# This is the title

FEICHTINGER CONJECTURES, $R_{\varepsilon}$-CONJECTURE AND WEAVER’S CONJECTURES
FOR BANACH SPACES
K. MAHESH KRISHNA

Post Doctoral Fellow

Statistics and Mathematics Unit

Indian Statistical Institute, Bangalore Centre

Karnataka 560 059 India

Email<EMAIL_ADDRESS>
Date:

Abstract: Motivated from two decades old famous Feichtinger conjectures for
frames, $R_{\varepsilon}$-conjecture and Weaver’s conjecture for Hilbert
spaces (and their solution by Marcus, Spielman, and Srivastava), we formulate
Feichtinger conjectures for p-approximate Schauder frames,
$R_{\varepsilon}$-conjecture, Weaver’s conjectures and Akemann-Weaver
conjectures for Banach spaces. We also formulate conjectures on p-approximate
Schauder frames based on the results of Casazza for frames for Hilbert spaces.
We state conjectures and problems for p-approximate Schauder frames based on
fundamental inequality for frames for Hilbert spaces and scaling problem for
Hilbert space frames. Based on Kothe-Lorch theorem for Riesz bases for Hilbert
spaces, we formulate a problem for p-approximate Riesz bases for Banach
spaces. We formulate dynamical sampling problem for p-approximate Schauder
frames for Banach spaces. We ask phase retrieval problem and norm retrieval
problem for p-approximate Schauder frames for Banach spaces. We also formulate
discretization problem for continuous p-approximate Schauder frames.

Keywords: Feichtinger conjectures, p-approximate Schauder Frame, p-approximate
Riesz basis, p-approximate Bessel sequence, p-orthonormal basis.

Mathematics Subject Classification (2020): 42C15, 11K38, 46B45, 46B04, 94A20,
46L05, 14Q15.

###### Contents

  1. 1 Introduction
  2. 2 Feichtinger conjectures, $R_{\varepsilon}$-conjecture, Weaver’s conjectures, scaling problem, dynamical sampling problems, phase and norm retrieval problems and discretization problem for p-approximate Schauder frames for Banach spaces

##  1\. Introduction

Let $\mathcal{H}$ be a Hilbert space. Recall that a sequence
$\\{\tau_{n}\\}_{n}$ in $\mathcal{H}$ is said to be a frame for $\mathcal{H}$
[47, 57, 74] if there exist $a,b>0$ such that

(1) $\displaystyle a\|h\|^{2}\leq\sum_{n=1}^{\infty}|\langle
h,\tau_{n}\rangle|^{2}\leq b\|h\|^{2},\quad\forall h\in\mathcal{H}.$

Constants $a$ and $b$ are called as lower frame bound and upper frame bound,
respectively. If

$\displaystyle\sum_{n=1}^{\infty}|\langle
h,\tau_{n}\rangle|^{2}=\|h\|^{2},\quad\forall h\in\mathcal{H},$

then we say $\\{\tau_{n}\\}_{n}$ is a Parseval frame for $\mathcal{H}$. If we
do not demand the first inequality in Inequality (1), then we say that
$\\{\tau_{n}\\}_{n}$ is a Bessel sequence for $\mathcal{H}$. Constant $b$ is
called as Bessel bound. Notion which is stronger than that of frames is that
of Riesz bases defined as follows.

######  Definition 1.1.

[14, 47, 13]

  1. (i)

A sequence $\\{\tau_{n}\\}_{n}$ in $\mathcal{H}$ is said to be a Riesz basis
for $\mathcal{H}$ if there exists a bounded invertible linear operator
$T:\mathcal{H}\to\mathcal{H}$ such that

$\displaystyle T\omega_{n}=\tau_{n},\quad\forall n\in\mathbb{N},$

where $\\{\omega_{n}\\}_{n}$ is an orthonormal basis for $\mathcal{H}$.

  2. (ii)

A sequence $\\{\tau_{n}\\}_{n}$ in $\mathcal{H}$ is said to be a Riesz
sequence for $\mathcal{H}$ if it is a Riesz basis for
$\overline{span}\\{\tau_{n}\\}_{n}$. Constants $a,b>0$ satisfying for all
$m\in\mathbb{N}$,

$\displaystyle
a\left(\sum_{n=1}^{m}|c_{n}|^{2}\right)^{\frac{1}{2}}\leq\left\|\sum_{n=1}^{m}c_{n}\tau_{n}\right\|\leq
b\left(\sum_{n=1}^{m}|c_{n}|^{2}\right)^{\frac{1}{2}}\quad,\forall
c_{1},\dots,c_{m}\in\mathbb{R}\text{ or }\mathbb{C}$

are called as lower Riesz bound and upper Riesz bound, respectively.

Towards the end of $20^{th}$ century, works of Casazza, Christensen, and
Lindner [30, 46, 49, 32, 28] showed that there are frames which do not contain
Schauder basis, in particular Riesz basis. On the other hand, work on Gabor
frames led Feichtinger to formulate the following conjectures in the beginning
years of $21^{th}$ century (see [48] for the history).

######  Conjecture 1.2.

[34](Feichtinger conjecture for frames) Let $\\{\tau_{n}\\}_{n}$ be a frame
for $\mathcal{H}$ such that

$\displaystyle 0<\inf_{n\in\mathbb{N}}\|\tau_{n}\|.$

Then $\\{\tau_{n}\\}_{n}$ can be partitioned into a finite union of Riesz
sequences. Moreover, what is the number of partitions required?

######  Conjecture 1.3.

[34](Feichtinger conjecture for Bessel sequences) Let $\\{\tau_{n}\\}_{n}$ be
a Bessel sequence for $\mathcal{H}$ such that

$\displaystyle 0<\inf_{n\in\mathbb{N}}\|\tau_{n}\|.$

Then $\\{\tau_{n}\\}_{n}$ can be partitioned into a finite union of Riesz
sequences. Moreover, what is the number of partitions required?

######  Conjecture 1.4.

[34](Finite dimensional Feichtinger conjecture for frames) Let $\mathcal{H}$
be a $d$-dimensional Hilbert space. For every real $b,c>0$, there exist a
natural number $M(b,c)$, a real $a(b,c)>0$ so that whenever
$\\{\tau_{j}\\}_{j=1}^{n}$ is a frame for $\mathcal{H}$ with upper frame bound
$b$ and $\|\tau_{j}\|\geq c$, $\forall 1\leq j\leq n$, then the set
$\\{1,2,\dots,n\\}$ can be partitioned into sets
$I_{1},I_{2},\dots,I_{M(b,c)}$ so that for each $1\leq k\leq M(b,c)$,
$\\{\tau_{j}\\}_{j\in I_{k}}$ is a Riesz sequence with lower Riesz bound
$a(b,c)$ and upper Riesz bound $b$. Moreover, what is the number of partitions
required?

######  Conjecture 1.5.

[34](Finite dimensional Feichtinger conjecture for Bessel sequences) Let
$\mathcal{H}$ be a $d$-dimensional Hilbert space. For every $b>0$, there
exists a natural number $M(b)$ and a real $a(b)>0$ so that for every (Bessel)
sequence $\\{\tau_{j}\\}_{j=1}^{n}$ for $\mathcal{H}$ with Bessel bound $b$
and $\|\tau_{j}\|=1$, $\forall 1\leq j\leq n$, can be written as a union of
$M(b)$ Riesz sequences each with lower Riesz bound $a(b)$. Moreover, what is
the number of partitions required?

After the formulation of Conjectures 1.2, 1.3, 1.4 and 1.5, several of their
equivalent conjectures were found and several particular cases of Conjectures
1.2, 1.3, 1.4 and 1.5, have been solved [10, 40, 98, 112, 87, 20, 86, 34, 65,
12, 21, 69]. Conjectures 1.2, 1.3, 1.4 and 1.5 received a lot of importance
after establishing their equivalence with Kadison-Singer conjectures [76, 39,
44, 26, 37, 35]. Finally, the Feichtinger conjectures have been solved fully
resolving Weaver’s conjecture by Marcus, Spielman, and Srivastava in 2013
using an entirely new method called “mixed characteristic polynomials” by them
[95, 19, 43, 96, 108, 94]. Later, Conjecture 1.2 has been set for tight
p-frames for $\ell^{p}(\mathbb{N})$ in [93] (see [50, 102, 101, 31, 25, 5] for
p-frames) and solved by using Clarkson’s inequalities [51] in the same paper.
In this paper, we formulate Conjectures 1.2, 1.3, 1.4 and 1.5 1.2 for
p-approximate Schauder frames for Banach spaces. We also formulate
$R_{\varepsilon}$-conjecture, Weaver’s conjectures and Akemann-Weaver
conjectures for Banach spaces. Three conjectures based on results of Casazza
are formulated. We state scaling problem, fundamental inequality problem,
Kothe-Lorch problem for p-approximate Schauder frames for Banach spaces. We
formulate dynamical sampling problem and phase and norm retrieval problems for
p-approximate Schauder frames. We also state some other problems including
discretization problem for continuous p-approximate Schauder frames for Banach
spaces.

##  2\. Feichtinger conjectures, $R_{\varepsilon}$-conjecture, Weaver’s
conjectures, scaling problem, dynamical sampling problems, phase and norm
retrieval problems and discretization problem for p-approximate Schauder
frames for Banach spaces

Let $p\in[1,\infty)$. Let $\\{e_{n}\\}_{n}$ be the standard Schauder basis for
$\ell^{p}(\mathbb{N})$ and $\\{\zeta_{n}\\}_{n}$ be the co ordinate
functionals associated with $\\{e_{n}\\}_{n}$. Throughout the paper,
$\mathcal{X}$ denotes a Banach space. In the process of characterization of
approximate Schauder frames and its duals (like that of Holub [75] and Li
[91]) for Banach spaces [62, 107, 24, 38], the notion of p-approximate
Schauder frames has been introduced by Krishna and Johnson whose detailed
study has been done in the thesis of the author [80] which reads as follows.

######  Definition 2.1.

[84, 82] Let $p\in[1,\infty)$. Let $\\{\tau_{n}\\}_{n}$ be a sequence in a
Banach space $\mathcal{X}$ and $\\{f_{n}\\}_{n}$ be a sequence in
$\mathcal{X}^{*}$ (dual of $\mathcal{X}$). The pair
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is said to be a p-approximate Schauder
frame (p-ASF) for $\mathcal{X}$ if the following conditions are satisfied.

  1. (i)

The map (frame operator)

$\displaystyle S_{f,\tau}:\mathcal{X}\ni x\mapsto
S_{f,\tau}x\coloneqq\sum_{n=1}^{\infty}f_{n}(x)\tau_{n}\in\mathcal{X}$

is a well-defined bounded linear, invertible operator.

  2. (ii)

The map (analysis operator)

$\displaystyle\theta_{f}:\mathcal{X}\ni
x\mapsto\theta_{f}x\coloneqq\\{f_{n}(x)\\}_{n}\in\ell^{p}(\mathbb{N})$

is a well-defined bounded linear operator.

  3. (iii)

The map (synthesis operator)

$\displaystyle\theta_{\tau}:\ell^{p}(\mathbb{N})\ni\\{a_{n}\\}_{n}\mapsto\theta_{\tau}\\{a_{n}\\}_{n}\coloneqq\sum_{n=1}^{\infty}a_{n}\tau_{n}\in\mathcal{X}$

is a well-defined bounded linear operator.

