# A survey on frame representations via dynamical sampling

Ole Christensen, Marzieh Hasannasab  Ole Christensen and Marzieh Hasannasab

###### Abstract

Dynamical sampling deals with representations of a frame
$\\{f_{k}\\}_{k=1}^{\infty}$ as an orbit $\\{T^{n}\varphi\\}_{n=0}^{\infty}$
of a linear and possibly bounded operator $T$ acting on the underlying Hilbert
space. It is known that the desire of boundedness of the operator $T$ puts
severe restrictions on the frame $\\{f_{k}\\}_{k=1}^{\infty}.$ The purpose of
the paper is to present an overview of the results in the literature and also
discuss various alternative ways of representing a frame; in particular the
class of considered frames can be enlarged drastically by allowing
representations using only a subset
$\\{T^{\alpha(k)}\varphi\\}^{\infty}_{k=1}$ of the operator orbit
$\\{T^{n}\varphi\\}_{n=0}^{\infty}$. In general it is difficult to specify
appropriate values for the scalars $\alpha(k)$ and the vector $\varphi;$
however, by accepting an arbitrarily small and controllable deviation between
the given frame $\\{f_{k}\\}_{k=1}^{\infty}$ and
$\\{T^{\alpha(k)}\varphi\\}_{k=1}^{\infty}$ we will be able to do so.

##  1 Introduction

The purpose of this paper is to give an overview of the research topic
dynamical sampling, seen from the frame perspective. We will also connect
dynamical sampling with certain developments within linear dynamics, e.g.,
hypercyclic operators and operator orbits.

Dynamical sampling deals with representations of a frame
$\\{f_{k}\\}_{k=1}^{\infty}$ as an orbit $\\{T^{n}\varphi\\}_{n=0}^{\infty}$
of a bounded linear operator $T$ acting on the underlying Hilbert space. We
will highlight a number of necessary conditions on the frame
$\\{f_{k}\\}_{k=1}^{\infty}$ for such a representation to exist, restrictions
that unfortunately exclude most classical frames. Motivated by this we discuss
a number of alternative operator representations of frames, e.g., using unions
of orbits. Most importantly, we will consider frame representations using
suborbits $\\{T^{\alpha(k)}\varphi\\}_{k=1}^{\infty}$ for certain integers
$\alpha(k),k\in\mathbb{N},$ a step that enlarge the class of relevant frames
considerably. As a final step we will consider approximate representations
using suborbits and show that by allowing a controllable and arbitrarily small
deviation between $\\{f_{k}\\}_{k=1}^{\infty}$ and
$\\{T^{\alpha(k)}\varphi\\}_{k=1}^{\infty}$ we will be able to specify
appropriate choices of the scalars $\alpha(k),\,k\in\mathbb{N}.$

The paper is presented in a problem-driven way, e.g., using Gabor frames and
shift-invariant frames as motivation for the development. In the rest of the
introduction we set the stage by providing the necessary background from
operator theory and frame theory. The material about (exact) operator
representations is collected in Section 2, while various methods for
approximate representations are considered in Section 3. In the entire paper
${\cal H}$ denotes a separable infinite-dimensional Hilbert space.

###  1.1 Operator representation of sequences

While our main interest is to analyse frames, several of our operator
theoretical tools also occur in a more general context in the literature. For
this reason we formulate the following definitions for general sequences in
Hilbert spaces.

###### Definition 1.1

Consider an ordered sequence
$\\{f_{k}\\}_{k=1}^{\infty}=\\{f_{1},f_{2},f_{3},\dots\\}$ of elements in
${\cal H}.$ If there exists a linear operator
$T:\text{span}\\{f_{k}\\}_{k=1}^{\infty}\to{\cal H}$ such that
$f_{k+1}=Tf_{k}$ for all $k\in\mathbb{N},$ we say that the sequence
$\\{f_{k}\\}_{k=1}^{\infty}$ is represented by the operator $T.$

Note that the sequence $\\{f_{k}\\}_{k=1}^{\infty}$ being represented by $T$
precisely means that

$\displaystyle\\{f_{k}\\}_{k=1}^{\infty}=\\{f_{1},f_{2},f_{3},\dots\\}=\\{f_{1},Tf_{1},T^{2}f_{1},\dots\\}=\\{T^{n}f_{1}\\}_{n=0}^{\infty}.$

In operator theoretical terms, this is phrased by saying that the sequence
$\\{f_{k}\\}_{k=1}^{\infty}$ is an orbit of the operator $T$:

###### Definition 1.2

Given a linear operator $T:{\cal H}\to{\cal H}$ and some $\varphi\in{\cal H},$
the sequence $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is called the orbit of
$\varphi$.

In general, the ordering of a given sequence $\\{f_{k}\\}_{k=1}^{\infty}$ is
crucial for the question whether or not it can be represented by a linear
operator. For example, if $f$ is an arbitrary nonzero vector in ${\cal H},$
the sequence $\\{f,0,0,0,\dots\\}$ is represented by the zero-operator, while
the sequence $\\{0,f,0,0,\dots\\}$ can not be represented by a linear
operator. We also observe that any linearly independent family
$\\{f_{k}\\}_{k=1}^{\infty}$ can be represented by a linear operator
$T:\text{span}\\{f_{k}\\}_{k=1}^{\infty}\to{\cal H},$ regardless of how the
sequence is ordered; indeed, we simply define the operator $T$ by
$Tf_{k}:=f_{k+1},\,k\in\mathbb{N},$ and extend it by linearity. It was shown
in [14] that for sequences $\\{f_{k}\\}_{k=1}^{\infty}$ spanning an infinite-
dimensional space, linear independence is actually equivalent with the
property of being representable by a linear operator:

###### Proposition 1.3

Consider any sequence $\\{f_{k}\\}_{k=1}^{\infty}$ in ${\cal H}$ for which
$\mbox{span}\\{f_{k}\\}_{k=1}^{\infty}$ is infinite-dimensional. Then the
following are equivalent:

  * (i)

$\\{f_{k}\\}_{k=1}^{\infty}$ is linearly independent.

  * (ii)

There exists a linear operator
$T:\mbox{span}\\{f_{k}\\}_{k=1}^{\infty}\to{\cal H}$ such that
$\\{f_{k}\\}_{k=1}^{\infty}=\\{T^{n}f_{1}\\}_{n=0}^{\infty}.$

In the current paper we will exclusively be interested in representability via
bounded operators. Even for linearly independent sequences, the possibility of
such a representation depends on the ordering, as the next example
demonstrates.

###### Example 1.4

Let $\\{e_{k}\\}_{k=1}^{\infty}$ denote an orthonormal basis for ${\cal H},$
and consider the sequence

$\displaystyle\\{f_{k}\\}_{k=1}^{\infty}=\\{e_{1},2e_{2},3e_{3},\dots\\}=\\{ke_{k}\\}_{k=1}^{\infty}.$

Then $\\{f_{k}\\}_{k=1}^{\infty}$ is represented by the linear operator $T$
defined by

$\displaystyle Te_{k}=\frac{k+1}{k}\,e_{k+1},\,k\in\mathbb{N};$

note that $T$ extends to a bounded operator on ${\cal H},$ with $||T||=2.$ On
the other hand, the following reordering of the elements

$\displaystyle\\{g_{k}\\}_{k=1}^{\infty}=\\{e_{1},10e_{10},2e_{2},10^{2}e_{10^{2}},3e_{3},10^{3}e_{10^{3}},\dots,9e_{9},10^{9}e_{10^{9}},11e_{11},10^{10}e_{10^{10}},\dots\\}$

clearly can not be represented by a bounded operator.

In the rest of the paper the focus will be on operator representations of
frames. Readers who are interested in corresponding results for general
sequences (in Banach spaces) are referred to [20].

###  1.2 Hypercyclic operators

The main focus of the paper is to consider exact and approximate
representations of frames. Parts of the flow of the paper is motivated by the
fact that there exist very special operators which have orbits that are dense
in the underlying Hilbert space and thus allow to approximate arbitrary
sequences.

###### Definition 1.5

A linear operator $T:{\cal H}\to{\cal H}$ is hypercyclic if there exists
$\varphi\in{\cal H}$ such that the orbit $\\{T^{n}\varphi\\}_{n=0}^{\infty}$
is dense in ${\cal H}.$ In this case the vector $\varphi$ is called a
hypercyclic vector.

Note that if $T:{\cal H}\to{\cal H}$ is a hypercyclic operator with
hypercyclic vector $\varphi,$ then the set
$\\{T^{n}\varphi\\}_{n=N}^{\infty}=\\{T^{n}\left(T^{N}\varphi\right)\\}_{n=0}^{\infty}$
is dense in ${\cal H}$ for all $N\in\mathbb{N};$ this implies that
$T^{N}\varphi$ is a hypercyclic vector for all $N\in\mathbb{N},$ and thus that
the set of hypercyclic vectors associated with $T$ is dense in ${\cal H}.$

The first example of a hypercyclic operator on a separable Hilbert space was
constructed by Rolewicz [35] in 1969. It deals with the left-shift operator
$L(x_{1},x_{2},x_{3},\dots):=(x_{2},x_{3},\dots)$ on ${\ell}^{2}(\mathbb{N})$:

###### Lemma 1.6

Assume that $a>1$ and let $L$ denote the left-shift operator on
${\ell}^{2}(\mathbb{N}).$ Then $T:=aL$ is a hypercyclic operator on
${\ell}^{2}(\mathbb{N}).$

More recently hypercyclic operators have also been constructed on
$L^{2}(\mathbb{R})$:

###### Example 1.7

In [11] it was proved that for any $a\in\mathbb{R}$ there exists functions
$\omega:\mathbb{R}\to\mathbb{R}$ such that the operator
$Tf(x):=\omega(x)f(x-a)$ is hypercyclic on $L^{2}(\mathbb{R})$. In particular,
for the case $a=0$ we can take any continuous decreasing function such that

$\displaystyle\omega(x)=\begin{cases}2\,&\mbox{if}\,\,x\leq 0\\\
1/2\,\,&\mbox{if}\,x\geq 1.\end{cases}$

For a survey of other hypercyclic operators we refer to [26]. While the
existence of hypercyclic operators is surprising in itself, the theory also
contains several beautiful and intriguing results. We collect a few of them
below, and refer to the literature [6, 25] for more results.

###### Theorem 1.8

Consider a bounded linear operator $T:{\cal H}\to{\cal H}.$ Then the following
hold:

  * (i)

[4] If $T$ is hypercyclic, then $T^{N}$ is hypercyclic for every
$N\in\mathbb{N};$ moreover, $T$ and $T^{N}$ have the same set of hypercyclic
vectors.

  * (ii)

[23, 33] If there exist a finite collection of vectors
$\varphi_{1},\dots,\varphi_{n}\in{\cal H}$ such that the union of their orbits
is dense in ${\cal H},$ then one of the orbits is dense in ${\cal H}$ by
itself, i.e., $T$ is hypercyclic.

