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Revista Matemática Complutense
https://doi.org/10.1007/s13163-024-00494-5
Quantum signal processing and nonlinear Fourier analysis
Michel Alexis1·Gevorg Mnatsakanyan1·Christoph Thiele1
Received: 6 November 2023 / Accepted: 2 April 2024
© The Author(s) 2024
Abstract
Elucidating a connection with nonlinear Fourier analysis (NLFA), we extend a well
known algorithm in quantum signal processing (QSP) to represent measurable sig-
nals by square summable sequences. Each coefficient of the sequence is Lipschitz
continuous as a function of the signal.
Mathematics Subject Classification 68Q12 ·81P68 ·34L25 ·42C99
1 Introduction
A signal in this paper is a function from the interval I=[0,1]to the interval
(−2−1
2,2−1
2). In quantum signal processing, one represents such a signal as the imag-
inary part of one entry of an ordered product of unitary matrices. The factors of this
product alternate between matrices depending on the functional parameter x∈Iand
matrices depending on a sequence of scalar parameters ψnwhich are tuned so that
the product represents a given signal. We are interested in the particular representation
of this type proposed by [14] and extended to infinite absolutely summable sequences
in [7]. Our main observation is that after some change of variables, the map sending
the sequence to the signal is identified as the nonlinear Fourier series described
in [23]. Indeed, this nonlinear Fourier series as well as variants including one with
SU(1,1)matrices in [24] have been studied for a long time in different contexts such
as orthogonal polynomials [21], Krein systems [6], scattering transforms [2,22]or
AKNS systems [1].
In particular, transferring knowledge from nonlinear Fourier analysis, we extend
the theory in [7] from absolutely summable to square summable using a nonlinear
version of the Plancherel identity. We obtain a representation of measurable signals
BGevorg Mnatsakanyan
gevorg@math.uni-bonn.de
Michel Alexis
alexis@math.uni-bonn.de
Christoph Thiele
thiele@math.uni-bonn.de
1Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
123
M. Alexis et al.
by square summable sequences . This representation extremizes a certain inequality
of Plancherel type.
To state our main result, Theorem 1, we make some formal definitions. Given >0,
define the signal space Sto be the set of real valued measurable functions fon [0,1]
that satisfy the bound
sup
x∈[0,1]|f(x)|≤2−1
2−. (1.1)
We equip Swith the metric induced by the Hilbert space norm
f≡⎛
⎝2
π
1
0
|f(x)|2dx
√1−x2⎞
⎠
1
2
.(1.2)
Let Pbe the space of sequences =(ψk)k∈Nof numbers ψk∈(−π
2,π
2).We
equip Pwith the metric induced by the L∞-norm
∞=sup
k∈N|ψk|.
For x∈[0,1], define
W(x):= xi
√1−x2
i√1−x2x,Z=10
0−1.(1.3)
For ∈Pand x∈[0,1], define recursively
U0(, x)=eiψ0Z(1.4)
and
Ud(, x)=eiψdZW(x)Ud−1(, x)W(x)eiψdZ.(1.5)
Define ud(, x)to be the upper left entry of Ud(, x).
Theorem 1 Let >0. For each f ∈S, there exists a unique sequence ∈Psuch
that
k∈Z
log(1+tan2ψ|k|)=−2
π1
0
log |1−f(x)2|dx
√1−x2(1.6)
and (ud(, x)) converges with respect to the norm (1.2) to the function f as d tends
to ∞. For two functions f ,˜
f∈Swith corresponding sequences , ˜
as above, we
have the Lipschitz bound
−˜
∞≤7.3−3
2f−˜
f.(1.7)
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Quantum signal processing...
Fig. 1 Illustration of QSP
Figure 1is a simplified cartoon of QSP, conflating for illustrative purpose the group
SO(3)with its double-cover SU(2)and ignoring for simplicity the reflection symme-
try in the product (1.5). For a given signal f, Theorem 1provides tuning parameters
ψjwith which we can then evaluate fat x=cos θas follows. We alternatingly rotate
the horizontal blue vector by θabout the vertical axis, an action generated by the Pauli
matrix Xdefined in Sect. 4, and by the consecutive tuning parameters ψjabout the
horizontal brown axis, an action generated by the Pauli matrix Z. The resulting rotated
blue vector has height f(x).
Our proof provides an algorithm to compute ψkvia a Banach fixed point iteration
that converges exponentially fast with rate depending on . The iteration step requires
the application of a Cauchy projection, which in practice may be computed using a
fast Fourier transform.