Constants $a>0$ and $b>0$ satisfying

$\displaystyle a\|x\|\leq\|S_{f,\tau}x\|\leq b\|x\|,\quad\forall
x\in\mathcal{X},$

are called as lower frame bound and upper frame bound, respectively. If
$S_{f,\tau}x=x$, $\forall x\in\mathcal{X}$, then we say that
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is a Parseval p-ASF for $\mathcal{X}$.
If $S_{f,\tau}x=\lambda x$, for some nonzero scalar $\lambda$, $\forall
x\in\mathcal{X}$, then we say that $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is a
tight p-ASF for $\mathcal{X}$. If we do not demand condition (i), then we say
that $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is a p-approximate Bessel sequence
(p-ABS) for $\mathcal{X}$. Constant $b$ is called as Bessel bound.

Now we define the notion of p-orthonormal sequence and basis for Banach
spaces. Our basic motivation is the notion of orthonormal basis for Hilbert
spaces and the standard canonical Schauder basis for $\ell^{p}(\mathbb{N})$.

######  Definition 2.2.

Let $p\in[1,\infty)$. Let $\mathbb{M}\subseteq\mathbb{N}$. Let
$\\{\tau_{n}\\}_{n\in\mathbb{M}}$ be a sequence in $\mathcal{X}$ and
$\\{f_{n}\\}_{n\in\mathbb{M}}$ be a sequence in $\mathcal{X}^{*}$. The pair
$(\\{f_{n}\\}_{n\in\mathbb{M}},\\{\tau_{n}\\}_{n\in\mathbb{M}})$ is said to be
a p-orthonormal sequence for $\mathcal{X}$ if the following conditions are
satisfied.

  1. (i)

$f_{n}(\tau_{m})=\delta_{n,m}$, $\forall n,m\in\mathbb{M}$.

  2. (ii)

For each $x\in\mathcal{X}$,

$\displaystyle\|x\|^{p}\geq\sum_{n\in\mathbb{M}}|f_{n}(x)|^{p}.$

  3. (iii)

For each $\\{a_{n}\\}_{n\in\mathbb{M}}\in\ell^{p}(\mathbb{M})$,

$\displaystyle\left\|\sum_{{n\in\mathbb{M}}}a_{n}\tau_{n}\right\|^{p}=\sum_{{n\in\mathbb{M}}}|a_{n}|^{p}.$

######  Definition 2.3.

Let $p\in[1,\infty)$. Let $\\{\tau_{n}\\}_{n}$ be a sequence in $\mathcal{X}$
and $\\{f_{n}\\}_{n}$ be a sequence in $\mathcal{X}^{*}$. The pair
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is said to be a p-orthonormal basis for
$\mathcal{X}$ if the following conditions hold.

  1. (i)

$\\{\tau_{n}\\}_{n}$ is a Schauder basis for $\mathcal{X}$.

  2. (ii)

$f_{n}(\tau_{m})=\delta_{n,m}$, $\forall n,m\in\mathbb{N}$, i.e.,
$\\{f_{n}\\}_{n}$ is the coordinate functionals associated with
$\\{\tau_{n}\\}_{n}$

  3. (iii)

For each $x\in\mathcal{X}$,

$\displaystyle\|x\|^{p}=\sum_{n=1}^{\infty}|f_{n}(x)|^{p}.$

  4. (iv)

For each $\\{a_{n}\\}_{n}\in\ell^{p}(\mathbb{N})$,

$\displaystyle\left\|\sum_{n=1}^{\infty}a_{n}\tau_{n}\right\|^{p}=\sum_{n=1}^{\infty}|a_{n}|^{p}.$

It is clear that a p-orthonormal basis is a p-orthonormal sequence and every
subset of p-orthonormal basis is a p-orthonormal sequence. Also note that
condition (iv) in Definition 2.3 says that if
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is a p-orthonormal basis for
$\mathcal{X}$, then $\\{\tau_{n}\\}_{n}$ is an unconditional Schauder basis
for $\mathcal{X}$. Definition 2.3 give the following observations:

$\displaystyle\|\tau_{n}\|=1=\|f_{n}\|,\quad\forall n\in\mathbb{N},$
$\displaystyle\|\tau_{j_{1}}+\cdots+\tau_{j_{n}}\|=n^{\frac{1}{p}},\quad\forall
j_{1},\dots,j_{n}.$

######  Remark 2.4.

Let $\mathcal{H}$ be a Hilbert space and $\\{\tau_{n}\\}_{n}$ be an
orthonormal basis for $\mathcal{H}$. Define $f_{n}:\mathcal{H}\ni
h\mapsto\langle h,\tau_{n}\rangle\in\mathbb{K}$, $\forall n$. Then
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is a 2-orthonormal basis for
$\mathcal{X}$.

######  Example 2.5.

Standard Schauder basis and its co ordinate functionals is a p-orthonormal
basis for $\ell^{p}(\mathbb{N})$, for each $p\in[1,\infty)$.

Gram-Schmidt orthonormalization converts every linearly independent set in a
Hilbert space into an orthonormal set and hence to an orthonormal basis [88,
55]. On the other hand, there is a Gram-Schmidt orthonormalization in Banach
spaces due to Lin [92]. We ask the following open problem.

######  Problem 2.6.

Let $\\{\tau_{j}\\}_{j=1}^{n}$ be a linearly independent collection in
$\mathcal{X}$ and $\\{f_{j}\\}_{j=1}^{n}$ be a linearly independent collection
in $\mathcal{X}^{*}$. Whether there is a way to convert
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ into a p-orthonormal
sequence, say $(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})$ such that

$\displaystyle\operatorname{span}\\{f_{1},\dots,f_{j}\\}=\operatorname{span}\\{g_{1},\dots,g_{j}\\}~{}\text{
and/or
}\operatorname{span}\\{\tau_{1},\dots,\tau_{j}\\}=\operatorname{span}\\{\omega_{1},\dots,\omega_{j}\\},\quad\forall
1\leq j\leq n.$

In particular, whether we can convert a
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ into a p-orthonormal basis?

Like orthonormal basis for Hilbert spaces, we can characterize all
p-orthonormal bases for Banach spaces.

######  Theorem 2.7.

Let $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ be a p-orthonormal basis for
$\mathcal{X}$. Then all p-orthonormal bases for $\mathcal{X}$ are precisely
the family $(\\{f_{n}V^{-1}\\}_{n},\\{V\tau_{n}\\}_{n})$, where
$V:\mathcal{X}\to\mathcal{X}$ is an invertible isometry.

###### Proof.

(i) $\Rightarrow$ (ii) Let $(\\{g_{n}\\}_{n},\\{\omega_{n}\\}_{n})$ be a
p-orthonormal basis for $\mathcal{X}$. Define

$\displaystyle
V:\mathcal{X}\ni\sum_{n=1}^{\infty}f_{n}(x)\tau_{n}\mapsto\sum_{n=1}^{\infty}f_{n}(x)\omega_{n}\in\mathcal{X}$

and

$\displaystyle
U:\mathcal{X}\ni\sum_{n=1}^{\infty}g_{n}(x)\omega_{n}\mapsto\sum_{n=1}^{\infty}g_{n}(x)\tau_{n}\in\mathcal{X}.$

Then $U$ and $V$ are well-defined and

$\displaystyle
UVx=U\left(\sum_{n=1}^{\infty}f_{n}(x)\omega_{n}\right)=\sum_{n=1}^{\infty}f_{n}(x)U\omega_{n}=\sum_{n=1}^{\infty}f_{n}(x)\tau_{n}=x,\quad\forall
x\in\mathcal{X},$ $\displaystyle
VUx=V\left(\sum_{n=1}^{\infty}g_{n}(x)\tau_{n}\right)=\sum_{n=1}^{\infty}g_{n}(x)V\tau_{n}=\sum_{n=1}^{\infty}g_{n}(x)\omega_{n}=x,\quad\forall
x\in\mathcal{X}.$

Hence $V$ is invertible. Now

$\displaystyle\|Vx\|=\left\|\sum_{n=1}^{\infty}f_{n}(x)\omega_{n}\right\|=\left(\sum_{n=1}^{\infty}|f_{n}(x)|^{p}\right)^{\frac{1}{p}}=\|x\|,\quad\forall
x\in\mathcal{X}.$

Hence $V$ is isometry. We now have $V\tau_{n}=\omega_{n}$, $\forall
n\in\mathbb{N}$, and

$\displaystyle
f_{n}(V^{-1}x)=f_{n}(Ux)=f_{n}\left(\sum_{m=1}^{\infty}g_{m}(x)\tau_{m}\right)=\sum_{m=1}^{\infty}g_{m}(x)f_{n}(\tau_{m})=g_{n}(x),\quad\forall
n\in\mathbb{N},\forall x\in\mathcal{X}.$

(ii) $\Rightarrow$ (i) Define $g_{n}\coloneqq f_{n}V^{-1},\omega_{n}\coloneqq
V\tau_{n}$, $\forall n\in\mathbb{N}$. We claim that
$(\\{g_{n}\\}_{n},\\{\omega_{n}\\}_{n})$ is a p-orthonormal basis for
$\mathcal{X}$. Since $V$ is invertible, $\\{\omega_{n}\\}_{n}$ is a Schauder
basis for $\mathcal{X}$. Now we verify remaining three conditions:

$\displaystyle
g_{n}(\omega_{m})=f_{n}(V^{-1}V\tau_{m})=f_{n}(\tau_{m})=\delta_{n,m},\quad\forall
n,m\in\mathbb{N},$
$\displaystyle\sum_{n=1}^{\infty}|g_{n}(x)|^{p}=\sum_{n=1}^{\infty}|f_{n}(V^{-1}x)|^{p}=\|V^{-1}x\|^{p}=\|x\|^{p},\quad\forall
x\in\mathcal{X},$
$\displaystyle\left\|\sum_{n=1}^{\infty}a_{n}\omega_{n}\right\|^{p}$
$\displaystyle=\left\|\sum_{n=1}^{\infty}a_{n}V\tau_{n}\right\|^{p}=\left\|V\left(\sum_{n=1}^{\infty}a_{n}\tau_{n}\right)\right\|^{p}$
$\displaystyle=\left\|\sum_{n=1}^{\infty}a_{n}\tau_{n}\right\|^{p}=\sum_{n=1}^{\infty}|a_{n}|^{p},\quad\\{a_{n}\\}_{n}\in\ell^{p}(\mathbb{N}).$

∎

######  Theorem 2.8.

If $\mathcal{X}$ admits a p-orthonormal basis, then $\mathcal{X}$ is
isometrically isomorphic to $\ell^{p}(\mathbb{N})$.

###### Proof.

Let $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ be a p-orthonormal basis for
$\mathcal{X}$. Define

$\displaystyle
V:\mathcal{X}\ni\sum_{n=1}^{\infty}f_{n}(x)\tau_{n}\mapsto\sum_{n=1}^{\infty}f_{n}(x)e_{n}\in\ell^{p}(\mathbb{N})$

and

$\displaystyle
U:\ell^{p}(\mathbb{N})\ni\sum_{n=1}^{\infty}\zeta_{n}(x)e_{n}\mapsto\sum_{n=1}^{\infty}\zeta_{n}(x)\tau_{n}\in\mathcal{X}.$

Other parts are similar to the proof of Theorem 2.7. ∎

######  Theorem 2.9.

If a Banach space $\mathcal{X}$ is isometrically isomorphic to
$\ell^{p}(\mathbb{N})$, then it admits a p-orthonormal basis.

