(τ)(1−2λ(τ))θ3(z)J(z)
g+(z)eiπnτ = =: K (τ,z),
n J(z)−J(τ) +
n=0
(18)
(cid:88)∞ θ(τ)J(τ)θ3(z)(1−2λ(z))
g−(z)eiπnτ = =: K (τ,z).
n J(z)−J(τ) −
n=1
Proof. The proof follows the same lines as the proof of Theorem 2 from [4]. We only
prove the statement for g+, since the case of g− is almost identical. From the q-expansion
n n
of J−1 and the fact that
J(z) (cid:88)
= Jn(τ)J−n(z),
J(z)−J(τ)
n≥0
it is clear that the g+ defined by (18) are also of the form θ3(z)P (J−1(z)) for some monic
n n
polynomial P of degree n. The only thing that we need to check is that they satisfy
n
g+(z) = q−n/2 +O(q1/2), z → i∞,
n
or, equivalently, that P = P+. By Cauchy’s theorem we know that
n n
1 (cid:90) τ0+2 1 (cid:73)
g+(z) = K (τ,z)q−n/2dτ = K (τ,z)q−(n+1)/2d(q1/2),
n 2 + τ 2πi + τ τ
τ0 C
where τ ∈ H has sufficiently large imaginary part and C is a small enough loop around 0
0
in the q1/2-plane. Using the identity
τ
dJ J(cid:48)(τ)
(19) q1/2 (τ) = = θ4(τ)(1−2λ(τ))J(τ)
τ d(q1/2) πi
τ
we get that
q1/2 dJ (τ)
K (τ,z) = τ d(qτ1/2) · θ3(z)J(z),
+ J(z)−J(τ) θ3(τ)J(τ)
and thus changing the variable of integration we get
1 (cid:73) (q1/2(j))−n θ3(z)J(z)
g+(z) = τ · dj.
n 2πi J(z)−j θ3(τ)j
C˜
Now recall that θ3(z)P+(J−1(z)) = q−n/2+O(q1/2), so that (θ3(τ)P+(j−1)−q−n/2(j))/j
n n τ
is holomorphic in some small neighborhood of 0 in the j-plane. Therefore, for some small
8 RADCHENKO AND VIAZOVSKA
˜
loop C around zero, we have
1 (cid:73) (q1/2(j))−n θ3(z)J(z) θ3(z) (cid:73) P+(j−1)
g+(z) = τ · dj = n J(z)dj
n 2πi J(z)−j θ3(τ)j 2πi j(cid:0)J(z)−j(cid:1)
C˜ C˜
θ3(z) (cid:73) P+(j−1)
= − n dj−1 = θ3(z)P+(J−1(z)).
2πi J(z)(cid:0)j−1 −J−1(z)(cid:1) n
C˜
The last sign is changed since the contour for j−1 in the last application of Cauchy’s
formula has the opposite orientation. (cid:3)
Remark. From (19) it also follows that K (τ,z) has a simple pole at z = τ with
ε
residue 1 for all τ ∈ H. We also record here the following identities for K :
iπ ε
K (τ,−1/z) = ε(−iz)3/2K (τ,z),
ε ε
(20)
K (−1/τ,z) = −ε(−iτ)1/2K (τ,z).
ε ε
Note that generating functions very similar to (18) have also been used in [17] in the
computation of traces of singular moduli.
4. Interpolation basis for even functions
Let us define a function bε : R → R by the integral
m
1 (cid:90) 1
(21) bε (x) = gε (z)eiπx2zdz,
m 2 m
−1
where the path of integration is chosen to lie in the upper half-plane and orthogonal to
the real line at the endpoints 1 and −1. Since gε has exponential decay at ±1, the above
m
integral converges. Note that bε is defined for m ≥ 0 if ε = +1 and for m ≥ 1 if ε = −1;
m
for convenience let us also define b−(x) = 0.
0
Proposition 1. The function bε : R → R is an even Schwartz function that satisfies
m
b(cid:99)ε (x) = εbε (x)
m m
and
√
bε ( n) = δ , n ≥ 1, m ≥ 0,
m n,m
where δ is the Kronecker delta. In addition, we have b+(0) = 1.
n,m 0
Proof. Clearly, bε is an even function, since e (x) = eiπx2z is even.
m z
Let us first prove that bε belongs to the Schwartz class. We will only consider the
m
case “ε = +”, but the same argument will work also in the case “ε = −”. Since g+(z) =
n
θ3(z)P+(J−1(z)), it is enough to prove that for each n ∈ N the integral
n
1 (cid:90) 1
β (x) = θ3(z)J−n(z)eiπx2zdz
n
2
−1
is a Schwartz function. On the circle arc from −1 to 1 the function J−1(z) takes real
values between 0 and 64, and moreover
J−1(±1+i/t) ≤ Cexp(−2πt), t → ∞, Re(t) > 0.