The weight (1−x2)−1
2in (1.6) has a singularity at one but not at zero. This
asymmetry arises because our theory works naturally with fextended to an even
function on [−1,1].
After developing the relevant parts of nonlinear Fourier analysis, we prove Theorem
1in Sect. 8. A relaxation of the threshold (1.1) will be discussed in a forthcoming paper.
The literature both on QSP and NLFA is extensive and we do not try to give a
complete overview here. Our first reference to the QSP algorithm discussed here is
[14], which was interested in an optimal algorithm for Hamiltonian simulation. Various
interesting properties of QSP are discussed in [5,9]. [18] introduces an SU (1,1)
variant of QSP. For the task of computing the potential (ψn)for a given target function
f, several algorithms have been proposed including the so-called factorization method
[3,11,12,26], an optimization algorithm [25], fixed point iteration [7] and Newton’s
method [8]. The factorization method in the context of nonlinear Fourier series is called
the layer-stripping formula and is discussed below. The papers [7] and [25] develop
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M. Alexis et al.
the 1and 2theories for QSP with many interesting theoretical results. Many of these
results are implicit in our discussion of NLFA in the present paper.
Discrete NLFA for the SU(1,1)model was studied in [24] with particular empha-
sis on transferring analytic estimates for the linear Fourier transform to the nonlinear
setting. For the SU (2)model, a similar discussion appears in [23]. Some important
contributions to the quest for analogs of classical linear inequalities were made by [4,
13,16,17,20], namely providing maximal and variational Hausdorff-Young inequal-
ities and discussing Carleson-type theorems for the SU (1,1)model of the nonlinear
Fourier transform and some variants. For a discussion of some recent results and open
questions see [19].
The interest of the third author and subsequently the other authors in quantum signal
processing was initiated during an inspiring talk by L. Lin at a delightful conference at
ICERM on Modern Applied and Computational Analysis. In particular, we dedicate
this result to R. Coifman, who anticipated at the conference that QSP is some sort
of nonlinear Fourier analysis. The third author acknowledges an invitation to the
Santaló Lecture 2022, where he gave an introduction to nonlinear Fourier analysis.
The authors acknowledge support by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1
– 390685813 as well as CRC 1060. We also thank Jiasu Wang for pointing out typos
in the first ArXiv posting of this article.
2 The nonlinear Fourier transform
We are mainly interested in nonlinear Fourier series. However, we start with an excur-
sion to the nonlinear Fourier transform on the real line, which is a multiplicative and
non-commutative version of the linear Fourier transform.
Recall the linear Fourier transform
f(ξ) := R
f(x)e−2πixξdx.
This integral is understood to be a Lebesgue integral if fis in L1(R).If
fis also
in L1(R), then both fand
fcan be seen to be in L2(R)and one has the Plancherel
identity
f2=f2.
The Plancherel identity holds for fin a dense subset of L2(R), and one can use it to
extend the Fourier transform to a unitary map from L2(R)to itself. This definition in
L2(R)coincides with the integral definition when fis in L2(R)∩L1(R).
The integral in the definition of the Fourier transform is an additive process over con-
tinuous time x. This process can alternatively be expressed by a differential evolution
equation for the partial Fourier integrals
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Quantum signal processing...
S(ξ, x)=x
−∞
f(t)e−2πitξdt,
namely
∂xS(ξ, x)=f(x)e−2πixξ
with the initial condition
S(ξ, −∞)=0
and the final state
S(ξ, ∞)=
f(ξ).
If f∈L1(R), the required analytic facts such as solvability of the differential equation
and limits as xtends to ±∞ can be elaborated with standard methods.
Exponentiation turns this additive process into a multiplicative process. Define
G(ξ, x)=eS(ξ,x).
Then Gsatisfies the differential equation
∂xG(ξ, x)=G(ξ, x)f(x)e−2πixξ(2.1)
with the initial condition
G(ξ, −∞)=1
and the final state
G(ξ, ∞)=e
f(ξ).
In the above scalar valued setting, the multiplicative perspective is of an artificial
nature. However, the multiplicative process allows for matrix valued generalizations,
which lead to substantially different nonlinear Fourier transforms. For these general-
izations, the complex factor f(x)e−2πixξin Cin (2.1) needs to be replaced by a matrix
factor. The most basic choices of such matrix factors come from real linear embed-
dings of Cinto three dimensional Lie algebras, in particular the ones associated with
the Lie groups SU (1,1)and SU (2).