  * (iii)

[8] If an orbit $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is somewhere dense in
${\cal H},$ it is automatically dense in ${\cal H},$ i.e., $T$ is hypercyclic
with hypercyclic vector $\varphi.$

Note in particular the result stated in Theorem 1.8 (i): phrased in words, it
says that if $T$ is hypercyclic with hypercyclic vector $\varphi,$ then the
subset $\\{\varphi,T^{N}\varphi,T^{2N}\varphi,\dots\\}$ is also dense in
${\cal H}$, for every $N\in\mathbb{N}!$

Let us conclude this section with an interesting connection between
hypercyclic operators and the celebrated invariant subspace problem, a
classical open problem that can be phrased as follows:

Q: Does there exist a bounded linear operator $T:{\cal H}\to{\cal H}$ having
no closed nontrivial invariant subspace?

Note that if $T:{\cal H}\to{\cal H}$ is a bounded linear operator, the closed
(sub)space $V:=\overline{\text{span}}\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is
invariant for any $\varphi\in{\cal H}.$ Thus, a hypercyclic operator for which
every nonzero vector is hypercyclic would provide evidence for the existence
of the type of operator asked for in the invariant subspace problem; indeed,
such an operator is the only candidate. Taking Theorem 1.8 (iii) into account
and following the formulation by Bourdon and Feldman in [8], we can phrase the
invariant subspace problem as follows:

Q: Does there exists a bounded linear operator $T:{\cal H}\to{\cal H}$ having
a nontrivial nowhere dense orbit?

###  1.3 Operator representations via suborbits

We already considered subsets of operator orbits in Section 1.2. This
motivates the next definition.

###### Definition 1.9

Consider a linear operator $T:{\cal H}\to{\cal H}$ and some $\varphi\in{\cal
H}.$ Given an increasing sequence of nonnegative integers,

$\displaystyle 0\leq\alpha(1)<\alpha(2)<\alpha(3)<\dots,$

the sequence $\\{T^{\alpha(k)}\varphi\\}_{k=1}^{\infty}$ is called a suborbit
of $\\{T^{n}\varphi\\}_{n=0}^{\infty}$.

The following deep theorem by Halperin et al. [28] shows that we gain a
significant freedom by asking for sequences to be represented by suborbits of
bounded operators rather than the full orbit.

###### Theorem 1.10

Consider any linearly independent sequence $\\{f_{k}\\}_{k=1}^{\infty}$ in a
Hilbert space ${\cal H}.$ Then there exists a bounded operator $T:{\cal
H}\to{\cal H}$ and an increasing sequence of nonnegative integers

$\displaystyle 0\leq\alpha(1)<\alpha(2)<\alpha(3)<\cdots$

such that

$\displaystyle\\{f_{k}\\}_{k=1}^{\infty}=\\{T^{\alpha(k)}f_{1}\\}_{k=1}^{\infty}.$
(1.1)

Theorem 1.10 is very appealing as a theoretical result, but it does not
explain how one can choose the operator $T$ and the integers $\alpha(k)$ in
practice. The proof in [28] involves the selection of a dense linearly
independent sequence of elements in the underlying Hilbert space ${\cal H}$
and does not give a feasible way of identifying $T$ and $\alpha(k).$ For
example, if $\\{e_{k}\\}_{k=1}^{\infty}$ is an ortnonormal basis for ${\cal
H},$ it is easy to see directly that
$\\{e_{k}\\}_{k=1}^{\infty}=\\{T^{n}e_{1}\\}_{n=0}^{\infty},$ where the
operator $T$ is defined by $Te_{k}:=e_{k+1},\,k\in\mathbb{N};$ however, even
in this simple case the proof of Theorem 1.10 yields a very complicated
procedure. One of the main goals of the current paper is to provide more
applicable and explicit methods for representing sequences
$\\{f_{k}\\}_{k=1}^{\infty}$ in an exact or approximate sense, with an
explicit given operator $T$ and direct access to the integers $\alpha(k).$

###  1.4 Basic frame theory

In this section we will give a very short presentation of the key elements in
frame theory. Only aspects that are directly relevant for the current paper
will be addressed; for more information we refer to [12] and [29].

A sequence $\\{f_{k}\\}_{k=1}^{\infty}$ in ${\cal H}$ is a frame for ${\cal
H}$ if there exist constants $A,B>0$ such that

$\displaystyle A\,||f||^{2}\leq\sum_{k=1}^{\infty}|\langle
f,f_{k}\rangle|^{2}\leq B\,||f||^{2},\,\forall f\in{\cal H};$

it is tight if we can choose $A=B,$ and it is a Bessel sequence if at least
the upper frame condition holds. A Riesz basis is a frame which is at the same
time a basis; alternatively, a Riesz basis is a frame
$\\{f_{k}\\}_{k=1}^{\infty}$ for which

$\displaystyle\sum_{k=1}^{\infty}c_{k}f_{k}=0,\,\\{c_{k}\\}_{k=1}^{\infty}\in{\ell}^{2}(\mathbb{N})\Rightarrow
c_{k}=0,\,\forall k\in\mathbb{N}.$

If $\\{f_{k}\\}_{k=1}^{\infty}$ is a Bessel sequence, the synthesis operator
is defined by

$\displaystyle U:{\ell}^{2}(\mathbb{N})\to{\cal
H},\,U\\{c_{k}\\}_{k=1}^{\infty}:=\sum_{k=1}^{\infty}c_{k}f_{k};$ (1.2)

it is well known that $U$ is well-defined and bounded. An important role is
played by the kernel of the operator $U,$ i.e., the subset of
${\ell}^{2}(\mathbb{N})$ given by

$\displaystyle{\cal
N}(U)=\left\\{\\{c_{k}\\}_{k=1}^{\infty}\in{\ell}^{2}(\mathbb{N})~{}\bigg{|}~{}\sum_{k=1}^{\infty}c_{k}f_{k}=0\right\\}.$
(1.3)

The frame operator is defined by

$\displaystyle S:{\cal H}\to{\cal H},\,Sf:=UU^{*}f=\sum_{k=1}^{\infty}\langle
f,f_{k}\rangle f_{k}.$

For a frame $\\{f_{k}\\}_{k=1}^{\infty},$ the frame operator is bounded,
bijective, and self-adjoint; these properties immediately lead to the
important frame decomposition

$\displaystyle f=SS^{-1}f=\sum_{k=1}^{\infty}\langle f,S^{-1}f_{k}\rangle
f_{k},\quad\forall f\in{\cal H}.$ (1.4)

The sequence $\\{S^{-1}f_{k}\\}_{k=1}^{\infty}$ is also a frame; it is called
the canonical dual frame. Frames that are not Riesz bases are overcomplete,
and there exists
$\\{g_{k}\\}_{k=1}^{\infty}\neq\\{S^{-1}f_{k}\\}_{k=1}^{\infty}$ such that

$\displaystyle f=\sum_{k=1}^{\infty}\langle f,g_{k}\rangle f_{k},\quad\forall
f\in{\cal H}.$ (1.5)

Any frame $\\{g_{k}\\}_{k=1}^{\infty}$ satisfying (1.5) for a given frame
$\\{f_{k}\\}_{k=1}^{\infty}$ is called a dual frame of
$\\{f_{k}\\}_{k=1}^{\infty}.$

The excess of a frame $\\{f_{k}\\}_{k=1}^{\infty}$ is the maximal number of
elements that can be removed yet leaving a frame. It is well-known that the
excess equals $\mbox{dim}\,{\cal N}(U),$ the dimension of the kernel of the
synthesis operator; see [5].

###  1.5 Structured function systems

In this section we will give a short introduction to a number of explicit
frames in the Hilbert space $L^{2}(\mathbb{R}).$ These frames will illustrate
and motivate several results throughout the paper. As for Section 1.4 only
results with direct relevance to the current paper are discussed.

In order to introduce the relevant frames we first need to consider a number
of operators on $L^{2}(\mathbb{R}).$ For $a\in\mathbb{R},$ define the
translation operator

$\displaystyle T_{a}:L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R}),\,T_{a}f(x):=f(x-a)$ (1.6)

and the modulation operator

$\displaystyle E_{a}:L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R}),\,E_{a}f(x):=e^{2\pi iax}f(x).$ (1.7)

The translation operators and the modulation operators are unitary.

Given a function $\varphi\in L^{2}(\mathbb{R})$ and some $b>0$, the associated
shift-invariant system is given by $\\{T_{kb}\varphi\\}_{k\in\mathbb{Z}}.$ The
frame properties of such systems are well understood, see, e.g., [7, 13, 5].
In particular, regardless of the choice of $\varphi\in L^{2}(\mathbb{R})$ and
the parameter $b>0,$ the system can not be a frame for all of
$L^{2}(\mathbb{R})$ but only for the subspace
$\overline{\text{span}}\\{T_{kb}\varphi\\}_{k\in\mathbb{Z}}.$ It is also known
that $\\{T_{kb}\varphi\\}_{k\in\mathbb{Z}}$ is linearly independent for all
$\varphi\neq 0.$

Classical examples of frames of the form
$\\{T_{kb}\varphi\\}_{k\in\mathbb{Z}}$ are obtained by taking $\varphi$ to be
the sinc-function,

$\displaystyle\mbox{sinc}(x):=\begin{cases}\frac{\sin x}{x}&\mbox{if}\,x\neq
0,\\\ 1&\mbox{if}\,x=0.\end{cases}$

Let us explain this in more detail. First, define the Fourier transform of
$f\in L^{1}(\mathbb{R})$ by

$\displaystyle\widehat{f}(\gamma):=\int_{-\infty}^{\infty}f(x)e^{-2\pi i\gamma
x}dx;$

we extend the Fourier transform in the standard way to a unitary operator on
$L^{2}(\mathbb{R}).$ Then the functions
$\\{T_{k}\mbox{sinc}\\}_{k\in\mathbb{Z}}$ form an orthnormal basis for the
Paley-Wiener space,

$\displaystyle PW:=\left\\{f\in
L^{2}(\mathbb{R})\,\big{|}\,\mbox{supp}\widehat{f}\subseteq[-\frac{1}{2},\frac{1}{2}]\right\\}.$

Also, the system $\\{T_{k/2}\mbox{sinc}\\}_{k\in\mathbb{Z}}$ can be considered
as the union of two orthonormal bases for $PW,$ and hence form a tight
overcomplete frame for $PW.$

It is known that a finite union of shift-invariant systems at most can form a
frame for a subspace for $L^{2}(\mathbb{R}).$ However, by acting on a shift-
invariant system with an infinite number of modulation operators, it is
possible to construct frames for $L^{2}(\mathbb{R}).$ This leads to the so-
called Gabor frames. More formaully, given some $a,b>0$ and a function $g\in
L^{2}(\mathbb{R})$, the associated Gabor system is the collection of functions
given by

$\displaystyle\\{E_{mb}T_{na}g\\}_{m,n\in\mathbb{Z}}=\\{e^{2\pi
imbx}g(x-na)\\}_{m,n\in\mathbb{Z}}.$

The following result collects the information about Gabor systems that is
needed in the current paper; much more information can be found, e.g., in [27,
12].