###### Proof.

Let $V:\mathcal{X}\to\ell^{p}(\mathbb{N})$ be an isometric isomorphism. Define
$f_{n}\coloneqq\zeta_{n}V,\tau_{n}\coloneqq V^{-1}e_{n}$, $\forall
n\in\mathbb{N}$. Then $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is a
p-orthonormal basis for $\mathcal{X}$. ∎

Now we want to define the notion of Riesz basis in accordance with Definition
1.1. We note that Riesz basis notion has been defined for Banach spaces using
a single sequence in [50, 5] which we do not consider in this paper. Our
motivation comes from the definition given in [81].

######  Definition 2.10.

Let $p\in[1,\infty)$. Let $\\{\tau_{n}\\}_{n}$ be a sequence in $\mathcal{X}$
and $\\{f_{n}\\}_{n}$ be a sequence in $\mathcal{X}^{*}$. The pair
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is said to be a p-approximate Riesz
basis for $\mathcal{X}$ if there exist bounded linear invertible operators
$U,V:\mathcal{X}\to\mathcal{X}$ such that

$\displaystyle f_{n}=g_{n}U,~{}\tau_{n}=V\omega_{n},\quad\forall
n\in\mathbb{N},$

where $(\\{g_{n}\\}_{n},\\{\omega_{n}\\}_{n})$ is a p-orthonormal basis for
$\mathcal{X}$.

######  Definition 2.11.

Let $p\in[1,\infty)$. Let $\\{\tau_{n}\\}_{n}$ be a sequence in $\mathcal{X}$
and $\\{f_{n}\\}_{n}$ be a sequence in $\mathcal{X}^{*}$. The pair
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is said to be a p-approximate Riesz
sequence for $\mathcal{X}$ if $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is a
p-approximate Riesz basis for $\overline{span}\\{\tau_{n}\\}_{n}$. Constants
$a,b>0$ satisfying for all $m\in\mathbb{N}$,

$\displaystyle
a\left(\sum_{n=1}^{m}|c_{n}|^{p}\right)^{\frac{1}{p}}\leq\left\|\sum_{n=1}^{m}c_{n}\tau_{n}\right\|\leq
b\left(\sum_{n=1}^{m}|c_{n}|^{p}\right)^{\frac{1}{p}},\quad\forall
c_{1},\dots,c_{m}\in\mathbb{K}$

are called as lower p-approximate Riesz bound and upper p-approximate Riesz
bound, respectively. Note that they exist. We say that
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is unit norm if
$\|f_{n}\|=\|\tau_{n}\|=|f_{n}(\tau_{n})|=1$, $\forall n\in\mathbb{N}$.

Note that Definition 2.10 of p-approximate Riesz basis is not the same as the
definition of p-approximate Riesz basis given in [81]. The following theorem
says that they are equivalent. In the paper, given a space $\mathcal{X}$, by
$I_{\mathcal{X}}$ we mean the identity operator on $\mathcal{X}$.

######  Theorem 2.12.

The following are equivalent.

  1. (i)

The pair $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is a p-approximate Riesz basis
for $\mathcal{X}$.

  2. (ii)

The pair $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is a p-ASF for $\mathcal{X}$
and

(2)
$\displaystyle\theta_{f}S_{f,\tau}^{-1}\theta_{\tau}=I_{\ell^{p}(\mathbb{N})}.$

###### Proof.

  1. (i)

$\Rightarrow$ (ii) Let $U,V:\mathcal{X}\to\mathcal{X}$ be bounded linear
invertible operators such that $f_{n}=g_{n}U,~{}\tau_{n}=V\omega_{n},\forall
n\in\mathbb{N}$, where $(\\{g_{n}\\}_{n},\\{\omega_{n}\\}_{n})$ is a
p-orthonormal basis for $\mathcal{X}$. We then have $\theta_{f}=\theta_{g}U$
and $\theta_{\tau}=V\theta_{\omega}$. Hence
$S_{f,\tau}=\theta_{\tau}\theta_{f}=V\theta_{\omega}\theta_{g}U=VU$ which is
invertible. Therefore $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is a p-ASF for
$\mathcal{X}$. We now find

$\displaystyle\theta_{f}S_{f,\tau}^{-1}\theta_{\tau}\\{a_{n}\\}_{n}$
$\displaystyle=\theta_{g}U(VU)^{-1}V\theta_{\omega}\\{a_{n}\\}_{n}=\theta_{g}UU^{-1}V^{-1}V\theta_{\omega}\\{a_{n}\\}_{n}$
$\displaystyle=\theta_{g}\theta_{\omega}\\{a_{n}\\}_{n}=\theta_{g}\left(\sum_{n=1}^{\infty}a_{n}\omega_{n}\right)=\sum_{n=1}^{\infty}a_{n}\theta_{g}\omega_{n}$
$\displaystyle=\sum_{n=1}^{\infty}a_{n}\sum_{m=1}^{\infty}g_{m}(\omega_{n})e_{m}=\sum_{n=1}^{\infty}a_{n}e_{n}=\\{a_{n}\\}_{n},\quad\forall\\{a_{n}\\}_{n}\in\ell^{p}(\mathbb{N}).$

  2. (ii)

$\Rightarrow$ (i) Let $(\\{g_{n}\\}_{n},\\{\omega_{n}\\}_{n})$ be a
p-orthonormal basis for $\mathcal{X}$. Define
$U\coloneqq\theta_{\omega}\theta_{f}$ and $V\coloneqq\theta_{\tau}\theta_{g}$.
Then

$\displaystyle g_{n}(Ux)$
$\displaystyle=g_{n}(\theta_{\omega}\theta_{f}x)=g_{n}\left(\theta_{\omega}\left(\sum_{m=1}^{\infty}f_{m}(x)e_{m}\right)\right)$
$\displaystyle=g_{n}\left(\sum_{m=1}^{\infty}f_{m}(x)\theta_{\omega}e_{m}\right)=g_{n}\left(\sum_{m=1}^{\infty}f_{m}(x)\omega_{m}\right)$
$\displaystyle=\sum_{m=1}^{\infty}f_{m}(x)g_{n}(\omega_{m})=f_{n}(x),\quad\forall
x\in\mathcal{X},$ $\displaystyle
V\omega_{n}=\theta_{\tau}\theta_{g}\omega_{n}=\theta_{\tau}\left(\sum_{m=1}^{\infty}g_{m}(\omega_{n})e_{m}\right)=\theta_{\tau}e_{n}=\tau_{n},\quad\forall
n\in\mathbb{N}.$

To complete the proof, we have to show that both $U$ and $V$ are bounded
invertible. First note that

$\displaystyle\theta_{g}\theta_{\omega}\\{a_{n}\\}_{n}$
$\displaystyle=\theta_{g}\left(\sum_{n=1}^{\infty}a_{n}\omega_{n}\right)=\sum_{n=1}^{\infty}a_{n}\theta_{g}\omega_{n}$
$\displaystyle=\sum_{n=1}^{\infty}a_{n}\sum_{m=1}^{\infty}g_{m}(\omega_{n})e_{m}=\sum_{n=1}^{\infty}a_{n}e_{n}$
$\displaystyle=\\{a_{n}\\}_{n},\quad\forall\\{a_{n}\\}_{n}\in\ell^{p}(\mathbb{N}).$

Hence

(3) $\displaystyle\theta_{g}\theta_{\omega}=I_{\ell^{p}(\mathbb{N})}.$

Now using Equation (2) and Equation (3) we find

$\displaystyle
U(S_{f,\tau}^{-1}\theta_{\tau}\theta_{g})=\theta_{\omega}\theta_{f}S_{f,\tau}^{-1}\theta_{\tau}\theta_{g}=\theta_{\omega}I_{\ell^{p}(\mathbb{N})}\theta_{g}=\theta_{\omega}\theta_{g}=I_{\mathcal{X}},$
$\displaystyle(S_{f,\tau}^{-1}\theta_{\tau}\theta_{g})U=S_{f,\tau}^{-1}\theta_{\tau}\theta_{g}\theta_{\omega}\theta_{f}=S_{f,\tau}^{-1}\theta_{\tau}I_{\ell^{p}(\mathbb{N})}\theta_{f}=S_{f,\tau}^{-1}\theta_{\tau}\theta_{f}=I_{\mathcal{X}}$

and

$\displaystyle
V(\theta_{\omega}\theta_{f}S_{f,\tau}^{-1})=\theta_{\tau}\theta_{g}\theta_{\omega}\theta_{f}S_{f,\tau}^{-1}=\theta_{\tau}I_{\ell^{p}(\mathbb{N})}\theta_{f}S_{f,\tau}^{-1}=\theta_{\tau}\theta_{f}S_{f,\tau}^{-1}=I_{\mathcal{X}},$
$\displaystyle(\theta_{\omega}\theta_{f}S_{f,\tau}^{-1})V=\theta_{\omega}\theta_{f}S_{f,\tau}^{-1}\theta_{\tau}\theta_{g}=\theta_{\omega}I_{\ell^{p}(\mathbb{N})}\theta_{g}=\theta_{\omega}\theta_{g}=I_{\mathcal{X}}.$

Hence $U$ and $V$ are invertible.

∎

Now we formulate Feichtinger conjectures for Banach spaces.

######  Conjecture 2.13.

(Feichtinger conjecture for p-approximate Schauder frames) Let
$(\\{f_{n}\\}_{n},$ $\\{\tau_{n}\\}_{n})$ be a p-ASF for $\mathcal{X}$ such
that

$\displaystyle
0<\inf_{n\in\mathbb{N}}\|\tau_{n}\|\leq\sup_{n\in\mathbb{N}}\|\tau_{n}\|<\infty,$
$\displaystyle
0<\inf_{n\in\mathbb{N}}\|f_{n}\|\leq\sup_{n\in\mathbb{N}}\|f_{n}\|<\infty.$

Then $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ can be partitioned into a finite
union of p-approximate Riesz sequences. Moreover, what is the number of
partitions required?

Note that using Feichtinger conjecture (result of Marcus, Spielman, and
Srivastava) for Hilbert spaces we get the following. Let $(\\{f_{n}\\}_{n},$
$\\{\tau_{n}\\}_{n})$ be a 2-ASF for a Hilbert space $\mathcal{H}$ such that

$\displaystyle
0<\inf_{n\in\mathbb{N}}\|\tau_{n}\|\leq\sup_{n\in\mathbb{N}}\|\tau_{n}\|<\infty,$
$\displaystyle
0<\inf_{n\in\mathbb{N}}\|f_{n}\|\leq\sup_{n\in\mathbb{N}}\|f_{n}\|<\infty.$

Riesz representation theorem says we can identify $\\{f_{n}\\}_{n}$ by a
sequence $\\{\omega_{n}\\}_{n}$ in $\mathcal{H}$. Therefore, we have
$\\{\tau_{n}\\}_{n}$ and $\\{\omega_{n}\\}_{n}$ are frames for $\mathcal{H}$.
Feichtinger conjecture for Hilbert spaces says that both $\\{\tau_{n}\\}_{n}$
and $\\{\omega_{n}\\}_{n}$ can be partitioned into a finite union of Riesz
sequences. However, this does not solve Conjecture 2.13 even in this special
case. The reason is that the number of partitions of $\\{\tau_{n}\\}_{n}$ and
$\\{\omega_{n}\\}_{n}$ may be different. We want a partition of the form

$\displaystyle(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})=(\\{f^{(1)}_{n}\\}_{n},\\{\tau^{(1)}_{n}\\}_{n})\cup\cdots\cup(\\{f^{(m)}_{n}\\}_{n},\\{\tau^{(m)}_{n}\\}_{n}),$

where each of
$(\\{f^{(1)}_{n}\\}_{n},\\{\tau^{(1)}_{n}\\}_{n}),\dots,(\\{f^{(m)}_{n}\\}_{n},\\{\tau^{(m)}_{n}\\}_{n})$
is a p-approximate Riesz sequence.