The most common SU (1,1)model of the nonlinear Fourier transform is described
by the differential equation
∂xG(ξ, x)=G(ξ, x)0f(x)e−2πixξ
f(x)e−2πixξ0(2.2)
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M. Alexis et al.
with the initial condition
G(ξ, −∞)=10
01
and the final state defined to be the SU (1,1)nonlinear Fourier transform of f,
G(ξ, ∞)=a(ξ) b(ξ )
b(ξ) a(ξ ) .(2.3)
As the matrix factor in (2.2) is in the Lie Algebra of SU (1,1), the solution to the
differential equation stays in SU (1,1). This explains the particular structure of the
matrix in (2.3) and we also have
|a(ξ)|2−|b(ξ)|2=1.
Analogous to the linear situation, solvability of the differential equation with limits
as xtends to ±∞ is elementary for f∈L1(R). By Picard iteration, a solution can be
written as the limit of recursively defined approximations Gkwith
G0(ξ, x)=10
01
and for k>0
Gk(ξ, x)=10
01
+x
−∞
Gk−1(ξ, tk)0f(tk)e−2πitkξ
f(tk)e−2πitkξ0dtk.
In particular, Gk−Gk−1is k-linear in f.Ifkis even, the k-linear term is diagonal
with upper left entry
−∞<t1<t2<···<tk<∞
k/2
j=1
f(t2j)f(t2j−1)e2πiξ(t2j−t2j−1)dt2j−1dt2j(2.4)
and lower right entry the complex conjugate of (2.4). If kis odd, then the k-linear term
is anti-diagonal with upper right entry
−∞<t1<t2<···<tk<∞
f(tk)e−2πiξtk
(k−1)/2
j=1
f(t2j)f(t2j−1)e2πiξ(t2j−t2j−1)dt2j−1dt2j
(2.5)
and lower left entry the complex conjugate of (2.5). Note that (2.4) and (2.5)are
the terms involved in the multilinear expansions of aand b, which have first order
approximation of the constant function 1 and the linear Fourier transform of f,
respectively.
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Quantum signal processing...
The entries (2.4) and (2.5) are bounded in absolute value by the integrals
−∞<t1<t2<···<tk<∞
k
j=1|f(tj)|dtj=1
k!fk
1.(2.6)
Hence the nonlinear Fourier transform is a real analytic map from L1(R)to the space
L∞(R,C2×2). Moreover, the inverse linear Fourier transform of (2.4) can be written
with the Dirac δand the functional variable xas
−∞<t1<t2<···<tk<∞
δ⎛
⎝x+
k/2
j=1
t2j−t2j−1⎞
⎠
k/2
j=1¯
f(t2j)f(t2j−1)dt2j−1dt2j,
(2.7)
and similarly for (2.5).Thefunction(2.7)isagaininL1(R)with norm bounded as
in (2.6). Hence the nonlinear Fourier transform is a real analytic map from L1(R)
to A(R,C2×2), the matrix valued functions with entries in the Wiener space A(R),
which is the linear Fourier transform of L1(R).
With more work, one can also show that the nonlinear Fourier transform extends to
an analytic map from Lp(R)into a suitable space [4,17]for1<p<2. At p=2, the
SU(1,1)nonlinear Fourier transform can be defined by a similar density argument as
in the linear case using the nonlinear analogue of the Plancherel identity
f2
2=2log |a(ξ)|dξ=R
log(1+|b(ξ)|2)dξ, (2.8)
which we will elaborate on below after (2.14). However, unlike the linear setting,
one obtains neither an injective map on L2(R), nor a real analytic map on L2(R)in
any suitable sense. See [24] in the discrete setting and [15] for references on these
respective phenomena.
The SU(2)model of the nonlinear Fourier transform is described by the solution
to the differential equation
∂xG(ξ, x)=G(ξ, x)0f(x)e−2πixξ
−f(x)e−2πixξ0(2.9)
with the initial condition
G(ξ, −∞)=10
01
and whose final state is the SU (2)nonlinear Fourier transform of f
G(ξ, ∞)=a(ξ) b(ξ )
−b(ξ) a(ξ ) .(2.10)
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M. Alexis et al.
Here, the matrix in (2.10)isinSU (2)for each ξ∈R, and in particular
|a(ξ)|2+|b(ξ)|2=1.