###### Proposition 1.11

Let $g\in L^{2}(\mathbb{R})\setminus\\{0\\}.$ Then the following hold:

  * (i)

$\\{E_{mb}T_{na}g\\}_{m,n\in\mathbb{Z}}$ is linearly independent.

  * (ii)

If $\\{E_{mb}T_{na}g\\}_{m,n\in\mathbb{Z}}$ is a frame for
$L^{2}(\mathbb{R}),$ then $ab\leq 1.$

  * (iii)

If $\\{E_{mb}T_{na}g\\}_{m,n\in\mathbb{Z}}$ is a frame for
$L^{2}(\mathbb{R}),$ then $\\{E_{mb}T_{na}g\\}_{m,n\in\mathbb{Z}}$ is a Riesz
basis if and only if $ab=1.$

  * (iv)

If $\\{E_{mb}T_{na}g\\}_{m,n\in\mathbb{Z}}$ is an overcomplete frame for
$L^{2}(\mathbb{R})$, then it has infinite excess.

The result in (i) was proved in [32] (hereby confirming a conjecture stated in
[30]); (ii) & (iii) are classical results [27, 12], and (iv) is proved in [5].

##  2 Frames and dynamical sampling

###  2.1 Frame properties of operator orbits

The analysis of the frame aspects related to dynamical sampling was initiated
in the papers [1, 2] by Aldroubi et al. The paper [2] gives a complete
treatment of the finite-dimensional case; we will not go into this topic as we
only deal with the infinite-dimensional case in the current paper.

The approach in [1, 2] puts operator theory at the central spot. The central
questions can be formulated as follows:

  * (i)

Given a bounded operator $T:{\cal H}\to{\cal H}$ belonging to a certain class,
is it possible to choose $\varphi\in{\cal H}$ such that the orbit
$\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is a basis?

  * (ii)

Given a bounded operator $T:{\cal H}\to{\cal H}$ belonging to a certain class,
is it possible to choose $\varphi\in{\cal H}$ such that the orbit
$\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is a frame?

The papers [1, 2, 14] provide positive answers to these questions in certain
cases, to be described in detail in Sections 2.2 & 2.3. However, except for
these cases, most results in the literature are negative. We state a number of
such results next.

###### Proposition 2.1

Consider a bounded opearor $T:{\cal H}\to{\cal H}.$ Then the following hold:

  * (i)

[2] If $T$ is normal, then $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is not a basis.

  * (ii)

[3] If $T$ is unitary, then $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is not a
frame.

  * (iii)

[19] If $T$ is compact, then $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is not a
frame.

If $T$ is a hypercyclic operator, clearly $\\{T^{n}\varphi\\}_{n=0}^{\infty}$
can not be a frame for any hypercyclic vector $\varphi\in{\cal H}.$ We will
now prove that $\\{T^{n}\eta\\}_{n=0}^{\infty}$ can not be a frame for any
choice of $\eta\in{\cal H}.$ For this purpose we need the following result by
Aldroubi and Petrosyan [3].

###### Lemma 2.2

Assume that $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is a frame for a bounded
operator $T:{\cal H}\to{\cal H}$ and some $\varphi\in{\cal H}.$ Then
$(T^{n})^{*}f\to 0$ as $n\to\infty,$ for all $f\in{\cal H}.$

Proof. Fix any $f\in{\cal H},$ and let $k\in\mathbb{N}.$ Then, denoting a
lower frame bound for $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ by $A,$ we have that

$\displaystyle
A\,||(T^{k})^{*}f||^{2}\leq\sum_{n=0}^{\infty}|\langle(T^{k})^{*}f,T^{n}\varphi\rangle|^{2}=\sum_{n=k}^{\infty}|\langle
f,T^{n}\varphi\rangle|^{2}\to 0\,\mbox{as}\,k\to\infty.$

Thus also $(T^{k})^{*}f\to 0$ as $k\to\infty,$ $\square$

###### Proposition 2.3

Assume that $T:{\cal H}\to{\cal H}$ is hypercyclic. Then
$\\{T^{n}\eta\\}_{n=0}^{\infty}$ and $\\{(T^{*})^{n}\eta\\}_{n=0}^{\infty}$
can not be frames for any choice of $\eta\in{\cal H}.$

Proof. Let $\varphi\in{\cal H}$ be a hypercyclic vector and consider any
$\eta\in{\cal H}\setminus\\{0\\}.$ Then the scalar sequence
$\\{\langle\eta,T^{n}\varphi\rangle\\}_{n=0}^{\infty}=\\{\langle(T^{*})^{n}\eta,\varphi\rangle\\}_{n=1}^{\infty}$
is unbounded, which implies that the sequence of norms
$\\{||(T^{*})^{n}\eta||\\}_{n=1}^{\infty}$ is unbounded. Using Lemma 2.2 it
now follows that the sequence $\\{T^{n}\eta\\}_{n=0}^{\infty}$ is not a frame.
For the proof of the second claim, considering again any $\eta\neq 0$ and
still letting $\varphi$ denote a hypercyclic vector,

$\displaystyle\sum_{n=0}^{\infty}|\langle\varphi,(T^{*})^{n}\eta\rangle|^{2}=\sum_{n=0}^{\infty}|\langle
T^{n}\varphi,\eta\rangle|^{2}=\infty,$

which implies that $\\{(T^{*})^{n}\eta\\}_{n=0}^{\infty}$ is not a frame.
$\square$

Yet we have not seen any examples of a frame with an operator representation
$\\{T^{n}\varphi\\}_{n=0}^{\infty}.$ However, let us prove already now that if
$\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is a frame for a surjective and bounded
operator $T,$ the frame property is preserved if an arbitrary finite number of
elements is removed from the frame:

###### Proposition 2.4

Assume that $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is a frame for ${\cal H}$ for
some bounded surjective operator $T:{\cal H}\to{\cal H}$ and some
$\varphi\in{\cal H}.$ Then $\\{T^{n}\varphi\\}_{n\in\mathbb{N}_{0}\setminus
I}$ is a frame for ${\cal H}$ for an arbitrary finite index set
$I\subset\mathbb{N}_{0}.$ In particular, $\\{T^{n}\varphi\\}_{n=0}^{\infty}$
has infinite excess.

Proof. Consider any $N\in\mathbb{N}.$ By removing the first $N$ elements of
the frame $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ we are left with the family
$\\{T^{n}\varphi\\}_{n=N}^{\infty}=\\{T^{N}T^{n}\varphi\\}_{n=0}^{\infty};$
since the operator $T^{N}$ is bounded and surjective, this family is a frame.
$\square$

###  2.2 The Carleson frame

The first construction of a frame of the form
$\\{T^{n}\varphi\\}_{n=0}^{\infty}$ was obtained by Aldroubi et all in [1] and
further discussed in [2, 3]. We will formulate the result in our setting of an
arbitrary separable infinite-dimensional Hilbert space.

###### Theorem 2.5

Let $\\{e_{k}\\}_{k=1}^{\infty}$ be an orthonormal basis for ${\cal H}$ and
consider a bounded operator $T:{\cal H}\to{\cal H}$ such that
$Te_{k}=\lambda_{k}e_{k}$ for a bounded sequence of complex scalars
$\lambda_{k}.$ Also, let $\varphi\in{\cal H}.$ Then
$\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is a frame for ${\cal H}$ if and only if
the following conditions are satisfied:

  * (i)

$|\lambda_{k}|<1$ for all $k\in\mathbb{N};$

  * (ii)

$|\lambda_{k}|\to 1$ as $k\to\infty;$

  * (iii)

The sequence $\\{\lambda_{k}\\}_{k=1}^{\infty}$ satisfies the Carleson
condition, i.e.,

$\displaystyle\displaystyle\inf_{n}\prod_{n\neq
k}\frac{|\lambda_{k}-\lambda_{n}|}{|1-\lambda_{k}\overline{\lambda_{n}}|}>0;$
(2.1)

  * (iv)

$\varphi=\sum_{k=1}^{\infty}m_{k}\sqrt{1-|\lambda_{k}|^{2}}e_{k}$ for a
scalar-sequence $\\{m_{k}\\}_{k=1}^{\infty}$ that is bounded below away from
zero and above.

For the sake of easy reference we will call the frames arising from the
conditions in Theorem 2.5 for Carleson frames. The following result (see,
e.g., [24, Thm. 9.2]) gives an easy verifiable criterion for the Carleson
condition (2.1) to hold.

###### Proposition 2.6

Let $\\{\lambda_{k}\\}_{k=1}^{\infty}$ be a sequence of distinct complex
numbers such that $\lambda_{k}|<1$ for all $k\in\mathbb{N}.$ If there exists
$c\in]0,1[$ such that

$\displaystyle\frac{1-|\lambda_{k+1}|}{1-|\lambda_{k}|}\leq c<1,\quad\forall
k\in\mathbb{N},$ (2.2)

then $\\{\lambda_{k}\\}_{k=1}^{\infty}$ satisfies the Carleson condition. If
$\\{\lambda_{k}\\}_{k=1}^{\infty}$ is positive and increasing, the condition
(2.2) is also necessary for $\\{\lambda_{k}\\}_{k=1}^{\infty}$ to satisfy the
Carleson condition.

Based on Proposition 2.6 it is easy to give an explict example of a sequence
satisfying the Carleson condition:

###### Corollary 2.7

For every $\alpha>1$, the sequence
$\\{\lambda_{k}\\}_{k=1}^{\infty}=\\{1-\alpha^{-k}\\}_{k=1}^{\infty}$
satisfies the Carleson condition.

Carleson frames have a number of very special features. For example, under a
very weak condition, arbitrary finite subsets can be removed without
destroying the frame property:

###### Corollary 2.8

Consider a Carleson frame $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ as in Theorem
2.5 and assume that $\lambda_{k}\neq 0$ for all $k\in\mathbb{N}.$ Then
$\\{T^{n}\varphi\\}_{n=N}^{\infty}$ is a frame for ${\cal H}$ for any
$N\in\mathbb{N}.$

Proof. Since $\lambda_{k}\neq 0$ for all $k\in\mathbb{N},$ the condition (ii)
in Theorem 2.5 implies that there is a $C>0$ such that $|\lambda_{k}|\geq C$
for all $k\in\mathbb{N};$ hence the operator $T$ in Theorem 2.5 is surjective.
The result now follows from Proposition 2.4. $\square$

###  2.3 Frames and orbit representations

We will now continue the theme from Section 2.1, but with an important change
of focus. While Section 2.1 was putting the operator $T$ in the central spot
and asking for frame properties of the associated orbits, we will now take a
frame $\\{f_{k}\\}_{k=1}^{\infty}$ as the starting point and analyze when and
how it can be represented as an orbit of a bounded operator. Recall from
Proposition 1.3 that a representation as the orbit of a linear operator is
possible if $\\{f_{k}\\}_{k=1}^{\infty}$ is linearly independent; thus, the
key issue is to determine when the operator $T$ can be chosen to be bounded.