######  Conjecture 2.14.

(Feichtinger conjecture for p-approximate Bessel sequences) Let
$(\\{f_{n}\\}_{n},$ $\\{\tau_{n}\\}_{n})$ be a p-ABS for $\mathcal{X}$ such
that

$\displaystyle
0<\inf_{n\in\mathbb{N}}\|\tau_{n}\|\leq\sup_{n\in\mathbb{N}}\|\tau_{n}\|<\infty,$
$\displaystyle
0<\inf_{n\in\mathbb{N}}\|f_{n}\|\leq\sup_{n\in\mathbb{N}}\|f_{n}\|<\infty.$

Then $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ can be partitioned into a finite
union of p-approximate Riesz sequences. Moreover, what is the number of
partitions required?

######  Conjecture 2.15.

(Finite dimensional Feichtinger conjecture for p-approximate Schauder frames)
Let $\mathcal{X}$ be a $d$-dimensional Banach space. For every real $b,c>0$,
there exist a natural number $M(b,c)$, a real $a(b,c)>0$ so that whenever
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ is a p-ASF for
$\mathcal{X}$ with upper frame bound $b$ and

$\displaystyle\|\tau_{j}\|\geq c\quad\text{ and }\quad\|f_{j}\|\geq
c,\quad\forall 1\leq j\leq n,$

then the set $\\{1,2,\dots,n\\}$ can be partitioned into sets
$I_{1},I_{2},\dots,I_{M(b,c)}$ so that for each $1\leq k\leq M(b,c)$,
$(\\{f_{j}\\}_{j\in I_{k}},\\{\tau_{j}\\}_{j\in I_{k}})$ is a p-approximate
Riesz sequence with lower p-approximate Riesz bound $a(b,c)$ and upper
p-approximate Riesz bound $b$. Moreover, what is the number of partitions
required?

######  Conjecture 2.16.

(Finite dimensional Feichtinger conjecture for p-approximate Bessel sequences)
Let $\mathcal{X}$ be a $d$-dimensional Banach space. For every $b>0$, there
exists a natural number $M(b)$ and a real $a(b)>0$ so that for every p-ABS
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ for $\mathcal{X}$ with
Bessel bound $b$ and

$\displaystyle\|\tau_{j}\|=1\quad\text{ and }\quad\|f_{j}\|=1,\quad\forall
1\leq j\leq n$

can be written as a union of $M(b)$ p-approximate Riesz basic sequences each
with lower p-approximate Riesz bound $a(b)$. Moreover, what is the number of
partitions required?

Casazza-Vershynin conjecture [39] known as $R_{\varepsilon}$-conjecture for
Hilbert spaces reads as follows.

######  Theorem 2.17.

[39] (Casazza-Vershynin conjecture or $R_{\epsilon}$-conjecture for Hilbert
spaces) For every $\varepsilon>0$, every unit norm Riesz sequence can be
written as a finite union of unit norm $\varepsilon$-Riesz sequences.

We next formulate Casazza-Vershynin conjecture
[39]/$R_{\varepsilon}$-conjecture for Banach spaces. First we need a
definition.

######  Definition 2.18.

Let $p\in[1,\infty)$. A p-approximate Riesz sequence
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ for $\mathcal{X}$ is said to be unit
norm $\varepsilon$-p-approximate Riesz sequence ($\varepsilon<1$) if the
following conditions hold.

  1. (i)

$\|f_{n}\|=\|\tau_{n}\|=|f_{n}(\tau_{n})|=1$, $\forall n\in\mathbb{N}$.

  2. (ii)

For all $m\in\mathbb{N}$,

$\displaystyle(1-\varepsilon)\left(\sum_{n=1}^{m}|c_{n}|^{p}\right)^{\frac{1}{p}}\leq\left\|\sum_{n=1}^{m}c_{n}\tau_{n}\right\|\leq(1+\varepsilon)\left(\sum_{n=1}^{m}|c_{n}|^{p}\right)^{\frac{1}{p}},\quad\forall
c_{1},\dots,c_{m}\in\mathbb{K}.$

######  Conjecture 2.19.

(Casazza-Vershynin conjecture or $R_{\epsilon}$-conjecture for Banach spaces)
For every $\varepsilon>0$, every unit norm p-approximate Riesz sequence can be
written as a finite union of unit norm $\varepsilon$-p-approximate Riesz
sequences.

Weaver’s conjecture for Hilbert spaces states as follows.

######  Theorem 2.20.

[111] (Weaver’s conjecture for Hilbert spaces) Let $\mathcal{H}$ be a
$d$-dimensional Hilbert space. There are universal constants $b\geq 2$ and
$b>\varepsilon>0$ such that the following holds. Let
$\\{\tau_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{H}$ satisfying:

$\displaystyle\|\tau_{j}\|\leq 1,\quad\forall 1\leq j\leq n$

and

$\displaystyle\sum_{j=1}^{n}|\langle h,\tau_{j}\rangle|^{2}\leq
b\|h\|^{2},\quad\forall h\in\mathcal{H}.$

Then there exists a partition $I_{1},\dots,I_{M}$ of $\\{1,2,\dots,n\\}$ such
that

$\displaystyle\sum_{j\in I_{k}}|\langle
h,\tau_{j}\rangle|^{2}\leq(b-\varepsilon)\|h\|^{2},\quad\forall
h\in\mathcal{H},\forall 1\leq k\leq M.$

Here is Weaver’s conjecture [111] for Banach spaces.

######  Conjecture 2.21.

(Weaver’s conjecture for Banach spaces - 1) Let $\mathcal{X}$ be a
$d$-dimensional Banach space. There are universal constants $b\geq 2$ and
$b>\varepsilon>0$ such that the following holds. Let
$\\{\tau_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}$ and
$\\{f_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}^{*}$ satisfying:

$\displaystyle\|f_{j}\|\leq 1,~{}\|\tau_{j}\|\leq 1,\quad\forall 1\leq j\leq
n$

and

$\displaystyle\left\|\sum_{j=1}^{n}f_{j}(x)\tau_{j}\right\|\leq
b\|x\|,\quad\forall x\in\mathcal{X}.$

Then there exists a partition $I_{1},\dots,I_{M}$ of $\\{1,2,\dots,n\\}$ such
that

$\displaystyle\left\|\sum_{j\in
I_{k}}f_{j}(x)\tau_{j}\right\|\leq(b-\varepsilon)\|x\|,\quad\forall
x\in\mathcal{X},\forall 1\leq k\leq M.$

Well, we can make variants of Conjecture 2.21 as follows.

######  Conjecture 2.22.

(Weaver’s conjecture for Banach spaces - 2) Let $\mathcal{X}$ be a
$d$-dimensional Banach space. There are universal constants $b\geq 2$ and
$b>\varepsilon>0$ such that the following holds. Let
$\\{\tau_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}$ and
$\\{f_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}^{*}$ satisfying:

$\displaystyle\|f_{j}\|\leq 1,~{}\|\tau_{j}\|\leq 1,\quad\forall 1\leq j\leq
n$

and

$\displaystyle\left\|\sum_{j=1}^{n}f_{j}(x)\tau_{j}\right\|=b\|x\|,\quad\forall
x\in\mathcal{X}.$

Then there exists a partition $I_{1},\dots,I_{M}$ of $\\{1,2,\dots,n\\}$ such
that

$\displaystyle\left\|\sum_{j\in
I_{k}}f_{j}(x)\tau_{j}\right\|\leq(b-\varepsilon)\|x\|,\quad\forall
x\in\mathcal{X},\forall 1\leq k\leq M.$

######  Conjecture 2.23.

(Weaver’s conjecture for Banach spaces - 3) Let $\mathcal{X}$ be a
$d$-dimensional Banach space. There are universal constants $b\geq 2$ and
$b>\varepsilon>0$ such that the following holds. Let
$\\{\tau_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}$ and
$\\{f_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}^{*}$ satisfying:

$\displaystyle\|f_{j}\|=\|\tau_{j}\|=|f_{j}(\tau_{j})|=1,\quad\forall 1\leq
j\leq n$

and

$\displaystyle\left\|\sum_{j=1}^{n}f_{j}(x)\tau_{j}\right\|\leq
b\|x\|,\quad\forall x\in\mathcal{X}.$

Then there exists a partition $I_{1},\dots,I_{M}$ of $\\{1,2,\dots,n\\}$ such
that

$\displaystyle\left\|\sum_{j\in
I_{k}}f_{j}(x)\tau_{j}\right\|\leq(b-\varepsilon)\|x\|,\quad\forall
x\in\mathcal{X},\forall 1\leq k\leq M.$

######  Conjecture 2.24.

(Weaver’s conjecture for Banach spaces - 4) Let $\mathcal{X}$ be a
$d$-dimensional Banach space. There are universal constants $b\geq 2$ and
$b>\varepsilon>0$ such that the following holds. Let
$\\{\tau_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}$ and
$\\{f_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}^{*}$ satisfying:

$\displaystyle\|f_{j}\|=\|\tau_{j}\|=|f_{j}(\tau_{j})|=1,\quad\forall 1\leq
j\leq n$

and

$\displaystyle\left\|\sum_{j=1}^{n}f_{j}(x)\tau_{j}\right\|=b\|x\|,\quad\forall
x\in\mathcal{X}.$

Then there exists a partition $I_{1},\dots,I_{M}$ of $\\{1,2,\dots,n\\}$ such
that

$\displaystyle\left\|\sum_{j\in
I_{k}}f_{j}(x)\tau_{j}\right\|\leq(b-\varepsilon)\|x\|,\quad\forall
x\in\mathcal{X},\forall 1\leq k\leq M.$

Previous four Weaver’s conjectures for Banach spaces can be stated in slightly
stronger form as follows.