The Lptheory for p<2 in so far as discussed above is largely analogous to the case
of SU(1,1)but with suitable changes of signs in the multi-linear terms. The analogue
of Plancherel however is the weaker information
f2
2=lim
ξ→i∞2πiξlog (a(ξ)), (2.11)
where ξtends to ∞along the imaginary axis in the upper half plane, or more generally
through any ray from the origin strictly in the upper half plane. This can be shown by
doing an asymptotic expansion
2πiξlog(a(ξ)) =c+O(|ξ|−1)
along such a ray as in [16] and observing that it is only the bilinear term in the
multilinear expansion of athat contributes to c. The bilinear term of a,nowthe
negative of the bilinear term of the SU(1,1)case, is equal to
−−∞<t1<t2<∞
f(t2)f(t1)e2πiξ(t2−t1)dt1dt2=−s>0t
f(t+s)f(t)e2πiξsdtds.
(2.12)
Multiplying by 2πiξand using that
−2πiξe2πiξs1{s>0}(2.13)
is an approximating unit converging to the Dirac delta as ξtends to infinity along a
ray in the upper half plane, we obtain
lim
ξ→i∞−2πiξs>0t
f(t+s)f(t)e2πiξsdtds =f(t)f(t)dt.(2.14)
This shows (2.11).
In the SU (1,1)setting, where log(a)has an analytic extension to the upper half
plane, one can use a contour integral over a large semicircle in the upper half plane to
express the analogue of the limit (2.11)byanintegralasin(2.8). Here, in the SU(2)
case, log(a)is in general not analytic in the upper half plane due to zeros of aand one
cannot as easily express the limit by an integral. Instead, one resorts to tools such as
factorization into inner and outer functions [10].
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Quantum signal processing...
3 Nonlinear Fourier series
Passing from functions on Rto sequences Fon Z, the Fourier transform, which we now
call Fourier series, no longer lives on Rbut on the unit circle T:= {z∈C:|z|=1}.
We slightly misuse the notion of Fourier series here, usually this notion is reserved for
the inverse of the map that we call Fourier series here.
There are nonlinear Fourier series with values in SU(1,1), this is discussed in [24],
and nonlinear Fourier series with values in SU(2)discussed in [23]. We focus here on
the SU(2) model, which is relevant to the QSP model in Theorem 1.
The linear Fourier series of a sequence F=(Fn)n∈Zwith finite support is defined
as
F(z)=
n∈Z
Fnzn.
The analogy with the Fourier transform becomes apparent when writing z=e−2πiξ
for some ξ∈R. Indeed, if we define a measure fon the real line as
f(x)=
n∈Z
Fnδ(x−n),
then
f(ξ) =R
n∈Z
Fnδ(x−n)e−2πiξxdx
=R
n∈Z
Fnδ(x−n)e−2πiξndx =
n∈Z
Fne−2πiξn=
F(z).
The nonlinear analog becomes an ordered product of matrices described below. We
will be interested in meromorphic extensions beyond the circle T, hence we consider
the Riemann sphere C∪{∞}where ∞is the reciprocal of 0. For a subset of the
Riemann sphere we define the reflected set
∗={z−1:z∈}.(3.1)
For a function aon we define a∗on ∗by
a∗(z)=a(z−1). (3.2)
We note that (∗)∗=and (a∗)∗=a.Ifz∈T, then a∗(z)=a(z). Define the open
unit disc
D≡{z∈C:|z|<1},
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M. Alexis et al.
The function ais analytic on D∗precisely if a∗is analytic on D.Wehave
a(∞)=a∗(0). (3.3)
If ais analytic on D∗and continuous up to the boundary Tof D∗, then we have the
mean value theorem
a(∞)=a∗(0)=T
a∗=T
a,(3.4)
where we denote by
T
a=1
0
a(e2πiθ)dθ
the mean value of aon T, i.e., the constant term in the Fourier expansion of a.
For a sequence F:Z→Cwith finite support, define the meromorphic matrix
valued function Gon the Riemann sphere by the recursive equation
Gk(z)=Gk−1(z)1
1+|Fk|21Fkzk
−Fkz−k1(3.5)
with the initial condition
lim
k→−∞ Gk(z)=10
01
,
and define the SU (2)nonlinear Fourier series
G(z)=lim
k→∞ Gk(z)=a(z)b(z)
−b∗(z)a∗(z).(3.6)
Existence of the limit as k→±∞is trivial thanks to the finite support of F, which
makes the sequence Gk(z)eventually constant in k. The matrix factors in (3.5)arein
SU(2)on Tand hence so is their product. In particular,
a(z)a∗(z)+b(z)b∗(z)=1
on Tand as well on the Riemann sphere by analytic continuation.