We first state a result appearing in [19]. We will need the right-shift
operator on ${\ell}^{2}(\mathbb{N}),$ defined by

$\displaystyle{\cal T}:{\ell}^{2}(\mathbb{N})\to{\ell}^{2}(\mathbb{N}),{\cal
T}\\{c_{k}\\}_{k=1}^{\infty}=\\{0,c_{1},c_{2},\dots\\}.$ (2.3)

###### Theorem 2.9

Consider a frame $\\{f_{k}\\}_{k=1}^{\infty}$ with frame bounds $A,B.$ Then
the following are equivalent:

  * (i)

The frame has a representation
$\\{f_{k}\\}_{k=1}^{\infty}=\\{T^{n}f_{1}\\}_{n=0}^{\infty}$ for some bounded
operator $T:{\cal H}\to{\cal H}.$

  * (ii)

For some dual frame $\\{g_{k}\\}_{k=1}^{\infty}$ (and hence all),

$\displaystyle f_{j+1}=\sum_{k=1}^{\infty}\langle f_{j},g_{k}\rangle
f_{k+1},\,\forall j\in\mathbb{N}.$ (2.4)

  * (iii)

The kernel ${\cal N}(U)$ of the synthesis operator $U$ is invariant under the
right-shift operator ${\cal T}.$

In the affirmative case, letting $\\{g_{k}\\}_{k=1}^{\infty}$ denote an
arbitrary dual frame of $\\{f_{k}\\}_{k=1}^{\infty},$ the operator $T$ has the
form

$\displaystyle Tf=\sum_{k=1}^{\infty}\langle f,g_{k}\rangle f_{k+1},\,\forall
f\in{\cal H},$ (2.5)

and $1\leq\|T\|\leq\sqrt{BA^{-1}}.$

Note that if $\\{f_{k}\\}_{k=1}^{\infty}$ is a Riesz basis, then
$\\{f_{k}\\}_{k=1}^{\infty}$ and the (unique) dual frame
$\\{g_{k}\\}_{k=1}^{\infty}$ form a biorthogonal system, and hence the
condition (2.4) is trivially satisfied. Thus Theorem 2.9 immediately shows
that every Riesz basis is the orbit of a bounded operator, a result that was
first proved directly in [14]:

###### Corollary 2.10

If $\\{f_{k}\\}_{k=1}^{\infty}$ is a Riesz basis, there exists a bounded
operator $T:{\cal H}\to{\cal H}$ such that
$\\{f_{k}\\}_{k=1}^{\infty}=\\{T^{n}f_{1}\\}_{n=0}^{\infty}.$

Note that any reordering of a Riesz basis is a Riesz basis itself; thus, no
matter how the elements in a Riesz basis are ordered, we can apply Corollary
2.10. On the other hand, we saw in Example 1.4 that for general linearly
independent sequences in Hilbert spaces, the existence of a bounded operator
representation depends on the ordering of the sequence. For general
overcomplete frames, the question of representability via the orbit of a
bounded operator is also extremely sensitive to reorderings of the elements,
as demonstrated by results in [19].

The simplicity of the proof of Corollary 2.10 could make us optimistic with
regard to construction of other classes of frames having an orbit
representation via a bounded operator, but unfortunately this is misleading.
Indeed, the Carleson frames and the Riesz bases are the only explicitly
available examples in the literature of frames having such a representation.
In particular, it was shown in [19] that a so-called near-Riesz basis (i.e., a
frame with finite excess) never can be represented via a bounded operator:

###### Proposition 2.11

Let $\\{f_{k}\\}_{k=1}^{\infty}$ denote a frame which is a union of a Riesz
basis and a finite non-empty collection of vectors. Then
$\\{f_{k}\\}_{k=1}^{\infty}$ can not be represented as the orbit of a bounded
operator.

Another serious restriction was proved in [18]. It excludes all overcomplete
shift-invariant frames and Gabor frames from having a representation as the
orbit of a bounded operator:

###### Theorem 2.12

Assume that $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is an overcomplete frame for
some $\varphi\in{\cal H}$ and a bounded operator $T:{\cal H}\to{\cal H}.$ Then
for each $f\in{\cal H}$ we have that

$\displaystyle T^{n}f\to 0\quad\text{as }n\to\infty.$ (2.6)

The result in Theorem 2.12 is very disappointing. We have already seen that
shift-invariant frames and Gabor frames automatically are linearly
independent, so by Proposition 1.3 such systems can be represented as an orbit
of a linear operator $T;$ however, in the overcomplete case $T$ is forced to
be unbounded. Since the problem is the boundedness and not the existence  of a
operator representation, it is natural to ask whether boundedness can be
achieved by allowing a “scaling factor” in the operator representation. We
formulate it as an open question:

Q: Do there exist overcomplete Gabor frames
$\\{E_{mb}T_{na}g\\}_{m,n\in\mathbb{Z}}$ such that an appropriate ordering
$\\{f_{k}\\}_{k=1}^{\infty}$ of the frame elements has a representation

$\displaystyle\\{f_{k}\\}_{k=1}^{\infty}=\\{a_{n}T^{n}\varphi\\}_{n=0}^{\infty},$
(2.7)

for some scalars $a_{n}\neq 0$, a bounded operator $T:L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R}),$ and some $\varphi\in L^{2}(\mathbb{R})$?

Let us end the section with result connecting frame orbits and shift-invariant
subspaces of the Hardy space. Recall that letting $\mathbb{D}$ denote the open
unit disc in the complex plane, the Hardy space $H^{2}(\mathbb{D})$ is defined
as

$H^{2}(\mathbb{D})=\\{f:\mathbb{D}\to\mathbb{C}:\,f(z)=\sum_{n=0}^{\infty}a_{n}z^{n},\\{a_{n}\\}_{n=0}^{\infty}\in\ell^{2}(\mathbb{N}_{0})\\}.$

The Hardy space is a Hilbert space, with the inner product defined by

$\langle
f,g\rangle=\sum_{n=0}^{\infty}a_{n}\overline{b}_{n},\,\,\mbox{for}\,\,f=\sum_{n=0}^{\infty}a_{n}z^{n},g=\sum_{n=0}^{\infty}b_{n}z^{n}.$

Furthermore the sequence of functions $\\{z^{n}\\}_{n=0}^{\infty}$ is an
orthonormal basis for $H^{2}(\mathbb{D})$. The shift operator  on the Hardy
space is defined by

$\displaystyle{\cal S}:H^{2}(\mathbb{D})\to H^{2}(\mathbb{D}),{\cal
S}f(z)=zf(z),\,z\in\mathbb{D}$

Finally let $1_{\mathbb{D}}$ denote the constant function, i.e.,
$1_{\mathbb{D}}(z)=1$.

###### Theorem 2.13

Consider a bounded operator $T:{\cal H}\to{\cal H}$ and let $\varphi\in{\cal
H}.$ Then $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is a frame if and only if there
exists a shift invariant subspace ${\cal F}\subset H^{2}(\mathbb{D})$ and a
bounded bijective operator $W:\cal F^{\perp}\to{\cal H}$ such that

$T=WP_{{\cal F}^{\perp}}{\cal S}W^{-1},\quad\varphi=WP_{{\cal
F}^{\perp}}1_{\mathbb{D}}.$

In the affirmative case, $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is a Riesz basis
if and only if ${\cal F}=\\{0\\}$.

Proof. In the entire proof we let $\langle\cdot,\cdot\rangle$ denote the inner
product on $H^{2}(\mathbb{D}).$ First assume that
$\\{T^{n}\varphi\\}_{n=0}^{\infty}$ is a frame. Consider the linear operator
$V:H^{2}(\mathbb{D})\to{\cal H}$ defined by

$\displaystyle Vf=\sum_{n=0}^{\infty}\langle f,z^{n}\rangle T^{n}\varphi,\
f\in H^{2}(\mathbb{D}).$ (2.8)

The operator $V$ is bounded; indeed, letting $B$ denote a Bessel bound for
$\\{T^{n}\varphi\\}_{n=0}^{\infty}$, we have

$\displaystyle\|Vf\|^{2}=\|\sum_{n=0}^{\infty}\langle f,z^{n}\rangle
T^{n}\varphi\|^{2}\leq B\sum_{n=0}^{\infty}|\langle
f,z^{n}\rangle|^{2}=B\|f\|^{2}.$

The frame assumption also implies that the operator $V$ is surjective.
Moreover, for $f\in H^{2}(\mathbb{D})$ we have

$\displaystyle TVf$ $\displaystyle=$ $\displaystyle
T\sum_{n=0}^{\infty}\langle f,z^{n}\rangle
T^{n}\varphi=\sum_{n=0}^{\infty}\langle f,z^{n}\rangle T^{n+1}\varphi$ (2.9)
$\displaystyle=$ $\displaystyle V\sum_{n=0}^{\infty}\langle f,z^{n}\rangle
z^{n+1}=V{\cal S}f.$

This shows in particular that $\ker V$ is a closed shift-invariant subspace of
$H^{2}(\mathbb{D})$. Now let ${\cal F}:=\ker V$ and consider the restriction
of the operator $V$ to the subspace ${\cal F}^{\bot},$ i.e., let
$W:=V_{|_{{\cal F}^{\perp}}}:{\cal F}^{\perp}\to{\cal H}$. Also let $P_{{\cal
F}^{\perp}}:H^{2}(\mathbb{D})\to{\cal F}^{\perp}$ be the orthogonal
projection. Then, using (2.9), on the subspace ${\cal F}^{\perp},$ we have

$\displaystyle TW=TV=V{\cal S}=VP_{{\cal F}^{\bot}}{\cal S}=WP_{{\cal
F}^{\bot}}{\cal S}$

Therefore the invertibility of $W$ yields that $T=WP_{{\cal F}^{\perp}}{\cal
S}W^{-1};$ also $\varphi=V1_{\mathbb{D}}=VP_{{\cal
F}^{\perp}}1_{\mathbb{D}}=WP_{{\cal F}^{\perp}}1_{\mathbb{D}}$.

Conversely, let ${\cal F}\subset H^{2}(\mathbb{D})$ be a closed shift-
invariant subspace and $W:{\cal F}^{\perp}\to{\cal H}$ be an invertible
bounded operator. Since $\\{{\cal S}^{n}1_{\mathbb{D}}\\}_{n=0}^{\infty}$ is
an orthonormal basis for $H^{2}(\mathbb{D})$, the sequence $\\{P_{{\cal
F}^{\perp}}{\cal S}^{n}1_{\mathbb{D}}\\}_{n=0}^{\infty}$ is a frame for ${\cal
F}^{\perp}$. Now let $T:=WP_{{\cal F}^{\perp}}{\cal S}W^{-1}$ and
$\varphi:=WP_{{\cal F}^{\perp}}1_{\mathbb{D}}$. Since ${\cal F}$ is invariant
under the action of ${\cal S}$, we have

$P_{{\cal F}^{\perp}}{\cal S}=P_{{\cal F}^{\perp}}{\cal S}P_{{\cal
F}^{\perp}}$

which implies that $(P_{{\cal F}^{\perp}}{\cal S})^{n}=P_{{\cal
F}^{\perp}}{\cal S}^{n}$, for every $n\in\mathbb{N}_{0}$. Thus

$\displaystyle T^{n}\varphi$ $\displaystyle=$ $\displaystyle W(P_{{\cal
F}^{\perp}}{\cal S})^{n}W^{-1}\varphi=W(P_{{\cal F}^{\perp}}{\cal
S})^{n}P_{{\cal F}^{\perp}}1_{\mathbb{D}}=WP_{{\cal F}^{\perp}}{\cal
S}^{n}1_{\mathbb{D}}$

Therefore the collection $\\{T^{n}\varphi\\}_{n=0}^{\infty}$= $\\{WP_{{\cal
F}^{\perp}}{\cal S}^{n}1_{\mathbb{D}}\\}_{n=0}^{\infty}$ is a frame for ${\cal
H}$. $\square$

Theorem 2.13 is theoretically appealing, but it is not instrumental in terms
of constructing more examples of explicitly given frames
$\\{T^{n}\varphi\\}_{n=0}^{\infty}.$ The difficulty in constructing such
frames is the motivation for the remaining sections in the paper.