######  Conjecture 2.25.

(Weaver’s conjecture for Banach spaces - strong form - 1) Let $\mathcal{X}$ be
a $d$-dimensional Banach space. There are universal constants $b\geq 2$ and
$b>\varepsilon>0$ such that the following holds. Let
$\\{\tau_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}$ and
$\\{f_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}^{*}$ satisfying:

$\displaystyle\|f_{j}\|\leq 1,~{}\|\tau_{j}\|\leq 1,\quad\forall 1\leq j\leq
n,$

the spectrum of the operator $S_{f,\tau}:\mathcal{X}\ni
x\mapsto\sum_{j=1}^{n}f_{j}(x)\tau_{j}\in\mathcal{X}$ lies in $[0,\infty)$ and

$\displaystyle\left\|\sum_{j=1}^{n}f_{j}(x)\tau_{j}\right\|\leq
b\|x\|,\quad\forall x\in\mathcal{X}.$

Then there exists a partition $I_{1},\dots,I_{M}$ of $\\{1,2,\dots,n\\}$ such
that

$\displaystyle\left\|\sum_{j\in
I_{k}}f_{j}(x)\tau_{j}\right\|\leq(b-\varepsilon)\|x\|,\quad\forall
x\in\mathcal{X},\forall 1\leq k\leq M.$

######  Conjecture 2.26.

(Weaver’s conjecture for Banach spaces - strong form - 2) Let $\mathcal{X}$ be
a $d$-dimensional Banach space. There are universal constants $b\geq 2$ and
$b>\varepsilon>0$ such that the following holds. Let
$\\{\tau_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}$ and
$\\{f_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}^{*}$ satisfying:

$\displaystyle\|f_{j}\|\leq 1,~{}\|\tau_{j}\|\leq 1,\quad\forall 1\leq j\leq
n,$

the spectrum of the operator $S_{f,\tau}:\mathcal{X}\ni
x\mapsto\sum_{j=1}^{n}f_{j}(x)\tau_{j}\in\mathcal{X}$ lies in $[0,\infty)$ and

$\displaystyle\left\|\sum_{j=1}^{n}f_{j}(x)\tau_{j}\right\|=b\|x\|,\quad\forall
x\in\mathcal{X}.$

Then there exists a partition $I_{1},\dots,I_{M}$ of $\\{1,2,\dots,n\\}$ such
that

$\displaystyle\left\|\sum_{j\in
I_{k}}f_{j}(x)\tau_{j}\right\|\leq(b-\varepsilon)\|x\|,\quad\forall
x\in\mathcal{X},\forall 1\leq k\leq M.$

######  Conjecture 2.27.

(Weaver’s conjecture for Banach spaces - strong form - 3) Let $\mathcal{X}$ be
a $d$-dimensional Banach space. There are universal constants $b\geq 2$ and
$b>\varepsilon>0$ such that the following holds. Let
$\\{\tau_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}$ and
$\\{f_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}^{*}$ satisfying:

$\displaystyle\|f_{j}\|=\|\tau_{j}\|=|f_{j}(\tau_{j})|=1,\quad\forall 1\leq
j\leq n,$

the spectrum of the operator $S_{f,\tau}:\mathcal{X}\ni
x\mapsto\sum_{j=1}^{n}f_{j}(x)\tau_{j}\in\mathcal{X}$ lies in $[0,\infty)$ and

$\displaystyle\left\|\sum_{j=1}^{n}f_{j}(x)\tau_{j}\right\|\leq
b\|x\|,\quad\forall x\in\mathcal{X}.$

Then there exists a partition $I_{1},\dots,I_{M}$ of $\\{1,2,\dots,n\\}$ such
that

$\displaystyle\left\|\sum_{j\in
I_{k}}f_{j}(x)\tau_{j}\right\|\leq(b-\varepsilon)\|x\|,\quad\forall
x\in\mathcal{X},\forall 1\leq k\leq M.$

######  Conjecture 2.28.

(Weaver’s conjecture for Banach spaces - strong form - 4) Let $\mathcal{X}$ be
a $d$-dimensional Banach space. There are universal constants $b\geq 2$ and
$b>\varepsilon>0$ such that the following holds. Let
$\\{\tau_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}$ and
$\\{f_{j}\\}_{j=1}^{n}$ be a collection in $\mathcal{X}^{*}$ satisfying:

$\displaystyle\|f_{j}\|=\|\tau_{j}\|=|f_{j}(\tau_{j})|=1,\quad\forall 1\leq
j\leq n,$

the spectrum of the operator $S_{f,\tau}:\mathcal{X}\ni
x\mapsto\sum_{j=1}^{n}f_{j}(x)\tau_{j}\in\mathcal{X}$ lies in $[0,\infty)$ and

$\displaystyle\left\|\sum_{j=1}^{n}f_{j}(x)\tau_{j}\right\|=b\|x\|,\quad\forall
x\in\mathcal{X}.$

Then there exists a partition $I_{1},\dots,I_{M}$ of $\\{1,2,\dots,n\\}$ such
that

$\displaystyle\left\|\sum_{j\in
I_{k}}f_{j}(x)\tau_{j}\right\|\leq(b-\varepsilon)\|x\|,\quad\forall
x\in\mathcal{X},\forall 1\leq k\leq M.$

Here are Akemann-Weaver conjectures for Banach space (see [18, 1] for Hilbert
spaces).

######  Conjecture 2.29.

(Akemann-Weaver conjecture for Banach spaces)  Let $\mathcal{X}$ be a Banach
space (finite or infinite dimensional). There exists a universal constant $c$
such that the following holds. Let $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ be a
p-approximate Bessel sequence with Bessel bound 1 for $\mathcal{X}$ satisfying

$\displaystyle\sup_{n\in\mathbb{N}}\|f_{n}\|^{q}\leq\varepsilon,\quad\sup_{n\in\mathbb{N}}\|\tau_{n}\|^{s}\leq\varepsilon,$

for some $\epsilon>0$, for some $q,s>0$. Let $\\{r_{n}\\}_{n}$ be any sequence
in [0,1]. Then there exists a subset $\mathbb{M}\subseteq\mathbb{N}$ and a
$d>0$ such that

$\displaystyle\left\|\sum_{n\in\mathbb{M}}f_{n}(x)\tau_{n}-\sum_{n=1}^{\infty}r_{n}f_{n}(x)\tau_{n}\right\|\leq
c\varepsilon^{\frac{1}{d}}\|x\|,\quad\forall x\in\mathcal{X}.$

i.e.,

$\displaystyle\left\|\sum_{n\in\mathbb{M}}f_{n}(\cdot)\tau_{n}-\sum_{n=1}^{\infty}r_{n}f_{n}(\cdot)\tau_{n}\right\|\leq
c\varepsilon^{\frac{1}{d}}.$

######  Conjecture 2.30.

(Akemann-Weaver conjecture for Banach spaces - strong form)  Let $\mathcal{X}$
be a Banach space (finite or infinite dimensional). There exists a universal
constant $c$ such that the following holds. Let
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ be a p-approximate Bessel sequence with
Bessel bound 1 for $\mathcal{X}$ satisfying

$\displaystyle\sup_{n\in\mathbb{N}}\|f_{n}\|^{q}\leq\varepsilon,\quad\sup_{n\in\mathbb{N}}\|\tau_{n}\|^{s}\leq\varepsilon,$

for some $\epsilon>0$, for some $q,s>0$ and the spectrum of the operator
$S_{f,\tau}:\mathcal{X}\ni
x\mapsto\sum_{n=1}^{\infty}f_{n}(x)\tau_{n}\in\mathcal{X}$ lies in
$[0,\infty)$. Let $\\{r_{n}\\}_{n}$ be any sequence in [0,1]. Then there
exists a subset $\mathbb{M}\subseteq\mathbb{N}$ and a $d>0$ such that

$\displaystyle\left\|\sum_{n\in\mathbb{M}}f_{n}(x)\tau_{n}-\sum_{n=1}^{\infty}r_{n}f_{n}(x)\tau_{n}\right\|\leq
c\varepsilon^{\frac{1}{d}}\|x\|,\quad\forall x\in\mathcal{X},$

i.e.,

$\displaystyle\left\|\sum_{n\in\mathbb{M}}f_{n}(\cdot)\tau_{n}-\sum_{n=1}^{\infty}r_{n}f_{n}(\cdot)\tau_{n}\right\|\leq
c\varepsilon^{\frac{1}{d}}.$

We now formulate three more conjectures which are motivated from results of
Casazza [27] stated as follows.

######  Theorem 2.31.

[27] Every Riesz basis for a Hilbert space can be written as a linear
combination of two orthonormal bases.

######  Theorem 2.32.

[27] Every frame for a Hilbert space is a multiple of a sum of three
orthonormal basis.

######  Theorem 2.33.

[27] Every frame for a Hilbert space is a multiple of a sum of an orthonormal
basis and a Riesz basis.

######  Conjecture 2.34.

Every p-approximate Riesz basis is a linear combination of p-orthonormal
bases. Moreover, there exists $M\in\mathbb{N}$ such that every p-approximate
Riesz basis can be written as a linear combination of $M$ p-orthonormal bases.

######  Conjecture 2.35.

Every p-ASF is a multiple of a finite sum of p-orthonormal bases. Moreover,
there exists $M\in\mathbb{N}$ such that every p-ASF can be written as a
multiple of sum of $M$ p-orthonormal bases.

######  Conjecture 2.36.

Every p-ASF is a multiple of a sum of p-orthonormal basis and a p-approximate
Riesz basis.

We next recall the famous fundamental inequality for finite frames for finite
dimensional Hilbert spaces due to Casazza, Fickus, Kovacevic, Leon, and
Tremain [36] and independently by Viswanath and Anantharam [110].

######  Theorem 2.37.

[36, 110] (Fundamental inequality) Let $n\geq d$ and $\mathcal{H}$ be a
$d$-dimensional Hilbert space and let $a_{1},\dots,a_{n}$ be a collection of
non negative real numbers. Then there exists a tight frame
$\\{\tau_{j}\\}_{j=1}^{n}$ for $\mathcal{H}$ with $\|\tau_{j}\|=a_{j}$,
$\forall 1\leq j\leq n$ if and only if

$\displaystyle\max_{1\leq j\leq
n}a_{j}^{2}\leq\frac{1}{d}\sum_{j=1}^{n}a_{j}^{2}.$

Here is fundamental inequality conjecture for Banach spaces.

######  Conjecture 2.38.

(Fundamental inequality conjecture for Banach spaces) Let $n\geq d$ and
$\mathcal{X}$ be a $d$-dimensional Banach space and let
$a_{1},\dots,a_{n},b_{1},\dots,b_{n},c_{1},\dots,c_{n}$ be a collection of non
negative real numbers. Then there exists a tight ASF
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ for $\mathcal{X}$ with

$\displaystyle\|f_{j}\|=a_{j},\quad\|\tau_{j}\|=b_{j},\quad|f_{j}(\tau_{j})|=c_{j},\quad\forall
1\leq j\leq n$

if and only if

$\displaystyle\max_{1\leq j\leq
n}a_{j}^{p}\leq\frac{1}{d}\sum_{j=1}^{n}a_{j}^{p},$ $\displaystyle\max_{1\leq
j\leq n}b_{j}^{q}\leq\frac{1}{d}\sum_{j=1}^{n}b_{j}^{q},$
$\displaystyle\max_{1\leq j\leq
n}c_{j}^{r}\leq\frac{1}{d}\sum_{j=1}^{n}c_{j}^{r}$

for some real $p,q,r>0$.

A result in Hilbert space frame theory which is along with fundamental
inequality is the following result.

######  Theorem 2.39.

[41, 71] Let $n\geq d$ and $\mathcal{H}$ be a $d$-dimensional Hilbert space
and let $a_{1},\dots,a_{n}$ be a collection of non negative real numbers. For
any self adjoint positive operator $S:\mathcal{H}\to\mathcal{H}$ with
eigenvalues $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{d}>0$, the
following are equivalent.