Under the analogous formal transformation as above, the nonlinear Fourier series
becomes the SU (2)nonlinear Fourier transform for the measure
f(x)=
n∈Z
fnδ(x−n),
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where fn=arctan(|Fn|)Fn|Fn|−1. This value of fnarises from the model
computation
exp 0f0
−f00=cos |f0|f0|f0|−1sin |f0|
−f0|f0|−1sin |f0|cos |f0|
=cos |f0|1f0|f0|−1tan |f0|
−f0|f0|−1tan |f0|1=1
1+|F0|21F0
−F01.
We write the SU (2)nonlinear Fourier series of the sequence Fon Zas
F:= (a,b)
with aand bas defined in (3.6). We identify the row vector (a,b)with the matrix
function as in (3.6). In particular, we write the product
(a,b)(c,d)=(ac −bd∗,ad +bc∗).
We describe some properties of the nonlinear SU (2)Fourier series, following [23]
and the analogous arguments in [24]. The first theorem describes some basic transfor-
mation properties analogous to transformation properties of the linear Fourier series.
To better understand the analogy, recall from the analogous discussion of the nonlinear
Fourier transform that the first order approximations of aand bare one and the linear
Fourier series, respectively.
Theorem 2 Let F,H be complex valued finitely supported sequences on Zand let
F(z)=(a,b).
If all entries of F except possibly the zeroth entry vanish, then
(a(z), b(z)) =(1+|F0|2)−1
2(1,F0). (3.7)
If Hn=Fn−1, then
H(z)=(a(z), zb(z)). (3.8)
If the support of F is entirely to the left of the support of H , then
F+H=
F
H.(3.9)
If |c|=1, then
cF =(a,cb). (3.10)
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M. Alexis et al.
If Hn=F−n, then
H(z)=(a∗(z−1), b(z−1)). (3.11)
If Hn=Fn, then
H(z)=(a∗(z−1), b∗(z−1)). (3.12)
Note that the properties in Theorem 2are sufficient to uniquely determine the
map from Fto
F. The next theorem describes the range of this map on the space
of sequences with finite support. Let l(M,N)be the space of all complex valued
sequences Fon Zwhich are supported on the interval M≤k≤Nin the strict sense
that F(M)= 0 and F(N)= 0.
Theorem 3 Let M ≤N . The SU (2)nonlinear Fourier series maps l(M,N)bijectively
to the space of pairs (a,b)such that b is the linear Fourier series of a sequence in
l(M,N)and a is the linear Fourier series of a sequence in l(M−N,0)with 0<a(∞)
and
aa∗+bb∗=1.(3.13)
Moreover, we have the identity
a(∞)=
n∈Z
(1+|Fn|2)−1/2.(3.14)
Note that (3.13) implies that aand bhave no common zeros in the Riemann sphere.
Moreover, |a|and |b|are bounded by 1 on Tand a(∞)≤1 with equality only if
b=0 and F=0.
Note that if adoes not have zeros in D∗, then log(a)is analytic in D∗and the real
part of (3.4)gives
log |a(∞)|=T
log |a|.(3.15)
Multiplying by −2 and using (3.14) and (3.13), we obtain
n∈Z
log(1+|Fn|2)=−T
log(1−|b|2)(3.16)
in analogy to (2.8). If ahas zeros in D∗, then we have only the inequality
n∈Z
log(1+|Fn|2)≥−T
log(1−|b|2), (3.17)
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Quantum signal processing...
which can be obtained by applying the mean value theorem (3.4) to the logarithm of
the quotient of a∗divided by the Blaschke product of its zeros [10].
As log(1+x)is comparable to xfor small x, under suitable pointwise smallness
assumptions on Fand band absence of zeros of ain D∗we obtain from (3.16) that
Fl2(Z)and bL2(T)are comparable, in analogy to the linear situation.
4 Quantum signal processing for finite sequences
In this section, we relate at the level of finite sequences the nonlinear Fourier series to
QSP.
Let be in Pas in Theorem 1.LetFnfor n∈Zbe defined by
Fn=itan(ψ|n|)(4.1)
and note that (Fn)is even and purely imaginary, that is, for all n∈Z,
F−n=Fn=−Fn.
For d≥0, let Gdbe the nonlinear Fourier series of the truncated sequence
Fn1{−d≤n≤d}.
We may write Gd(z)for z∈T, using the symmetries of (Fn), recursively as
G0(z)=1
1−F2
01F0
F01,(4.2)
Gd(z)=1
1−F2
d1Fdz−d
Fdzd1Gd−1(z)1Fdzd
Fdz−d1.(4.3)
Define Xand Mand recall Zas follows:
X=01
10
,M=2−1
211
1−1,Z=10
0−1.(4.4)
Observe that M2is the identity matrix, that
XM =2−1
21−1
11
=MZ,(4.5)
and hence also MZM =Xand MXM =Z.