###  2.4 Frames and bi-orbit representations

In this section we will consider a frame, indexed as
$\\{f_{k}\\}_{k\in\mathbb{Z}};$ clearly, we can bring any frame into this form
by a reordering and reindexing.

A seemingly innocent modification of the topic from Section 2.3 would be to
ask for representations of a given frame $\\{f_{k}\\}_{k\in\mathbb{Z}}$ of the
form $\\{T^{k}f_{0}\\}_{k\in\mathbb{Z}}$ for an invertible operator
$T:\text{span}\\{f_{k}\\}_{k\in\mathbb{Z}}\to\text{span}\\{f_{k}\\}_{k\in\mathbb{Z}};$
such a representation is called a bi-orbit representation. Interestingly, this
leads to a theory that in certain aspects is very similar to the operator
representations discussed in the previous sections, but also is completely
different at some points. In this section we will highlight some of the
similarities and differences for the two ways of representing a frame.

First, it was proved in [15] that Proposition 1.3 has a completely similar
version for bi-orbit representations: any linearly independent frame
$\\{f_{k}\\}_{k\in\mathbb{Z}}$ has a representation
$\\{f_{k}\\}_{k\in\mathbb{Z}}=\\{T^{k}f_{0}\\}_{k\in\mathbb{Z}}$ for some
linear and invertible operator
$T:\text{span}\\{f_{k}\\}_{k\in\mathbb{Z}}\to\text{span}\\{f_{k}\\}_{k\in\mathbb{Z}}.$
It was also proved that Theorem 2.9 has a parallel version: indeed, if
additionally the kernel of the synthesis operator is invariant under right and
left-shifts, the operator $T$ extends to a bijective and bounded operator
$T:{\cal H}\to{\cal H}.$

However, when it comes to concrete manifestations of frames of the form
$\\{T^{k}f_{0}\\}_{k\in\mathbb{Z}}$ as well as other properties for such
frames, there are remarkable difference between orbit/bi-orbit
representations. This becomes apparent already from the following example.

###### Example 2.14

As discussed in Section 1.5, the shift-invariant system
$\\{T_{k}\mbox{sinc}\\}_{k\in\mathbb{Z}}$ is an orthnormal basis for the
Paley-Wiener space $PW$ of functions in $L^{2}(\mathbb{R})$, while
$\\{T_{k/2}\mbox{sinc}\\}_{k\in\mathbb{Z}}$ is an overcomplete frame for the
same space. Since
$\\{T_{kb}\varphi\\}_{k\in\mathbb{Z}}=\\{(T_{b})^{k}\varphi\\}_{k\in\mathbb{Z}},$
such frames are born to have a bi-orbit representation. However, since
$||T_{kb}\varphi||=||\varphi||$ for all $k\in\mathbb{Z}$ it is clear from
Theorem 2.12 that an overcomplete shift-invariant frame can not have a
representation $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ for a bounded operator $T.$

Recall that Lemma 2.2 and Theorem 2.12 provided necessary conditions for an
orbit $\\{T^{n}\varphi\\}_{n=0}^{\infty}$ to be a frame for a bounded operator
$T.$ The next result, proved in [18], shows that the corresponding conditions
on a bi-orbit $\\{T^{n}\varphi\\}_{n\in\mathbb{Z}}$ are fundamentally
different:

###### Theorem 2.15

Assume that $\\{T^{n}\varphi\\}_{n\in\mathbb{Z}}$ is a frame with bounds $A,B$
for some $\varphi\in{\cal H}$ and a bounded invertible operator $T:{\cal
H}\to{\cal H}.$ Then for each $f\in{\cal H}$ we have that

$\|T^{n}f\|\,\geq\,\sqrt{\frac{A}{B}}\,\|f\|\qquad\text{and}\qquad\|(T^{*})^{n}f\|\,\geq\,\sqrt{\frac{A}{B}}\,\|f\|.$

Theorem 2.15 makes it natural to ask whether overcomplete Gabor frames have
representations as bi-orbits of a bounded operator. We phrase the question as
stated originally in [16]:

Q: Do there exist overcomplete Gabor frames
$\\{E_{mb}T_{na}g\\}_{m,n\in\mathbb{Z}}$ such that an appropriate ordering
$\\{f_{k}\\}_{k\in\mathbb{Z}}$ of the frame elements has a representation

$\displaystyle\\{f_{k}\\}_{k\in\mathbb{Z}}=\\{T^{n}\varphi\\}_{n=-\infty}^{\infty},$

for some bounded operator $T:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ and some
$\varphi\in L^{2}(\mathbb{R})$?

In [22] Corso provided a partial solution to the question, by showing that the
answer is negative if $ab\notin\mathbb{Q},$ as well as under certain support
conditions on the function $g.$ However, the general question is still open.

###  2.5 Frame representations via multi-orbits

The difficulty in constructing explicitly given frames
$\\{T^{n}\varphi\\}_{n=0}^{\infty},$ discussed in detail in Section 2.3, makes
it natural to explore various modifications of the basic idea. In this section
we will discuss the additional flexibility that is obtained by allowing a
frame representation using a finite number of operator orbits. Let us first
connect the topics in Section 2.3 and Section 2.4 by showing that any frame
with a bi-orbit representation $\\{T^{n}\varphi\\}_{n\in\mathbb{Z}}$ can be
considered as the union of two orbits:

###### Example 2.16

Consider any frame that has a representation as a bi-infinite orbit
$\\{T^{n}\varphi\\}_{n\in\mathbb{Z}}$ for a bounded and bijective operator
$T:{\cal H}\to{\cal H}.$ Then

$\displaystyle\\{T^{n}\varphi\\}_{n\in\mathbb{Z}}=\\{T^{n}\varphi\\}_{n=0}^{\infty}\cup\\{(T^{-1})^{n}T^{-1}\varphi\\}_{n=0}^{\infty},$
(2.10)

i.e., the frame $\\{T^{n}\varphi\\}_{n\in\mathbb{Z}}$ can be represented as a
union of two orbits, generated by the bounded operators $T$ and $T^{-1}.$

The following result, originally proved in [14], shows that the idea of multi-
orbits significantly enlarges the class of frames that can be analyzed: for
example, all the classical frames in harmonic analysis like Gabor frames and
wavelet frames have such representations!

###### Theorem 2.17

Consider a frame $\\{f_{k}\\}_{k=1}^{\infty}$ which is norm-bounded below.
Then there is a finite collection of vectors from
$\\{f_{k}\\}_{k=1}^{\infty},$ to be called $\varphi_{1},\dots,\varphi_{J},$
and corresponding bounded operators $T_{j}:{\cal H}\to{\cal H}$ with closed
range, such that

$\displaystyle\\{f_{k}\\}_{k=1}^{\infty}=\bigcup_{j=1}^{J}\\{T_{j}^{n}\varphi_{j}\\}_{n=0}^{\infty}.$
(2.11)

While Theorem 2.17 is theoretically appealing, its practical applicability is
limited by the fact that the proof is based on the Feichtinger Theorem and
thus does not give direct access to the operators $T_{j}.$ More results on
multi-orbit representations can be found in [10].

##  3 Approximate operator representations

In Section 2 the focus is on frames $\\{f_{k}\\}_{k=1}^{\infty}$ having an
exact match with an operator orbit $\\{T^{n}\varphi\\}_{n=0}^{\infty};$
unfortunately, the results show that this is a very restrictive condition and
that the standard frames in $L^{2}(\mathbb{R})$ are excluded. In the current
section we will consider two relaxations on the orbit representations, which
in combination yield a theory covering a much larger class of frames. To be
more precise, recall that Theorem 1.10 by Halperin et all showed that suborbit
representations $\\{T^{\alpha(k)}\varphi\\}_{k=1}^{\infty}$ via bounded
operators are available for any linearly independent family
$\\{f_{k}\\}_{k=1}^{\infty};$ however, in general we do not have a feasible
procedure to identify the operator $T$ and the appropriate integers
$\alpha(k).$ In this section we will show that for a large class of frames we
can obtain approximate suborbit representations, where it is possible to
specify as well the operator $T$ as the integers $\alpha(k).$ In the entire
section we follow the presentation in [17]; we refer to that paper for proofs
and more information.

We begin with a formal definition, explaining the exact meaning of
“approximation.” In words, a frame $\\{f_{k}\\}_{k=1}^{\infty}$ is
approximated by a sequence $\\{\widetilde{f_{k}}\\}_{k=1}^{\infty}$ if
$\\{\widetilde{f_{k}}\\}_{k=1}^{\infty}$ satisfies the standard perturbation
condition within frame theory:

###### Definition 3.1

Let $\\{f_{k}\\}_{k=1}^{\infty}$ be a frame for ${\cal H}.$ Given any
$\epsilon>0,$ a sequence $\\{\widetilde{f_{k}}\\}_{k=1}^{\infty}\subset{\cal
H}$ is called an $\epsilon$-approximation of $\\{f_{k}\\}_{k\in I}$ if

$\displaystyle\left|\left|\sum
c_{k}(f_{k}-\widetilde{f_{k}})\right|\right|^{2}\leq\epsilon\,\sum|c_{k}|^{2}$
(3.1)

for all finite sequences $\\{c_{k}\\}.$

For sufficiently small values of $\epsilon,$ an $\epsilon$-approximation
$\\{\widetilde{f_{k}}\\}_{k=1}^{\infty}$ of a frame
$\\{f_{k}\\}_{k=1}^{\infty}$ is itself a frame and shares several key
properties of the frame, e.g., its excess. Furthermore, the synthesis operator
and the frame operator for $\\{f_{k}\\}_{k=1}^{\infty}$ are approximated
“well” by the corresponding operators associated with
$\\{\widetilde{f_{k}}\\}_{k=1}^{\infty}$:

###### Theorem 3.2

Consider a frame $\\{f_{k}\\}_{k=1}^{\infty}$ for ${\cal H}$ with frame bounds
$A,B$, and assume that $\\{\widetilde{f_{k}}\\}_{k=1}^{\infty}\subset{\cal H}$
is an $\epsilon$-approximation of $\\{f_{k}\\}_{k=1}^{\infty}$ for some
$\epsilon\in]0,A[.$ Then the following hold:

  * (i)

$\\{\widetilde{f_{k}}\\}_{k=1}^{\infty}$ is a frame with bounds
$A(1-\sqrt{\frac{\epsilon}{A}})^{2}$ and $B(1+\sqrt{\frac{\epsilon}{B}})^{2},$
with the same excess as $\\{f_{k}\\}_{k=1}^{\infty}.$

  * (ii)

Denoting the synthesis operators and frame operators of
$\\{f_{k}\\}_{k=1}^{\infty}$ and $\\{\widetilde{f_{k}}\\}_{k=1}^{\infty}$ by
$U,\widetilde{U}$, respectively, $S,\widetilde{S}$, we have

$\|U-\widetilde{U}\|\leq\sqrt{\epsilon},\quad\|S-\widetilde{S}\|\leq\sqrt{\epsilon
B}\left(2+\sqrt{\frac{\epsilon}{B}}\right),$

and

$\|S^{-1}-\widetilde{S}^{-1}\|\leq\frac{\sqrt{\epsilon
B}(2+\sqrt{\frac{\epsilon}{B}})}{A^{2}(1-\sqrt{\frac{\epsilon}{A}})^{2}}.$

The connection to approximate frame representations using suborbits of a
bounded operator is explained in the next result, which provides a natural way
to satisfy the condition (3.1):

###### Corollary 3.3

Let $\\{f_{k}\\}_{k=1}^{\infty}$ be a frame for ${\cal H}$ with lower frame
bound $A$. Also, let $\varphi\in{\cal H}$ and consider a bounded operator
$T:{\cal H}\to{\cal H}$. Assume that for a given $\epsilon\in]0,A[,$ and for
any $k\in\mathbb{N}$ there exists a nonnegative integer
$\alpha(k)\in\mathbb{N}_{0}$ such that

$\displaystyle\|f_{k}-T^{\alpha(k)}\varphi\|^{2}\leq\frac{\epsilon}{2^{k}}.$
(3.2)

Then $\\{T^{\alpha(k)}\varphi\\}_{k=1}^{\infty}$ is an
$\epsilon$-approximation of $\\{f_{k}\\}_{k=1}^{\infty}$.

Note that Corollary 3.3 applies to any frame and any hypercyclic operator:

###### Example 3.4

Let $T:{\cal H}\to{\cal H}$ be a hypercyclic operator with hypercyclic vector
$\varphi.$ Then, for any frame $\\{f_{k}\\}_{k=1}^{\infty}$ for ${\cal H}$ and
any given $\epsilon>0$ there exists nonnegative integers
$\alpha(k),\,k\in\mathbb{N},$ such that (3.2) holds. Thus, for sufficiently
small values of $\epsilon,$ we obtain a frame
$\\{T^{\alpha(k)}\varphi\\}_{k=1}^{\infty}$ with the same excess as the frame
$\\{f_{k}\\}_{k=1}^{\infty}$ and approximating $\\{f_{k}\\}_{k=1}^{\infty}$ in
the sense of Definition 3.1.

In general we do not have direct access to the numbers
$\alpha(k),\,k\in\mathbb{N},$ such that (3.2) holds, not even for hypercyclic
operators. However, under natural conditions we can be very explicit about the
choices of appropriate scalars $\alpha(k)$ for certain choices of the operator
$T.$ In the rest of the section we will use the following

General setup: Fix an orthonormal basis $\\{e_{k}\\}_{k=1}^{\infty}$ for the
Hilbert space ${\cal H},$ and let $\\{f_{k}\\}_{k=1}^{\infty}$ denote a frame
for ${\cal H}$. Fix some $\lambda>1$, and define the scaled left/right-shift
operators on ${\cal H}$ by

$\displaystyle
T\left(\sum_{k=1}^{\infty}c_{k}e_{k}\right):=\lambda\sum_{k=1}^{\infty}c_{k+1}e_{k},\,\,U\left(\sum_{k=1}^{\infty}c_{k}e_{k}\right):=\lambda^{-1}\sum_{k=1}^{\infty}c_{k}e_{k+1},\,\,\\{c_{k}\\}_{k=1}^{\infty}\in{\ell}^{2}(\mathbb{N}).$

For appropriately chosen nonnegative integers $\alpha(k),\,k\in\mathbb{N},$
let

$\displaystyle\varphi:=\sum_{n=1}^{\infty}U^{\alpha(n)}f_{n}.$ (3.3)

For the case of finitely supported vectors $\\{f_{k}\\}_{k=1}^{\infty}$
(meaning that each vector $f_{k}$ is a finite linear combination of elements
from $\\{e_{k}\\}_{k=1}^{\infty}$) the following result specifies how to
choose the powers $\alpha(k)$ such that the vector $\varphi$ in (3.3) is well-
defined and (3.2) holds:

###### Theorem 3.5

Under the conditions in the general setup, assume that
$\\{f_{k}\\}_{k=1}^{\infty}$ consists of finitely supported vectors in ${\cal
H};$ for $k\in\mathbb{N},$ let $m(k)$ denote the largest index for a nonzero
coordinate in $f_{k}.$ Let $\\{\alpha(k)\\}_{k=1}^{\infty}$ be a strictly
increasing sequence of nonnegative integers such that $\alpha(1)=0$ and
$\alpha(k+1)-\alpha(k)\geq m(k)$ for all $k\in\mathbb{N}.$ Then, for any
$k\in\mathbb{N},$

$\displaystyle||f_{k}-T^{\alpha(k)}\varphi||^{2}\leq\frac{B\lambda^{2}}{\lambda^{2}-1}\lambda^{-2[\alpha(k+1)-\alpha(k)]}.$
(3.4)

In particular, choosing the nonnegative integers $\alpha(k)$ such that
$\alpha(1)=0$ and for a given $\epsilon\in]0,A[,$

$\displaystyle\alpha(k+1)-\alpha(k)\geq\max\left(m(k),\frac{k\,\ln(2)+\ln\left(\frac{B}{\epsilon}\right)+\ln\left(\frac{\lambda^{2}}{\lambda^{2}-1}\right)}{2\ln(\lambda)}\right),$
(3.5)

the condition (3.2) is satisfied, i.e., the conclusions in Theorem 3.2 hold.

For certain values of the scaling parameter $\lambda$ we can be completely
specific about how to choose the integers $\alpha(k):$

###### Corollary 3.6

In the setup of Theorem 3.5, let $\lambda=\sqrt{2},$ take an upper frame bound
of the form $B=2^{N}$ for some $N\in\mathbb{N}$ and a tolerance
$\epsilon=2^{-j}$ for some $j\in\mathbb{N}.$ Then the following hold.

  * (i)

Without any restriction on the support sizes $m(k)$ of the vectors $f_{k},$
the condition (3.5) is satisfied if $\alpha(1)=0$ and

$\displaystyle\alpha(k)=(k-1)\left[N+j+1+\frac{k}{2}\right]+\sum_{\ell=1}^{k-1}m(\ell),\,\,k\in\mathbb{N}\setminus\\{1\\}.$
(3.6)

  * (ii)

If $m(k)\leq N+j+1+k$ for all $k\in\mathbb{N},$ the condition (3.5) is
satisfied if

$\displaystyle\alpha(k)=(k-1)\left[N+j+1+\frac{k}{2}\right],\,\,k\in\mathbb{N}.$
(3.7)

The main condition in Theorem 3.5, namely, that the frame
$\\{f_{k}\\}_{k=1}^{\infty}$ is finitely supported, can be relaxed: indeed,
similar but slightly more technical results can be proved for localized
frames. We refer to [17] for the details. Note also that generalizations to
Banach spaces (for sequences $\\{f_{k}\\}_{k=1}^{\infty}$ without assuming any
frame property) have been obtained in [20].

## References

  * [1] Aldroubi, A., Cabrelli, C., Molter, U., and Tang, S.: Dynamical sampling. Appl. Harm. Anal. Appl. 42 no. 3 (2017), 378-–401.
  * [2] Aldroubi, A., Cabrelli, C., Çakmak, A. F., Molter, U., and Petrosyan, A.: Iterative actions of normal operators. J. Funct. Anal. 272 no. 3 (2017), 1121–1146.
  * [3] Aldroubi, A. and Petrosyan, A., Dynamical sampling and systems from iterative actions of operators. In “Frames and Other Bases in Abstract and Function Spaces.” Eds. Mhaskar, H., Pesenson, I. Zhou, D. X., Le Gia, Q. T., and Mayeli, A., Birkhäuser, Boston, 2017
  * [4] Ansari, S. I.: Hypercyclic and cyclic vectors. J. Funct. Anal. 128 (1995), no. 2, 374–-383.
  * [5] Balan, R., Casazza, P., Heil, C. and Landau, Z.: Deficits and excesses of frames. Adv. Comp. Math. 18 (2002), 93–116.
  * [6] Bayart, F., and Matheron, E.: Dynamics of linear operators. Cambridge University Press, 2009.
  * [7] Benedetto, J. and Li, S.: The theory of multiresolution analysis frames and applications to filter banks. Appl. Comp. Harm. Anal. 5 (1998), 389–427.
  * [8] Bourdon, P. S., and Feldman, N. S.: Somewhere dense orbits are everywhere dense. Indiana Univ. Math. J. 52 no. 3 (2003), 811–819.
  * [9] Bownik, M., and Speegle, D.: Linear independence of Parseval wavelets. Illinois J. Math. 54 (2010), 771–785.
  * [10] Cabrelli, C., Molter, U, Paternostro, V., and Philipp, F.: Dynamical sampling on finite index sets. J. Anal. Math. 140 no.2 (2020), 637–667
  * [11] Chen, K. Y.: On aperiodicity and hypercyclic weighted translation operators. J. Math. Anal. Appl. 462 (2018), no. 2, 1669–1678
  * [12] Christensen, O.: An introduction to frames and Riesz bases. Second expanded edition. Birkhäuser (2016)
  * [13] Christensen, O., Deng, B., and Heil, C.: Density of Gabor frames. Appl. Comp. Harm. Anal. 7 (1999), 292–304.
  * [14] Christensen, O. and Hasannasab, M.: Frame properties of systems arising via iterative actions of operators. Appl. Comp. Harm. Anal. 46 (2019), 664-673.
  * [15] Christensen, O., and Hasannasab, M.:  Operator representations of frames: boundedness, duality, and stability Integral Equations and Operator Theory 88 (2017), no. 4, 483–499.
  * [16] Christensen, O., Hasannasab, M.: Frames, operator representations, and open problems.  In “The diversity and beauty of applied operator theory“, 155–165, Oper. Theory Adv. Appl., 268, Birkhäuser, Cham, 2018.
  * [17] Christensen, O., Hasannasab, M.: Frame representations via suborbits of bounded operators, accepted for publication in Studia Math., 2021.
  * [18] Christensen, O., Hasannasab, M. and Philipp, F.: Frame Properties of Operator Orbits. Math. Nach. 293 (2020), 52–66.
  * [19] Christensen, O., Hasannasab, M., and Rashidi, E.: Dynamical sampling and frame representations with bounded operators. J. Math. Anal. Appl. 463 no. 2 (2018), 634–644.
  * [20] Christensen, O., Hasannasab, M., and Steidl, G.: On approximate operator representations of sequences in Banach spaces. Accepted for publication in Complex Analysis and Operator Theory, 2021.
  * [21] Christensen, O., Hasannasab, M., and Stoeva, D. T.: Operator representations of sequences and dynamical sampling.  Sampling Theory in Signal and Image Processing 17 no. 1 (2018), 29–42.
  * [22] Corso, R.: Orbits of bounded bijective operators and Gabor frames. Ann. Mat. Pura Appl. 200 no. 1 (2021), 137–148.
  * [23] Costakis, G.: On a conjecture of D. Herrero concerning hypercyclic operators. C. R. Acad. Sci. Paris 330 (2000), 179–182.
  * [24] Duren, P. L.: Theory of $H^{p}$-spaces. New York and London: Academic Press (1970).
  * [25] Grosse-Erdmann, K. G., and Manguillot, A. P.: Linear chaos. Springer Science $\&$ Business Media, (2011).
  * [26] Grosse-Erdmann K. G., Universal families and hypercyclic operators Bulletin of the American Mathematical Society. 1999;36(3):345–81.
  * [27] Gröchenig, K.: Foundations of time-frequency analysis. Birkhäuser, Boston, 2000.
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  * [29] Heil, C.: A basis theory primer. Expanded edition. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York, 2011.
  * [30] Heil, C., Ramanathan, J. and Topiwala, P.: Linear independence of time-frequency translates. Proc. Amer. Math. Soc. 124 (1996), 2787–2795.
  * [31] Hilden, H. M.; Wallen, L. J. Some cyclic and non-cyclic vectors of certain operators. Indiana Univ. Math. J. 23 (1973/74), 557–-565.
  * [32] Linnell, P.: Von Neumann algebras and linear independence of translates. Proc. Amer. Math. Soc. 127 no. 11 (1999), 3269–3277.
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Ole Christensen
DTU Compute
Technical University of Denmark
Building 303
2800 Lyngby
Denmark
Email<EMAIL_ADDRESS>
Marzieh Hasannasab
Institut für Mathematik
TU Berlin
Straße des 17. Juni 136
10623 Berlin
Germany
Email<EMAIL_ADDRESS>