  1. (i)

Then exists a frame $\\{\tau_{j}\\}_{j=1}^{n}$ for $\mathcal{H}$ whose frame
operator $S_{\tau}=S$ and $\|\tau_{j}\|=a_{j}$, $\forall 1\leq j\leq n$.

  2. (ii)

For all $1\leq m\leq d$,

$\displaystyle\sum_{k=1}^{m}a_{k}^{2}\leq\sum_{k=1}^{m}\lambda_{k}\quad\text{and}\quad\sum_{j=1}^{n}a_{j}^{2}=\sum_{k=1}^{d}\lambda_{k}.$

Theorem 2.39 leads to the following conjecture.

######  Conjecture 2.40.

Let $n\geq d$ and $\mathcal{X}$ be a $d$-dimensional Banach space and let
$a_{1},\dots,a_{n},$ $b_{1},\dots,b_{n}$, $c_{1}\dots,c_{n}$ be a collection
of non negative real numbers. For any operator $S:\mathcal{X}\to\mathcal{X}$
with eigenvalues $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{d}>0$, the
following are equivalent.

  1. (i)

Then exists an ASF $(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ for
$\mathcal{X}$ whose frame operator $S_{f,\tau}=S$ and $\|f_{j}\|=a_{j}$,
$\|\tau_{j}\|=b_{j}$, $|f_{j}(\tau_{j})|=c_{j}$, $\forall 1\leq j\leq n$.

  2. (ii)

For all $1\leq m\leq d$,

$\displaystyle\sum_{k=1}^{m}a_{k}^{p}\leq\sum_{k=1}^{m}\lambda_{k}\quad\text{and}\quad\sum_{j=1}^{n}a_{j}^{p}=\sum_{k=1}^{d}\lambda_{k},$
$\displaystyle\sum_{k=1}^{m}b_{k}^{q}\leq\sum_{k=1}^{m}\lambda_{k}\quad\text{and}\quad\sum_{j=1}^{n}b_{j}^{q}=\sum_{k=1}^{d}\lambda_{k},$
$\displaystyle\sum_{k=1}^{m}c_{k}^{r}\leq\sum_{k=1}^{m}\lambda_{k}\quad\text{and}\quad\sum_{j=1}^{n}c_{j}^{r}=\sum_{k=1}^{d}\lambda_{k},$

for some real $p,q,r>0$.

Celebrated scaling problem for frames for Hilbert spaces reads as follows and
to state it, first we need a definition.

######  Definition 2.41.

[85] A frame $\\{\tau_{n}\\}_{n}$ for a Hilbert space $\mathcal{H}$ is said to
be scalable if there exists a sequence of (non negative) scalars
$\\{a_{n}\\}_{n}$ such that

$\displaystyle\\{a_{n}\tau_{n}\\}_{n}\text{ is a Parseval frame for
}\mathcal{H}.$

######  Problem 2.42.

[85, 42] (Scaling problem) Classify frames $\\{\tau_{n}\\}_{n}$ for a Hilbert
space $\mathcal{H}$ so that there is a sequence of scalars $\\{a_{n}\\}_{n}$
such that $\\{a_{n}\tau_{n}\\}_{n}$ is a Parseval frame $\mathcal{H}$, i.e.,
the frame $\\{\tau_{n}\\}_{n}$ is scalable.

Here we state scaling problem for Banach spaces with the introduction of
scaling for p-approximate Schauder frames.

######  Definition 2.43.

A p-ASF $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ for a Banach space
$\mathcal{X}$ is said to be p-scalable if there exist sequences of scalars
$\\{a_{n}\\}_{n}$, $\\{b_{n}\\}_{n}$ such that

$\displaystyle(\\{a_{n}f_{n}\\}_{n},\\{b_{n}\tau_{n}\\}_{n})\text{ is a
Parseval p-ASF for }\mathcal{X}.$

######  Problem 2.44.

(Scaling problem for Banach spaces) Classify p-ASFs
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ for a Banach space $\mathcal{X}$ so
that there are sequences of scalars $\\{a_{n}\\}_{n}$, $\\{b_{n}\\}_{n}$ such
that $(\\{a_{n}f_{n}\\}_{n},\\{b_{n}\tau_{n}\\}_{n})$ is a Parseval p-ASF for
$\mathcal{X}$, i.e., p-ASF $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is
p-scalable.

Let us now recall the Kothe-Lorch theorem for Riesz basis for Hilbert spaces.

######  Theorem 2.45.

[73, 67] (Kothe-Lorch theorem) For a collection $\\{\tau_{n}\\}_{n}$ in a
Hilbert space $\mathcal{H}$, the following are equivalent.

  1. (i)

$\\{\tau_{n}\\}_{n}$ is a Riesz basis for $\mathcal{H}$.

  2. (ii)

$\\{\tau_{n}\\}_{n}$ is an unconditional Schauder basis for $\mathcal{H}$, and

$\displaystyle
0<\inf_{n\in\mathbb{N}}\|\tau_{n}\|\leq\sup_{n\in\mathbb{N}}\|\tau_{n}\|<\infty.$

Based on Theorem 2.45 we formulate the following problem.

######  Problem 2.46.

(Kothe-Lorch problem for Banach spaces) For a collection $\\{\tau_{n}\\}_{n}$
in a Banach space $\mathcal{X}$ and a collection $\\{f_{n}\\}_{n}$ in
$\mathcal{X}^{*}$, whether the following are equivalent?

  1. (i)

$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is a p-approximate Riesz basis for
$\mathcal{X}$.

  2. (ii)

$\\{\tau_{n}\\}_{n}$ is an unconditional Schauder basis for $\mathcal{X}$,

$\displaystyle\phi(x)=\sum_{n=1}^{\infty}\phi(\tau_{n})f_{n}(x),\quad\forall
x\in\mathcal{X},\forall\phi\in\mathcal{X}^{*}$

and

$\displaystyle
0<\inf_{n\in\mathbb{N}}\|\tau_{n}\|\leq\sup_{n\in\mathbb{N}}\|\tau_{n}\|<\infty,$
$\displaystyle
0<\inf_{n\in\mathbb{N}}\|f_{n}\|\leq\sup_{n\in\mathbb{N}}\|f_{n}\|<\infty.$

Note that we always have (i) $\Rightarrow$ (ii) in Problem 2.46.

For frames for Hilbert spaces there are three algorithms to get approximation
to every element of the space. They are Duffin-Schaeffer algorithm [57],
Grochenig-Chebyshev algorithm [68] and Grochenig conjugate gradient algorithm
[68]. For p-ASFs we derive a Duffin-Schaeffer algorithm with stronger
assumption and ask others as problems.

######  Theorem 2.47.

Let $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ be a p-ASF for $\mathcal{X}$ with
bounds $a$ and $b$. For $x\in\mathcal{X}$, define

$x_{0}\coloneqq 0,~{}x_{n}\coloneqq
x_{n-1}+\frac{2}{a+b}S_{f,\tau}(x-x_{n-1}),\quad\forall n\geq 1.$

If $\|I_{\mathcal{X}}-\frac{2}{b+a}S_{f,\tau}\|\leq\frac{b-a}{b+a}$, then

$\|x_{n}-x\|\leq\left(\frac{b-a}{b+a}\right)^{n}\|x\|,\quad\forall n\geq 1.$

In particular, $x_{n}\to x$ as $n\to\infty$.

###### Proof.

Using the definition of $x_{n}^{\prime}$s, we see that

$\displaystyle x-x_{n}$
$\displaystyle=x-x_{n-1}-\frac{2}{a+b}S_{f,\tau}(x-x_{n-1})$
$\displaystyle=\left(I_{\mathcal{X}}-\frac{2}{b+a}S_{f,\tau}\right)(x-x_{n-1})$
$\displaystyle=\cdots=\left(I_{\mathcal{X}}-\frac{2}{b+a}S_{f,\tau}\right)^{n}x,\quad\forall
x\in\mathcal{X},\forall n\geq 1.$

Therefore

$\displaystyle\|x_{n}-x\|\leq\left\|I_{\mathcal{X}}-\frac{2}{b+a}S_{f,\tau}\right\|^{n}\|x\|\leq\left(\frac{b-a}{b+a}\right)^{n}\|x\|,\quad\forall
n\geq 1.$

∎

######  Problem 2.48.

(Frame algorithm problems for Banach spaces) Whether there is a

  1. (i)

Duffin-Schaeffer algorithm for p-ASFs?

  2. (ii)

Grochenig-Chebyshev algorithm for p-ASFs?

  3. (iii)

Grochenig conjugate gradient algorithm for p-ASFs?

In the groundbreaking paper [3], Aldroubi, Cabrelli, Molter, and Tang
introduced the notion of dynamical sampling for Hilbert spaces and solved for
finite dimensional Hilbert spaces. The problem reads as follows.

######  Problem 2.49.

[3, 4, 2, 56] (Dynamical sampling problem for Hilbert spaces) Let
$\mathcal{H}$ be a Hilbert space (finite or infinite dimensional). Let
$\mathbb{K},\mathbb{M}\subseteq\mathbb{N}$, $T:\mathcal{H}\to\mathcal{H}$ be a
bounded linear operator and $\\{\tau_{k}\\}_{k\in\mathbb{K}}$ be a sequence in
$\mathcal{H}$. Find conditions on $\mathbb{K},\mathbb{M}$ and $T$ such that

  1. (i)

$\\{T^{m}\tau_{k}\\}_{k\in\mathbb{K},m\in\mathbb{M}}$ is an orthonormal basis
for $\mathcal{H}$.

  2. (ii)

$\\{T^{m}\tau_{k}\\}_{k\in\mathbb{K},m\in\mathbb{M}}$ is a Riesz basis for
$\mathcal{H}$.

  3. (iii)

$\\{T^{m}\tau_{k}\\}_{k\in\mathbb{K},m\in\mathbb{M}}$ is a Riesz sequence for
$\mathcal{H}$.

  4. (iv)

$\\{T^{m}\tau_{k}\\}_{k\in\mathbb{K},m\in\mathbb{M}}$ is a frame for
$\mathcal{H}$.

  5. (v)

$\\{T^{m}\tau_{k}\\}_{k\in\mathbb{K},m\in\mathbb{M}}$ is a Parseval frame for
$\mathcal{H}$.

  6. (vi)

$\\{T^{m}\tau_{k}\\}_{k\in\mathbb{K},m\in\mathbb{M}}$ is a Bessel sequence for
$\mathcal{H}$.

If the operator $T$ is normal, then the set $\mathbb{M}$ can be replaced by
any subset of $[0,\infty)$.

We formulate Problem 2.49 for Banach spaces as follows.

######  Problem 2.50.

(Dynamical sampling problem for Banach spaces) Let $\mathcal{X}$ be a Banach
space (finite or infinite dimensional). Let
$\mathbb{K},\mathbb{M}\subseteq\mathbb{N}$, $U:\mathcal{X}\to\mathcal{X}$,
$V:\mathcal{X}\to\mathcal{X}$ be bounded linear operators,
$\\{\tau_{k}\\}_{k\in\mathbb{K}}$ be a sequence in $\mathcal{X}$ and
$\\{f_{k}\\}_{k\in\mathbb{K}}$ be a sequence in $\mathcal{X}^{*}$. Find
conditions on $\mathbb{K},\mathbb{M}$, $U$ and $V$ such that

  1. (i)

$(\\{f_{k}U^{m}\\}_{k\in\mathbb{K},m\in\mathbb{M}},\\{V^{m}\tau_{k}\\}_{k\in\mathbb{K},m\in\mathbb{M}})$
is a p-orthonormal basis for $\mathcal{X}$.