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M. Alexis et al.
Lemma 1 For x ∈[0,1]let θbe the unique number in [0,π
2]so that cos θ=x and
set z =e2iθ. We have for every d ≥0and Udas in Theorem 1,
MUd(, x)M=eidθ0
0e−idθGd(z)eidθ0
0e−idθ.
Note that the factor two in the exponent of the definition of zdiffers from the convention
in [7].
Proof We prove the Lemma by induction on d.Fork∈N,wehave
MeiψkZM=eiψkMZM (4.6)
=eiψkX=cos(ψk)isin(ψk)
isin(ψk)cos(ψk)=cos(ψk)1itan(ψk)
itan(ψk)1
(4.7)
=1
1+tan(ψk)21itan(ψk)
itan(ψk)1=1
1−F2
k1Fk
Fk1.
(4.8)
Applying this with k=0 and using (4.2) and (1.4) in the form
Meiψ0ZM=MU0(, x)M(4.9)
verifies the base case d=0 of the induction.
Now let d≥1 and assume the induction hypothesis is true for d−1. Noting that
similarly as in (4.7),
W(x)=eiarccos(x)X,
we have
MW(x)M=eiarccos(x)MXM =eiθZ=eiθ0
0e−iθ.(4.10)
Hence, with (4.6),
1−F2
dMW(x)eiψdZ=eiθ0
0e−iθ1Fd
Fd1M(4.11)
and
1−F2
deiψdZW(x)M=M1Fd
Fd1eiθ0
0e−iθ.(4.12)
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We obtain with the recursive definition (1.5) and induction hypothesis,
(1−F2
d)MUd(, x)M(4.13)
=(1−F2
d)MeiψdZW(x)M(MUd−1M)M(, x)W(x)eψdZM
=1Fd
Fd1eidθ0
0e−idθGd−1(z)eidθ0
0e−idθ1Fd
Fd1
=eidθ0
0e−idθ 1Fdz−d
Fdzd1Gd−1(z)1Fdzd
Fdz−d1eidθ0
0e−idθ
=(1−F2
d)eidθ0
0e−idθGd(z)eidθ0
0e−idθ.(4.14)
This proves the induction step for dand completes the proof of Lemma 1.
Lemma 2 Let d ≥0and set
Gd(z)=: a(z)b(z)
−b∗(z)a∗(z).
For x ∈[0,1],letθbe the unique number in [0,π
2]such that cos θ=x and set
z=e2iθ. We have for d ≥1and udas in Theorem 1,
i(ud(, x)) =b(z).
Proof We use Lemma 1to obtain
Ud(, x)=Meidθ0
0e−idθGd(z)eidθ0
0e−idθM
=Ma(z)zdb(z)
−b∗(z)a∗(z)z−dM.
We then compute the upper left corner
ud(, x)=1
211
a(z)zdb(z)
−b∗(z)a∗(z)z−d1
1=1
2(a(z)zd+a∗(z)z−d+b(z)−b∗(z)).
(4.15)
As zis in T, the last display becomes
(a(z)zd)+i(b(z)).
As (Fn)is purely imaginary and even, the symmetries of the nonlinear Fourier series
imply that bis also purely imaginary. In particular, (4.15) gives Lemma 2.
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The above proof gives b(z)=b(z−1)for z∈Tand that (ud(x, )) extends to
an even function in x∈[−1,1].
We note that with this correspondence between NLFA and QSP established, Theo-
rem 31 and Theorem 5 in [25] observe a version of the comparability of Fl2(Z)and
bL2(T)discussed in the remarks to (3.16) in the previous section.
5 Nonlinear Fourier series of summable sequences
While our focus in this paper is on square summable sequences, we briefly comment
on the analytically simpler theory of nonlinear Fourier series of elements in the space
1(Z)of absolutely summable sequences on Z. The linear Fourier series maps 1(Z)to
the space C(T)of continuous functions on T, a closed subspace of L∞(T). The actual
image of 1(Z)under the linear Fourier series is the Wiener algebra A(T). Similar
mapping properties are true for the nonlinear Fourier series.