# Reliable Micro-Transfer Printing Method for Heterogeneous Integration of
Lithium Niobate and Semiconductor Thin Films

Tom Vandekerckhove1,2,∗, Tom Vanackere1,2, Jasper De Witte1, Stijn Cuyvers1,
Luis Reis1,2, Maximilien Billet1, Günther Roelkens1, Stéphane Clemmen1,2,3,
and Bart Kuyken1

(1\. Photonics Research Group, INTEC, Ghent University - imec, 9052 Ghent,
Belgium
2\. OPERA-Photonique CP 194/5, Université Libre de Bruxelles, 1050 Brussels,
Belgium
3\. Laboratoire d’Information Quantique, Université Libre de Bruxelles, 1050
Brussels, Belgium
<EMAIL_ADDRESS>
###### Abstract

High-speed Pockels modulation and second-order nonlinearities are key
components in optical systems, but CMOS-compatible platforms like silicon and
silicon nitride lack these capabilities. Micro-transfer printing of thin-film
lithium niobate offers a solution, but suspending large areas of thin films
for long interaction lengths and high-Q resonators is challenging, resulting
in a low transfer yield. We present a new source preparation method that
enables reliable transfer printing of thin-film lithium niobate. We
demonstrate its versatility by successfully applying it to gallium phosphide
and silicon, and provide an estimate of the transfer yield by subsequently
printing 25 lithium niobate films without fail.

## 1\. Introduction

Photonic integrated circuits enable the construction of complex optical
systems with small footprints in a scalable and controlled way, with
applications ranging from telecommunications [1, 2] to quantum technologies
[3, 4]. Silicon (Si) and silicon nitride (SiN) photonics have established
themselves as mature photonic platforms due to their CMOS compatibility, but
lack certain key components such as optical gain, fast Pockels modulation and
$\chi^{(2)}$ nonlinearities. A promising approach to circumvent these
shortcomings is to heterogeneously integrate other materials at a back-end
level to remain CMOS-compatible [5]. Integrating direct band gap III-V
materials brings gain to the integrated circuits [6], enabling fabrication of
on-chip lasers [7, 8] and detectors [9, 10]. Introducing thin-film lithium
niobate and other ferro-electric materials gives access to high-speed Pockels
modulation [11, 12] and second order optical nonlinearities [13].

A mature heterogeneous integration technique consists of die-to-wafer or
wafer-to-wafer bonding [14, 15]. The wafer containing the device material is
bonded on a target wafer, followed by a substrate removal step and device
patterning. While lithographic alignment accuracy is obtained in the
fabrication of optical devices, the large bonding footprint combined with the
required back-end processing makes co-integration of multiple materials
challenging. A less mature but promising heterogeneous integration method for
integrated photonics is micro-transfer printing ($\upmu$TP)[16, 17, 18].
Devices are pre-fabricated on a source wafer, after which they are picked with
a PDMS stamp and printed on the target wafer through kinetically controlled
adhesion. While the alignment is determined through the printing process, with
state-of-the-art tools providing $\pm 0.5$ $\upmu$m $3\sigma$ alignment
accuracy [18], dense co-integration of different materials is possible with
limited processing required after printing. Moreover, arrays of devices can be
transferred in a single step, giving a high throughput.

To obtain a high transfer yield in micro-transfer printing, the pre-fabricated
devices are suspended on the source wafer, only connected at several points to
the source substrate. In case of semiconductor optical amplifiers and diodes,
the devices are typically several microns thick, limiting their mechanical
flexibility. In contrast, for thin films like lithium niobate (LN) or gallium
phosphide (GaP), the layers are only several hundreds of nanometers in
thickness. At the same time, large areas are desirable to enable long
interaction lengths and the definition of larger structures such as racetrack
resonators. However, the suspension of large areas of thin films is generally
challenging. In this paper, we developed a new source preparation method to
reliably micro-transfer print large areas of thin films based on pillar-like
support structures. We discuss the optimization of the method and demonstrate
reliable transfer-printing of thin-film lithium niobate. Once optimized, the
thin film areas can easily be enlarged without added process development.
While optimized with pillar-like structures, we show the method works with
other support configurations and give an estimate of the yield by printing 25
lithium niobate thin films without fail onto silicon nitride waveguides.
Lastly, we demonstrate the versatility of the method by successfully applying
it on large area thin film gallium phosphide and silicon.

## 2\. Micro-transfer printing and challenges

Micro-transfer printing is a pick-and-place method that allows for the
heterogeneous integration of many materials, with limited post-processing
required. This makes it compatible with CMOS technology as the integration
only happens on a back-end level, keeping the front-end process flow
unaffected and avoiding any contamination. The micro-transfer printing process
is schematically illustrated in Fig.1.

Figure 1: Micro-transfer printing concept: a suspended coupon (red) is picked
with a PDMS stamp from the source wafer, breaking the tethers (green) in a
controlled way, after which it is printed onto the target wafer.

A source wafer is fabricated where the to-be-integrated devices (red in
Fig.1), from now on referred to as coupons, are prepared. These can range from
optical amplifiers and detectors, to modulators, but consist of thin-film
lithium niobate in the framework of this paper. The layer stack contains a
release layer underneath the coupon, which can be selectively etched.
Connections, referred to as tethers (green in Fig.1) keep the coupons attached
to the wafer. In a last preparation step, the coupons are suspended by etching
away the release layer and are ready to be picked. In parallel, the target
wafer containing e.g. the photonic circuit is fabricated. An elastomeric PMDS
stamp is then used to pick the coupon from the source wafer by breaking the
tethers in a controlled manner, and print the device onto the target wafer
through direct or adhesive bonding.

For thin films such as thin-film lithium niobate, the suspension of large
areas can be challenging due to their mechanical flexibility. To demonstrate
this, we start from a 300 nm x-cut LN/2 $\upmu$m SiO2/LN layer stack (NanoLN)
and pattern simple rectangles with hourglass-shaped tethers to the side in the
300 nm LN device layer. The underlying SiO2 layer serves as release layer,
which can be selectively etched with hydrofluoric (HF) acid. The release etch
is typically optimized to ease the suspension by avoiding the liquid-to-vapor
transition, where capillary forces cause the coupons to collapse to the
substrate. A first option consists of using a vapor-phase HF tool where the
etchant is never in a liquid phase. However, the non-volatile byproduct
lithium fluoride (LiF) is redeposited on the top and bottom of the coupon [19,
20]. Alternatively, a liquid HF etchant can be used in combination with
critical point drying (CPD), such that any byproducts are rinsed away in the
liquid phase. While successful for small area coupons, from a certain size the
coupons will collapse due their flexibility, as shown in Fig.2. While
occasionally the edges remain suspended, the coupons themselves collapse. The
strong adhesion with the substrate impedes the coupons from being picked or
creates significant stress in the thin film during pick-up, leading to crack
formation. Fig.2 depicts such printed coupon on a SiN waveguide, showing
cracks through the coupon with non-ideal breaking of the tethers.

To overcome these issues, we developed a method that relies on pillar-like
support structures to keep the LN from collapsing to the substrate,
independent of the coupon area. Therefore, the coupons do not adhere to the
substrate, avoiding stresses during pick-up and resulting in an excellent
transfer yield.

Figure 2: Challenges with micro-transfer printing of thin films. Left:
Collapsed coupons on the source wafer with a corresponding cross section.
Right: Resulting printed LN coupon on a SiN waveguide showing cracks
(indicated by arrows) and bad tether-breaking.

## 3\. Pillar-based transfer printing method

Figure 3: Process flow of the pillar-based micro-transfer printing method

The process flow is depicted in Fig.3. The fabrication starts from a layer
stack containing a thin film with a release layer underneath (Fig.3.1). For
lithium niobate, this consists of 300 nm x-cut LN/2 $\upmu$m SiO2/LN layer
stack (NanoLN). A coupon is patterned in the device layer through standard
optical lithography and dry etching (Fig.3.2). Tethers are defined connecting
the coupon to the sides, with holes within the coupon providing access to the
release layer underneath. Next, the release layer is partially etched through
the holes to create voids that will serve as a mold (Fig.3.3). While HF-based
wet etching is used, dry etching is also a possibility. These molds are then
filled with a support material like photoresist, creating a pillar underneath
each patterned hole (Fig.3.4). Additionally, covering the coupon with support
material increases its mechanical strength. From this point, the coupon is
ready to be released from the substrate.

Releasing the coupon is done with an HF-based wet etch, without a CPD step as
typical intermediate fluids would dissolve the photoresist. While previously
the LN collapsed to the substrate, the thin film is now suspended by the
pillars (Fig.3.5). Due to the limited contact area between pillars and
substrate, the adhesion of the coupon to the substrate is significantly
reduced, enabling an easy pick-up with the stamp. In parallel to the source
preparation, dedicated recesses are patterned in the target wafer next to the
photonic circuit. With both source and target wafer prepared, the coupon is
picked and printed with the pillars placed within the recesses, such that the
thin film bonds directly to the target (Fig.3.6 & 3.7). Lastly, the
photoresist is removed with a standard cleaning procedure, leaving the thin
film printed on top of the photonic circuit (Fig.3.8).