By taking the k-th derivative of β (x) with respect to x under the integral we obtain
n
1 (cid:90) 1
β(k)(x) = θ3(z)J−n(z)Q (x,z)eiπx2zdz,
n 2 k
−1
FOURIER INTERPOLATION ON THE REAL LINE 9
where Q (x,z) are polynomials that satisfy Q = 1 and
k n,0
∂
Q (x,z) = Q (x,z)+(2πizx)Q (x,z).
k+1 k k
∂x
Clearly, there exists a constant C such that
k
|Q (x,z)| ≤ C (1+|x|2)k(1+|z|2)k,
k k
thus we get
(cid:90) 1/2
|β(k)(x)| ≤ π2k+3C (1+|x|2)k J−n(eiπt)e−πx2sin(πt)dt.
n k
0
Here we used a rather crude estimate |θ(eiπt)| < 2 for t ∈ (0,1/2). When |x| is small, we
estimate the above integral by 64n, for all other values of x we estimate the integral by
√
splitting it into two parts (where we take δ = ( πx)−1):
(cid:90) 1/2 (cid:90) δ (cid:90) 1/2
J−n(eiπt)e−πx2sin(πt)dt = J−n(eiπt)e−πx2sin(πt)dt+ J−n(eiπt)e−πx2sin(πt)dt
0 0 δ
√ √
≤ Cδe−2/δ +64ne−2πδx2 = e−2 πx(64n +C/(x π)),
from which it follows that β is a Schwartz function.
n
To check that b(cid:99)ε = εbε we will use the fact that e = (−iz)−1/2e and the trans-
m m (cid:98)z −1/z
formation property (15):
1 (cid:90) 1
b(cid:99)ε (x) = gε (z)(−iz)−1/2eiπx2(−1/z)dz
m 2 m
−1
1 (cid:90) 1
= −gε (z)(−iz)3/2eiπx2(−1/z)d(−1/z)
2 m
−1
1 (cid:90) −1
= εgε (−1/z)eiπx2(−1/z)d(−1/z) = εbε (x).
2 m m
1
In the above computations we always choose the branch of (−iz)k/2 that takes positive
values for z on the imaginary semiaxis. Finally, note that
√ 1 (cid:90) 1
bε ( n) = gε (z)eiπnzdz
m 2 m
−1
issimplythecoefficientofq−n/2 intheq-expansionofgε , sothat(16)immediatelyimplies
√ m
bε ( n) = δ and b+(0) = 1. (cid:3)
m n,m 0
Remark. Note that (16) also implies that b+(0) = δ , and using the explicit for-
m m,0
mula (18) for the kernel K , we also get
−
(cid:40)
−2, m ≥ 1 is a square,
b−(0) =
m 0, otherwise.
Alternatively, this last equation follows from the Poisson summation formula
(cid:88) (cid:88) (cid:88)
b−(n) = b(cid:99)−(n) = − b−(n).
m m m
n∈Z n∈Z n∈Z
To establish other properties of the sequences {bε (x)} we will need to work with
m m
generating functions. Let D be the standard fundamental domain for the group Γ (as
θ
defined in (4)). For a fixed x define a function F (τ,x) on the set
ε
{τ ∈ H : ∀k ∈ Z, |τ −2k| > 1} ⊃ D+2Z
10 RADCHENKO AND VIAZOVSKA
by
1 (cid:90) 1
(22) F (τ,x) = K (τ,z)eiπx2zdz,
ε ε
2
−1
where the contour is the semicircle in the upper half-plane that passes through −1 and 1.
Note that for Im(τ) > 1 we have
∞
(cid:88)
(23) F (τ,x) = bε(x)eiπnτ,
ε n
n=0
and the series converges absolutely. Our next task is to show that this identity holds for
all τ ∈ H.
Proposition 2. For any ε ∈ {+,−} and x ∈ R the function F (τ;x) admits an analytic
ε
continuation to H. Moreover, the analytic continuation satisfies the functional equations
F (τ;x)−F (τ +2;x) = 0,
ε ε
(24) (cid:16) 1 (cid:17)
F (τ;x)+ε(−iτ)−1/2F − ;x = eiπτx2 +ε(−iτ)−1/2eiπ(−1/τ)x2.
ε ε
τ
Proof. To prove the theorem, it is enough to show that there exists an analytic contin-
uation to some open set Ω containing the boundary of D, on which the equations (24)
hold. Indeed, we can then choose Ω such that
D ⊂ Ω ⊂ D∪SD∪T2D∪T−2D
Since ∪ gΩ = H, we can construct a continuation by repeatedly using (24). Since H
g∈Γ
θ
is simply-connected, this gives a well-defined analytic function on H.
Thefirstfunctionalequationin(24)isclearlysatisfied, sincetheintegralthatdefinesF
ε
automaticallydefinesa2-periodicfunctionontheopenset{τ ∈ H : ∀k ∈ Z, |τ−2k| > 1}
that contains the vertical lines Im(τ) = ±1.
Hence,weonlyneedtodealwiththesecondfunctionalequation. Wecangetananalytic
continuation of F to some neighborhood of {z ∈ H : |z| = 1,z (cid:54)= i} by changing the
ε
contour of integration in (22). First, we rewrite the integral as
(cid:90) i (cid:90) 1
2F (τ,x) = K (τ,z)eiπx2zdz + K (τ,z)eiπx2zdz
ε ε ε
−1 i
(cid:90) i (cid:90) i
(25) = K (τ,z)eiπx2zdz − K (τ,−1/z)eiπx2(−1/z)z−2dz
ε ε
−1 −1
(cid:90) i
= K (τ,z)(eiπx2z +ε(−iz)−1/2eiπx2(−1/z))dz,
ε
−1
where we have used the transformation property (20). Note, that if τ belongs to D∪SD,
then the only poles of K (τ,z) (as a function of z) inside D ∪ SD are at z = τ and
ε
z = −1/τ. Let γ be the circle arc from −1 to i, and let γ be a simple smooth path
1 2
from −1 to i that lies inside SD and strictly below γ . Denote by F the region enclosed
1
between γ and γ . We will now build a continuation of F to F and show that it satisfies
1 2 ε
the functional equation. We define a continuation by the contour integral
(cid:90)
1
F˜ (τ;x) = K (τ,z)(eiπx2z +ε(−iz)−1/2eiπx2(−1/z))dz.
ε ε
2
γ2