  2. (ii)

$(\\{f_{k}U^{m}\\}_{k\in\mathbb{K},m\in\mathbb{M}},\\{V^{m}\tau_{k}\\}_{k\in\mathbb{K},m\in\mathbb{M}})$
is a p-approximate Riesz basis for $\mathcal{X}$.

  3. (iii)

$(\\{f_{k}U^{m}\\}_{k\in\mathbb{K},m\in\mathbb{M}},\\{V^{m}\tau_{k}\\}_{k\in\mathbb{K},m\in\mathbb{M}})$
is a p-approximate Riesz sequence for $\mathcal{X}$.

  4. (iv)

$(\\{f_{k}U^{m}\\}_{k\in\mathbb{K},m\in\mathbb{M}},\\{V^{m}\tau_{k}\\}_{k\in\mathbb{K},m\in\mathbb{M}})$
is a p-approximate Schauder frame for $\mathcal{X}$.

  5. (v)

$(\\{f_{k}U^{m}\\}_{k\in\mathbb{K},m\in\mathbb{M}},\\{V^{m}\tau_{k}\\}_{k\in\mathbb{K},m\in\mathbb{M}})$
is a Parseval p-approximate Schauder frame for $\mathcal{X}$.

  6. (vi)

$(\\{f_{k}U^{m}\\}_{k\in\mathbb{K},m\in\mathbb{M}},\\{V^{m}\tau_{k}\\}_{k\in\mathbb{K},m\in\mathbb{M}})$
is a p-approximate Bessel sequence for $\mathcal{X}$.

The set $\mathbb{M}$ can be replaced by subsets $\Sigma\subseteq\mathbb{C}$
whenever $U^{\alpha}$ and $V^{\alpha}$ make sense for $\alpha\in\Sigma$. For
instance, if $T$ is in the class of bounded linear operators whose resolvent
contains $(-\infty,0]$ (sectorial operators) [79, 70], then we can take
$\Sigma$ to be any subset of $\mathbb{C}$.

Now we recall the phase retrieval and norm retrieval in Hilbert spaces which
originated from the fundamental work of Balan, Casazza, and Edidin [9].

######  Definition 2.51.

[9, 8, 22, 17]

  1. (i)

Let $\mathcal{H}$ be a Hilbert space (finite or infinite dimensional). A frame
$\\{\tau_{n}\\}_{n}$ for $\mathcal{H}$ does phase retrieval for $\mathcal{H}$
if $x,y$ are any two elements of $\mathcal{H}$ satisfying

$\displaystyle|\langle x,\tau_{n}\rangle|=|\langle
y,\tau_{n}\rangle|,\quad\forall n\in\mathbb{N},$

then $y=\alpha x$, for some $|\alpha|=1$.

  2. (ii)

Let $\mathcal{H}$ be a Hilbert space (finite or infinite dimensional). A frame
$\\{\tau_{n}\\}_{n}$ for $\mathcal{H}$ does norm retrieval for $\mathcal{H}$
if $x,y$ are any two elements of $\mathcal{H}$ satisfying

$\displaystyle|\langle x,\tau_{n}\rangle|=|\langle
y,\tau_{n}\rangle|,\quad\forall n\in\mathbb{N},$

then $\|y\|=\|x\|$.

Following is the phase retrieval and norm retrieval problems for Hilbert
spaces.

######  Problem 2.52.

[9, 8, 22, 17]

  1. (i)

(Phase retrieval problem for Hilbert spaces) For a given Hilbert space
$\mathcal{H}$, classify frames $\\{\tau_{n}\\}_{n}$ for $\mathcal{H}$ which
does phase retrieval for $\mathcal{H}$.

  2. (ii)

(Norm retrieval problem for Hilbert spaces) For a given Hilbert space for
$\mathcal{H}$, classify frames $\\{\tau_{n}\\}_{n}$ for $\mathcal{H}$ which
does norm retrieval for $\mathcal{H}$.

Phase retrieval problem poses challenges even in finite dimensions. For
instance, phase retrieval can be classified only for real Hilbert spaces using
complement property [11]. Number of vectors needed and a full description of
them to do phase retrieval in arbitrary dimension is still open [45, 52, 109,
16, 60].
We now state the phase retrieval and norm retrieval notions and problems for
p-approximate Schauder frames for Banach space.

######  Definition 2.53.

  1. (i)

Let $\mathcal{X}$ be a Banach space (finite or infinite dimensional). A
p-approximate Schauder frame $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ for
$\mathcal{X}$ does phase retrieval for $\mathcal{X}$ if $x,y$ are any two
elements of $\mathcal{X}$ and $\phi,\psi$ are any two elements of
$\mathcal{X}^{*}$ satisfying

$\displaystyle|\phi(\tau_{n})|=|\psi(\tau_{n})|\quad\text{and}\quad|f_{n}(x)|=|f_{n}(y)|,\quad\forall
n\in\mathbb{N},$

then $y=\alpha x$, for some $|\alpha|=1$ and $\psi=\beta\phi$, for some
$|\beta|=1$.

  2. (ii)

Let $\mathcal{X}$ be a Banach space (finite or infinite dimensional). A
p-approximate Schauder frame $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ for
$\mathcal{X}$ does norm retrieval for $\mathcal{X}$ if $x,y$ are any two
elements of $\mathcal{X}$ and $\phi,\psi$ are any two elements of
$\mathcal{X}^{*}$ satisfying

$\displaystyle|\phi(\tau_{n})|=|\psi(\tau_{n})|\quad\text{and}\quad|f_{n}(x)|=|f_{n}(y)|,\quad\forall
n\in\mathbb{N},$

then $\|y\|=\|x\|$ and $\|\phi\|=\|\psi\|$.

######  Problem 2.54.

  1. (i)

(Phase retrieval problem for Banach spaces) For a given Banach space
$\mathcal{X}$, classify p-approximate Schauder frames
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ for $\mathcal{X}$ which does phase
retrieval for $\mathcal{X}$.

  2. (ii)

(Norm retrieval problem for Banach spaces) For a given Banach space
$\mathcal{X}$, classify p-approximate Schauder frames
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ for $\mathcal{X}$ which does norm
retrieval for $\mathcal{X}$.

Our next problem is motivated from the works of Dykema and Strawn in Hilbert
spaces [23, 105, 97, 103, 58, 15, 66, 104] and Corach, Pacheco, and Stojanoff
[53].

######  Problem 2.55.

(Algebraic geometry and topological problems for approximate Schauder frames
for Banach spaces) Let $n\geq d$ be fixed natural numbers and $\mathcal{X}$ be
a fixed $d$-dimensional Banach space. Define

$\displaystyle\mathscr{F}^{n}(\mathbb{K}^{d})\coloneqq\\{($
$\displaystyle\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n}):(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})\text{
is a tight ASF for }\mathcal{X}\text{ such that}$
$\displaystyle\|f_{j}\|=\|\tau_{j}\|=|f_{j}(\tau_{j})|=1,\forall 1\leq j\leq
n\\}.$

  1. (i)

Whether there is an explicit description of $\mathscr{F}^{n}(\mathbb{K}^{d})$?

  2. (ii)

What is a (good) topology on $\mathscr{F}^{n}(\mathbb{K}^{d})$? In particular,
whether $\mathscr{F}^{n}(\mathbb{K}^{d})$ is path-connected?

  3. (iii)

Whether $\mathscr{F}^{n}(\mathbb{K}^{d})$ is a manifold?

######  Problem 2.56.

(Topological problem for p-approximate Schauder frames for Banach spaces) Let
$p\in[1,\infty),$ and $\mathcal{X}$ be a fixed Banach space (finite or
infinite dimensional). Define

$\displaystyle\mathscr{F}^{p}(\mathcal{X})\coloneqq\\{(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n}$
$\displaystyle:(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})\text{ is a p-ASF for
}\mathcal{X}\\}.$

  1. (i)

Whether there is an explicit description of $\mathscr{F}^{p}(\mathcal{X})$?

  2. (ii)

What is a (good) topology on $\mathscr{F}^{p}(\mathcal{X})$? In particular,
whether $\mathscr{F}^{p}(\mathcal{X})$ is path-connected?

  3. (iii)

Whether $\mathscr{F}^{p}(\mathcal{X})$ is a Banach manifold?

We observe that there are two important results in the theory of frames for
Hilbert spaces, one connects Bessel sequences to frames and another connects
frames to Riesz bases. First one, due to Li and Sun [89] states that every
Bessel sequence can be extended to a frame by adding extra elements, if
necessary. Second one, known as Naimark-Han-Larson dilation theorem [72, 54,
78] states that a frame for a Hilbert space can be dilated to a Riesz basis.
For p-approximate Schauder frames (even for approximate Schauder frames),
using a result (which used approximation properties [29]) of Casazza and
Christensen [33] it is proved by Krishna and Johnson [83] that p-approximate
Bessel sequences may not be always extended to p-approximate Schauder frames.
However, using explicit construction, Krishna and Johnson [81] proved that
p-approximate Schauder frame can always be dilated to a p-approximate Riesz
basis. It is also proved in [81] that the dilation is optimal, i.e., it will
not add anything extra if the given collection is already p-approximate Riesz
basis.

We next introduce the notion of continuous p-approximate Schauder frames. Our
motivation is from the notion of continuous frames for Hilbert spaces
introduced independently by Ali, Antoine, and Gazeau, [6] and Kaiser [77],
(see [64, 99]) and from the notion of continuous Schauder frames by Eisner and
Freeman [59] (also see [90]). All integrals which appear in sequel are in weak
sense (Pettis integrals [106]).

######  Definition 2.57.

Let $(\Omega,\mu)$ be a measure space and $p\in[1,\infty)$. Let
$\\{\tau_{\alpha}\\}_{\alpha\in\Omega}$ be a collection in a Banach space
$\mathcal{X}$ and $\\{f_{\alpha}\\}_{\alpha\in\Omega}$ be a collection in
$\mathcal{X}^{*}$ (dual of $\mathcal{X}$). The pair
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
is said to be a continuous p-approximate Schauder frame (continuous p-ASF) for
$\mathcal{X}$ if the following conditions are satisfied.

  1. (i)

For each $x\in\mathcal{X}$, the map $\Omega\ni\alpha\mapsto
f_{\alpha}(x)\in\mathbb{R}\text{ or }\mathbb{C}$ is measurable.

  2. (ii)

For each $u\in\mathcal{L}^{p}(\Omega,\mu)$, the map $\Omega\ni\alpha\mapsto
u(\alpha)\tau_{\alpha}\in\mathcal{X}$ is measurable.

  3. (iii)

The map (continuous analysis operator)

$\displaystyle\theta_{f}:\mathcal{X}\ni
x\mapsto\theta_{f}\in\mathcal{L}^{p}(\Omega,\mu);\quad\theta_{f}x:\Omega\ni\alpha\mapsto(\theta_{f}x)(\alpha)\coloneqq
f_{\alpha}(x)\in\mathbb{R}\text{ or }\mathbb{C}$

is a well-defined bounded linear operator.