We first recall the Theorem below of [23]forthe L∞bounds. Consider a metric on
SU(2)induced by the operator norm, i.e.,
dist(T,T):= T−Top (5.1)
and let C(T,SU(2)) be the metric space of all continuous G:T→SU(2)with
metric defined by
dist(G,G):= sup
z∈T
dist(G(z), G(z)). (5.2)
Theorem 4 ([23, Theorem 2.5]) The SU (2)nonlinear Fourier series extends uniquely
to a Lipschitz map 1(Z)→C(T,SU(2)) with Lipschitz constant at most 3.
The use of the operator norm is of no particular relevance except possibly for the
value of the Lipschitz constant, because all norms on the finite dimensional space of
2×2 matrices are equivalent. Let band bbe the second entries of the first row of
Fand
F, respectively. Then Theorem 4in particular implies
b−bL∞≤3F−F1.(5.3)
We next turn to the Wiener algebra A(T). Recall that the linear Fourier series is
injective from 1(Z)onto A(T)and the norm .Aon the Wiener algebra is defined
so that the linear Fourier series is an isometry from 1(Z)to the Wiener algebra. Let
A(T,C2)be the space of pairs (a,b)of functions in A(T)and let the norm of (a,b)
be defined as aA+bA.
Theorem 5 The SU (2)nonlinear Fourier series is a real analytic map from 1(Z)to
A(T,C2).
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Let R ≥0,F1,F1≤R. If b and bare the second entries of the nonlinear
Fourier series of F and F , respectively, then
b−bA≤eRF−F1.(5.4)
If additionally R ≤0.36, then
F−F1≤2b−bA.(5.5)
Proof We begin with a finite sequence Fand write the nonlinear Fourier series as an
ordered product
(a(z), b(z)) =∞
j=−∞
(1+|Fj|2)−1
2(1,Fjzj), (5.6)
where the non-commutative product is understood in the sense of jincreasing from
left to right. We decompose (1,Fjzj)=(1,0)+(0,Fjzj)and apply the distributive
law. The terms resulting from the distributive law are parameterized by increasing
sequences j1<···<jnof indices, for which Fjzjappears in the term. Hence we
write the right side of (5.6)as
C(F)⎛
⎝∞
n=0
j1<j2<···<jn
n
k=1
(0,Fjkzjk)⎞
⎠(5.7)
with
C(F)=∞
j=−∞
(1+|Fj|2)−1
2.
The n-thterminthesumof(5.7) is diagonal for even nand anti-diagonal for odd
n. Setting
Tn(F1,...,Fn)(z):=
j1<j2<···<jn
⎛
⎜
⎝
1≤k≤n
kis odd
Fk
jkzjk⎞
⎟
⎠⎛
⎜
⎝
1≤k≤n
kis even
−Fk
jkz−jk⎞
⎟
⎠,(5.8)
we obtain
a=C(F)∞
n=0
T2n(F,...,F). (5.9)
b=C(F)∞
n=0
T2n+1(F,...,F). (5.10)
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M. Alexis et al.
We will show that both the function Cand the multilinear expansions in (5.9) and
(5.10) extend to analytic maps in 1(Z), thereby proving that aand bextend to analytic
maps in the argument F∈1(Z).
We first discuss C(F). For a sequence Hjof non-negative numbers, we have
∞
j=−∞
(1+Hj)=1+∞
n=1
j1<···<jn
n
k=1
Hjk≤1+∞
n=1
1
n!H1(Z),(5.11)
which we recognize as a multi-linear expansion with infinite radius of convergence.
As the map Fj→FjFjis real analytic from 1(Z)to itself and the −1
2-th power is
real analytic from [1,∞)to (0,1], the function Cextends to a real analytic map from
1(Z)to (0,1].
As for Tn, taking all Fj=Fand summing (5.8) in absolute value over all permu-
tations of the indices j1to jn, the sum separates into a product of sums and one can
estimate for |z|=1
|Tn(F,...,F)(z)|≤ 1
n!Fn
1.(5.12)
Thus the multilinear expansions in the expression (5.9) and (5.10)ofaand bhave
infinite radius of convergence in 1(Z)and extend to real analytic maps from 1to
L∞(T,C2). Moreover, (5.8) is the linear Fourier series of the sequence given by
(ˇ
Tn(F1,...,Fn)) j=
j1<j2<···<jn
n
k=1−(−1)kjk=j
⎛
⎜
⎝
1≤k≤n
kis odd
Fk
jk⎞
⎟
⎠⎛
⎜
⎝
1≤k≤n
kis even
−Fi
ji⎞
⎟
⎠.