## 4\. Optimization of the method

Figure 4: Optimization of the pillar-based method: I Optical microscope images
of tether design: (a) new tether design with crack barrier, (b) Broken tether
without barrier, (c) Broken tether with barrier. II Optical microscope image
of pillar spacing effect: (d) Large pillar spacing, (e) Small pillar spacing.
III AFM measurements of a pillar: (f) a single pillar underneath the coupon,
(g) AFM roughness measurement on top of a pillar.

To ensure a high transfer yield, some further optimizations can be done, as
summarized in Fig.4. A first improvement is found in the tether design. A
simple hourglass design would be expected to break at its thinnest point.
Unfortunately, lithium niobate does not cleave well and sporadically a crack
arises that can propagate through the coupon. By adding elongated holes as so-
called crack barriers [21] to the design (Fig.4a), the cracks are shielded and
remain localized at the tether. Fig.4b and 4c show the effect without and with
crack barrier, respectively. Furthermore, in conventional micro-transfer
printing methods the tethers must be strong enough to suspend the coupon,
while weak enough to break during pick-up. In this method, the tethers should
no longer be strong and numerous for suspension, but only serve to keep the
coupons attached to the source wafer during the release step. The tether
design therefore becomes less critical and a small number suffices.

A further optimization is done by choosing the correct pillar configuration.
Due to the sparse layout of typical photonic circuits, pillars and
corresponding target recesses can easily be placed around waveguides without
putting restrictions on circuit design. Nonetheless, it is important that the
pillar spacing is not too large, to avoid collapse between the pillars due to
capillary forces, as shown in Fig.4d and 4e. Depending on the layer stack and
the pillar height, which can be controlled through the partial etch of the
release layer, the maximal spacing varies.

A last optimization is found in the pillar definition itself. Although the
lithium niobate is no longer in contact with the substrate after release, the
pillars themselves are. To guarantee a successful pick-up, the adhesion
between pillars and substrate can be minimized by increasing the roughness on
top of the pillars. This in itself is determined by the roughness of the mold,
i.e. by the partial etch of the release layer in Fig.3.3. For the HF wet etch,
the roughness can be worsened by increasing the HF concentration and therefore
the etch rate. An atomic force microscopy (AFM) measurement on the bottom side
of the coupons allows us to image a single pillar, shown in Fig.4f. The
resulting rms roughness on top of the pillar is measured to be 24 nm (Fig.4g),
which guarantees a bad adhesion with the source substrate.

## 5\. Micro-transfer printing of lithium niobate

Figure 5: Pillar-based $\upmu$TP of LN: (a) Optical microscope image of
suspended 60 $\upmu$m wide coupon, (b) Optical microscope image of suspended
120 $\upmu$m wide coupon, (c) AFM measurement of bottom side of a coupon, (d)
AFM roughness measurement on bottom side of a coupon, (e) Optical microscope
image of a prepared target with pillar recesses, (f) Optical microscope image
of a printed LN coupon on the target.

Having optimized the method, we now demonstrate the micro-transfer printing of
300 nm LN thin films. Fig.5a shows a 60 $\upmu$m x 1 mm suspended LN coupon on
the source chip, ready to be picked. The pillars are configured in a
triangular lattice with a spacing of 20 $\upmu$m, where horizontal waveguides
can be hosted in between pillars. The green color originates from thin film
interference in the air below the coupon, confirming that the lithium niobate
is in fact suspended. To demonstrate the area scalability of the method, the
same source chip contains 120 $\upmu$m wide coupons, shown in Fig.5b with the
blue color indicating its suspension. This confirms that by adding pillars
while maintaining the same pillar spacing, larger coupons are readily
suspended and no added or thicker support layers are required. While
demonstrated with 120 $\upmu$m wide coupons, the suspended area can be as
large as required for high-Q ring or racetrack resonators with sufficiently
large bending radii. An AFM measurement reveals the triangular grid of pillars
underneath the coupon (Fig.5c). Moreover, the bottom side of the LN coupon is
measured to give an rms roughness of 0.22 nm (Fig.5d), ideal for direct
bonding and low propagation losses. The target chip is prepared by etching
pillar recesses deeper than the pillars are high to ensure contact between the
coupon and the chip (Fig.5e). During printing, the pillars are placed within
the recesses and LN is directly bonded to the target. Fig.5f shows a printed
coupon, where the uniform purple color indicates a good and uniform direct
bonding between coupon and substrate. In a last step, the photoresist is
removed, leaving the bonded lithium niobate thin film.

Figure 6: $\upmu$TP of LN with a different pillar design: (a) Optical
microscope image of suspended LN coupon with side-pillars, (b) Cross section
sketch of the LN coupon with side-pillars, (c) Optical microscope image of SiN
waveguide surrounded by side-pillar recesses, (d) Optical microscope image of
printed LN coupon, (e) False-color SEM cross section of the hybrid LN/SiN
waveguide, (f) Optical microscope image of 4 out of 25 subsequently printed LN
coupons on SiN waveguides.

Until now, we relied on a grid of separated pillar-like supports to suspend
the coupon. However, the support configuration is defined through optical
lithography, which suggests that any kind of support layout can be employed,
as long as the maximal spacing is maintained to avoid collapse. For instance,
in case the coupon serves to only cover a straight waveguide, long connected
pillars along the side of the coupon instead of a grid of separated pillars
can be used. Such a suspended coupon is seen in Fig.6a, with a sketch of its
cross section in Fig.6b. This design freedom allows the pillar configuration
to be adapted to the photonic circuit and can be used for stress engineering
within the coupon to avoid possible cracks originating from capillary forces
during the release etch. With the current pillar design, two long recesses are
etched next to a SiN waveguide, as shown in Fig.6c. Again, the coupon is
printed while placing these side-pillars in the corresponding recesses. After
removal of the photoresist, the LN coupon is uniformly bonded to the SiN
(Fig.6d), creating a hybrid LN/SiN waveguide. A SEM image of a corresponding
cross section is given in Fig.6e, with platinum deposited on top for FIB
milling. These resulting hybrid waveguides were used to fabricate high-speed
modulators with beyond 50 GHz bandwidth [22], and in combination with periodic
poling, achieve high-efficiency second harmonic generation with a normalized
conversion efficiency of 2500 %/Wcm2 [23], attesting to their good optical
quality. Lastly, to give an estimate of the transfer yield, 25 LN coupons were
micro-transfer printed subsequently on top of SiN waveguides without fail, of
which four are shown in Fig.6f. Optical microscope images of all 25 coupons
are found in the supplementary material.

Figure 7: $\upmu$TP of gallium phosphide and silicon: optical microscope image
of (a) suspended 60 $\upmu$m wide GaP coupon, (b) suspended 120 $\upmu$m wide
GaP coupon, (c) suspended 60 $\upmu$m wide Si coupon, and (d) suspended 120
$\upmu$m wide Si coupon.

## 6\. Micro-transfer printing of gallium phosphide and silicon

To demonstrate the versatility of the method, we apply it on other thin-film
materials than lithium niobate and fabricate suspended coupons. A first
interesting material consists of gallium phosphide, as it possesses a strong
$\chi^{(2)}$ and $\chi^{(3)}$ response [24]. This makes heterogeneous
integration onto CMOS-compatible platforms interesting for the same reasons as
integrating lithium niobate [25, 26]. Starting from a layer stack of 300 nm
GaP/1 $\upmu$m AlGaP/GaP, the same steps are followed as depicted in Fig.3.
The intermediate AlGaP layer serves as release layer, as the high aluminium
content allows for selective etching compared to the GaP device layer with an
HF-based etch. The GaP is patterned into coupons, after which the pillar molds
are etched and filled with photoresist. Next, the release layer is etched,
resulting in suspended GaP thin films. A 60 $\upmu$m x 1 mm and 120 $\upmu$m x
1 mm sized coupon are shown in Fig.7a and Fig.7b, respectively. While the
focus of this work is integration onto CMOS-compatible platforms, integration
of single-crystal silicon onto other platforms is also interesting e.g. as
intermediate coupling layer on a silicon nitride [27, 28] or lithium-niobate-
on-insulator (LNOI) platform [29, 30]. Starting from a silicon-on-insulator
(SOI) chip with a layer stack of 220 nm Si/2 $\upmu$m SiO2/Si, the same
procedure is followed. The oxide layer serves as release layer, which can
again be selectively etched with an HF-based etch. The resulting suspended
coupons are shown in Fig.7c and Fig.7d, with a 60 $\upmu$m x 1 mm and 120
$\upmu$m x 1 mm coupon size, respectively. Thus, the same procedure can be
followed to readily suspend large areas of both GaP and Si thin films, showing
the versatility of the method.

## 7\. Conclusion

Micro-transfer printing of thin-film lithium niobate is a promising approach
to heterogeneously integrate functionalities that mature CMOS-compatible
platforms lack, such as high-speed Pockels modulation and $\chi^{(2)}$
nonlinearities. However, suspending large areas of thin films is challenging,
resulting in a low transfer yield. We have developed a new source preparation
method to reliably micro-transfer print thin films based on pillar-like
support structures. We discussed the optimization of the method by introducing
crack barriers in the tether design, controlling the pillar spacing and
engineering the pillar roughness. This enables the micro-transfer printing of
coupons with varying surface areas, without any additional process
development. Furthermore, different support configurations are possible, where
pillars along the sides of the coupon were shown for integration onto straight
waveguides. An estimate of the transfer yield was given by subsequently
printing 25 LN coupons onto SiN waveguides without fail. Moreover, the
versatility of the method was demonstrated by fabricating suspended coupons
for both gallium phosphide and silicon thin films. Importantly, this method is
complementary to existing LNOI processing. The suspended coupons can contain
etched LN waveguides and electrodes, forming optical components that as a
whole can be integrated onto a CMOS-compatible platform. Accordingly, this
enables the co-integration of LNOI-based high-speed modulators [22, 31] with
integrated lasers [32] and ultra-fast detectors [33] on the same chip. By
periodically poling the lithium niobate, also $\chi^{(2)}$ processes like
second harmonic generation are introduced [23, 34, 35]. As such, micro-
transfer printing of thin films brings us one step closer to build versatile
photonic integrated circuits on CMOS-compatible platforms.

Funding. Research Foundation Flanders (11H6723N, 11F5320N, 11F8122N,
1S69123N), European Research Council (No.759483), Fonds de la Recherche
Scientifique (MIS F.4506.20).

Acknowledgments. Tom Vandekerckhove, Tom Vanackere, Stijn Cuyvers and Jasper
De Witte are PhD fellows of the Research Foundation Flanders (FWO). Stéphane
Clemmen is a research associate of the Fonds de la Recherche Scientifique
(FNRS). We also thank the European Research Council (ERC) for the funding in
the context of the ELECTRIC project.

Supplemental document. See Supplement 1 for supporting content.

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