Absolutely summing over jas well as over permutations of the indices from j1to jn
yields that
Tn(F1,...,Fn)A≤1
n!
n
j=1Fj1.(5.13)
Hence the multilinear expansions in (5.9) and (5.10) extend to real analytic maps from
1(Z)to A(T). The nonlinear Fourier series extends to a real analytic map from 1(Z)
to A(T,C2). This proves the first statement of Theorem 5.
We turn to the proof of (5.4). By an n-fold application of the triangle inequality and
(5.13) above,
Tn(F,...,F)−Tn(F,...,F)A
≤
n
j=1Tn(F,...,F,F−F,F,...,F)A
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≤F−F1Rn−1
(n−1)!,(5.14)
where in the middle term the difference F−Foccurs in the j-th entry. On the other
hand, by a telescoping sum as in (5.14) using that all factors of C(F)are bounded by
1, we get
|C(F)−C(F)|≤ ∞
j=−∞ (1+|Fj|2)−1
2−(1+|F
j|2)−1
2
≤∞
j=−∞ (1+|Fj|2)1
2−(1+|F
j|2)1
2≤∞
j=−∞ |Fj−F
j|=F−F1(Z).
(5.15)
Thus, by the triangle inequality,
b−bA≤|C(F)−C(F)|∞
n=0
R2n+1
(2n+1)!+C(F)∞
n=0
F−F1R2n
(2n)!(5.16)
≤(sinh(R)+cosh(R))F−F1=eRF−F1.(5.17)
This proves (5.4).
We turn to the proof of (5.5). We first note a lower bound for C(F).Wehave
−2log(C(F)) =∞
j=−∞
log(1+|Fj|2)) ≤|Fj|2≤F2
1≤R2(5.18)
and hence
C(F)≥e−1
2R2.(5.19)
The key observation is now that T1(F)is the linear Fourier series of F. We will
isolate this term in (5.10) by the triangle inequality as follows:
b−bA+ ∞
n=1
C(F)T2n+1(F,...,F)−C(F)T2n+1(F,...,F)A
≥C(F)T1(F)−C(F)T1(F)A
≥C(F)T1(F)−T1(F)A−|C(F)−C(F)|T1(F)A
≥e−1
2R2F−F1−F−F1F1≥(e−1
2R2−R)F−F1.
(5.20)
Here we have used (5.19) and (5.15).
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M. Alexis et al.
Estimating the second term on the far left-hand side of (5.20) analogously to (5.16)
yields
b−bA+(eR−1−R)F−F1≥(e−1
2R2−R)F−F1
and hence
b−bA≥F−F11+e−1
2R2−eR,
where the last term in parentheses is larger than 1
2for R≤0.36.
In [7], the authors investigate similar inequalities. Up to comparing the constants,
Theorem 3, Corollaries 18 and 20 of [7] state the same inequalities as (5.4) and (5.5).
In fact, constants in [7] are better than the ones we obtain. This is due to the fact
that we only use the triangle inequality and absorb all the multilinear terms into the
first term, whereas [7] carries out a more subtle estimate through the Jacobian of
∞
n=0T2n+1(F,...,F).
6 Nonlinear Fourier series of one sided square summable sequences
In this section, we largely follow [23] while giving a self-contained presentation.
For 1 ≤p≤∞,letHp(D)be the classical Hardy space associated to the disc D,
that is the set of functions fin Lp(T)which are the linear Fourier series of a sequence
supported in [0,∞). The linear Fourier series of a Hardy space function fprovides
an analytic extension of fto Dwhich has non-tangential limits almost everywhere
on Tequal to the function f. We denote the value of the extension of fat a point
z∈Dby f(z). The anti-Hardy space Hp(D∗)consists of the functions fon Tfor
which f∗∈Hp(D). The mean value theorem in the form of (3.4) continues to hold
for functions a∈Hp(D∗). In particular, values of functions ain H(D∗)and H(D)
respectively at ∞and 0, if real, are the average of the real part of the function on T.
If f∈Hp(D)is bounded by 1, then its extension to Dis bounded by 1. If fhas
modulus 1 almost everywhere on T, then fis called inner. If f(0)>0, then log |f|
is integrable on Tand
T
log |f|≥log f(0), (6.1)
and fis called outer if equality holds in (6.1).
If f∈Hp(D), and fvanishes at 0, then the imaginary part of fis the Hilbert
transform Hwith respect to the circle of the real part of f.If f∈Hp(D∗), and f
vanishes at ∞, then the imaginary part of fis the negative of the Hilbert transform of
the real part of f. The Hilbert transform has operator norm one in L2(T).
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