  4. (iv)

The map (continuous synthesis operator)

$\displaystyle\theta_{\tau}:\mathcal{L}^{p}(\Omega,\mu)\ni
u\mapsto\theta_{\tau}u\coloneqq\int_{\Omega}u(\alpha)\tau_{\alpha}\,d\mu(\alpha)\in\mathcal{X}$

is a well-defined bounded linear operator.

  5. (v)

The map (continuous frame operator)

$\displaystyle S_{f,\tau}:\mathcal{X}\ni x\mapsto
S_{f,\tau}x\coloneqq\int_{\Omega}f_{\alpha}(x)\tau_{\alpha}\,d\mu(\alpha)\in\mathcal{X}$

is a well-defined bounded linear, invertible operator.

Constants $a>0$ and $b>0$ satisfying

$\displaystyle a\|x\|\leq\|S_{f,\tau}x\|\leq b\|x\|,\quad\forall
x\in\mathcal{X},$

are called as lower frame bound and upper frame bound, respectively. If
$S_{f,\tau}x=x$, $\forall x\in\mathcal{X}$, then we say that
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
is a Parseval continuous p-ASF for $\mathcal{X}$. If $S_{f,\tau}x=\lambda x$,
for some nonzero scalar $\lambda$, $\forall x\in\mathcal{X}$, then we say that
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
is a tight continuous p-ASF for $\mathcal{X}$. If we do not demand condition
(v), then we say that
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
is a continuous p-approximate Bessel sequence (p-ABS) for $\mathcal{X}$.
Constant $b$ is called as Bessel bound.

Here is a genuine example.

######  Example 2.58.

Let $p\in(1,\infty)$ and $q$ be the conjugate index of $p$. Define
$\mathcal{X}\coloneqq\mathbb{R}^{2}$ equipped with $\|\cdot\|_{1}$ norm.
Define $\Omega\coloneqq[0,2\pi]$ equipped with the usual Lebesgue measure. Now
define

$\displaystyle\tau_{\alpha}\coloneqq(\cos\alpha,\sin\alpha),\quad\forall\alpha\in[0,2\pi].$

and

$\displaystyle f_{\alpha}:\mathcal{X}\ni(x,y)\mapsto f_{\alpha}(x,y)\coloneqq
x\cos\alpha+y\sin\alpha\in\mathbb{R},\quad\forall\alpha\in[0,2\pi].$

Then

$\displaystyle\int_{\Omega}f_{\alpha}(x,y)\tau_{\alpha}\,d\mu(\alpha)$
$\displaystyle=\int_{0}^{2\pi}(x\cos\alpha+y\sin\alpha)(\cos\alpha,\sin\alpha)\,d\alpha$
$\displaystyle=\left(\int_{0}^{2\pi}(x\cos\alpha+y\sin\alpha)\cos\alpha\,d\alpha,\int_{0}^{2\pi}(x\cos\alpha+y\sin\alpha)\sin\alpha\,d\alpha\right)$
$\displaystyle=\pi(x,y),\quad\forall(x,y)\in\mathbb{R}^{2},$

$\displaystyle\|\theta_{f}(x,y)\|$
$\displaystyle=\left(\int_{0}^{2\pi}|(\theta_{f}(x,y))|^{p}\,d\alpha\right)^{\frac{1}{p}}=\left(\int_{0}^{2\pi}|f_{\alpha}(x,y))|^{p}\,d\alpha\right)^{\frac{1}{p}}$
$\displaystyle=\left(\int_{0}^{2\pi}|x\cos\alpha+y\sin\alpha|^{p}\,d\alpha\right)^{\frac{1}{p}}$
$\displaystyle\leq\left(\int_{0}^{2\pi}|x\cos\alpha|^{p}\,d\alpha\right)^{\frac{1}{p}}+\left(\int_{0}^{2\pi}|y\sin\alpha|^{p}\,d\alpha\right)^{\frac{1}{p}}$
$\displaystyle=|x|\left(\int_{0}^{2\pi}|\cos\alpha|^{p}\,d\alpha\right)^{\frac{1}{p}}+|y|\left(\int_{0}^{2\pi}|\sin\alpha|^{p}\,d\alpha\right)^{\frac{1}{p}}$
$\displaystyle=(2\pi)^{\frac{1}{p}}(|x|+|y|)=(2\pi)^{\frac{1}{p}}\|(x,y)\|_{1},\quad\forall(x,y)\in\mathbb{R}^{2},$

and

$\displaystyle\|\theta_{\tau}u\|_{1}$
$\displaystyle=\left\|\int_{0}^{2\pi}u(\alpha)\tau_{\alpha}\,d\alpha\right\|_{1}=\left\|\int_{0}^{2\pi}u(\alpha)(\cos\alpha,\sin\alpha)\,d\alpha\right\|_{1}$
$\displaystyle=\left\|\left(\int_{0}^{2\pi}u(\alpha)\cos\alpha\,d\alpha,\int_{0}^{2\pi}u(\alpha)\sin\alpha\,d\alpha\right)\right\|_{1}$
$\displaystyle=\left|\int_{0}^{2\pi}u(\alpha)\cos\alpha\,d\alpha\right|+\left|\int_{0}^{2\pi}u(\alpha)\sin\alpha\,d\alpha\right|$
$\displaystyle\leq\left(\int_{0}^{2\pi}|u(\alpha)|^{p}\,d\alpha\right)^{\frac{1}{p}}\left(\int_{0}^{2\pi}|\cos\alpha|^{q}\,d\alpha\right)^{\frac{1}{q}}+\left(\int_{0}^{2\pi}|u(\alpha)|^{p}\,d\alpha\right)^{\frac{1}{p}}\left(\int_{0}^{2\pi}|\sin\alpha|^{q}\,d\alpha\right)^{\frac{1}{q}}$
$\displaystyle\leq
2(2\pi)^{\frac{1}{q}}\left(\int_{0}^{2\pi}|u(\alpha)|^{p}\,d\alpha\right)^{\frac{1}{p}}=2(2\pi)^{\frac{1}{q}}\|u\|_{p},\quad\forall
u\in\mathcal{L}^{p}[0,2\pi].$

Therefore
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
is a continuous p-ASF for $\mathbb{R}^{2}$. Note that
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
is also continuous 1-ASF.

Here is discretization problem for continous p-approximate Schauder frames for
Banach spaces. For frames for Hilbert spaces this was asked by Ali, Antoine,
and Gazeau [7] and solved by Fornasier, Rauhut, Freeman, and Speegle [63, 61,
100]. The problem for continuous Schauder frames has been asked by Eisner and
Freeman is still open [59].

######  Problem 2.59.

(Discretization problem for continuous p-approximate Schauder frames for
Banach spaces) When a continuous p-approximate Schauder frame for a Banach
space can be sampled to obtain a (discrete) p-approximate Schauder frame?

We now introduce the notion of continuous p-approximate Riesz basis for Banach
spaces. Motivation comes from Theorem 2.12.

######  Definition 2.60.

A continuous p-ASF
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
for $\mathcal{X}$ is said to be continuous p-approximate Riesz basis if

$\displaystyle\theta_{f}S_{f,\tau}^{-1}\theta_{\tau}=I_{\mathcal{L}^{p}(\Omega,\mu)}.$

######  Definition 2.61.

Let $\\{\tau_{\alpha}\\}_{\alpha\in\Omega}$ be a collection in a Banach space
$\mathcal{X}$ and $\\{f_{\alpha}\\}_{\alpha\in\Omega}$ be a collection in
$\mathcal{X}^{*}$ (dual of $\mathcal{X}$). The pair
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
is said to be a continuous p-approximate Riesz family  for $\mathcal{X}$ if
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
is a continuous p-approximate Riesz basis for
$\overline{span}\\{\tau_{\alpha}\\}_{\alpha\in\Omega}$.

Now we can formulate most of the notions, conjectures and problems stated
earlier for continuous p-approximate Schauder frames. Here are some samples.
Throughout, $(\Omega,\mu)$ is a measure space and
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
is at least continuous p-approximate Bessel family whenever integrals appear.

######  Conjecture 2.62.

(Feichtinger conjecture for continuous p-approximate Schauder frames) Let
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
be a continuous p-ASF for $\mathcal{X}$ such that

$\displaystyle
0<\inf_{\alpha\in\Omega}\|\tau_{\alpha}\|\leq\sup_{\alpha\in\Omega}\|\tau_{\alpha}\|<\infty,$
$\displaystyle
0<\inf_{\alpha\in\Omega}\|f_{\alpha}\|\leq\sup_{\alpha\in\Omega}\|f_{\alpha}\|<\infty.$

Then
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
can be partitioned into a finite union of continuous p-approximate Riesz
families. Moreover, what is the number of partitions required?

######  Conjecture 2.63.

(Continuous Weaver’s conjecture for Banach spaces) Let $\mathcal{X}$ be a
$d$-dimensional Banach space. There are universal constants $b\geq 2$ and
$b>\varepsilon>0$ such that the following holds. Let
$\\{\tau_{\alpha}\\}_{\alpha\in\Omega}$ be a family in $\mathcal{X}$ and
$\\{f_{\alpha}\\}_{\alpha\in\Omega}$ be a family in $\mathcal{X}^{*}$
satisfying:

$\displaystyle\|f_{\alpha}\|\leq 1,~{}\|\tau_{\alpha}\|\leq
1,\quad\forall\alpha\in\Omega$

and

$\displaystyle\left\|\int_{\Omega}f_{\alpha}(x)\tau_{\alpha}\,d\mu(\alpha)\right\|\leq
b\|x\|,\quad\forall x\in\mathcal{X}.$

Then there exists a measurable partition $\Delta_{1},\dots,\Delta_{M}$ of
$\Omega$ such that

$\displaystyle\left\|\int_{\Delta_{k}}f_{\alpha}(x)\tau_{\alpha}\,d\mu(\alpha)\right\|\leq(b-\varepsilon)\|x\|,\quad\forall
x\in\mathcal{X},\forall 1\leq k\leq M.$

######  Conjecture 2.64.

(Continuous Weaver’s conjecture for Banach spaces - strong form) Let
$\mathcal{X}$ be a $d$-dimensional Banach space. There are universal constants
$b\geq 2$ and $b>\varepsilon>0$ such that the following holds. Let
$\\{\tau_{\alpha}\\}_{\alpha\in\Omega}$ be a family in $\mathcal{X}$ and
$\\{f_{\alpha}\\}_{\alpha\in\Omega}$ be a family in $\mathcal{X}^{*}$
satisfying:

$\displaystyle\|f_{\alpha}\|\leq 1,~{}\|\tau_{\alpha}\|\leq
1,\quad\forall\alpha\in\Omega,$

the spectrum of the operator $S_{f,\tau}:\mathcal{X}\ni
x\mapsto\int_{\Omega}f_{\alpha}(x)\tau_{\alpha}\,d\mu(\alpha)\in\mathcal{X}$
lies in $[0,\infty)$ and

$\displaystyle\left\|\int_{\Omega}f_{\alpha}(x)\tau_{\alpha}\,d\mu(\alpha)\right\|\leq
b\|x\|,\quad\forall x\in\mathcal{X}.$

Then there exists a measurable partition $\Delta_{1},\dots,\Delta_{M}$ of
$\Omega$ such that

$\displaystyle\left\|\int_{\Delta_{k}}f_{\alpha}(x)\tau_{\alpha}\,d\mu(\alpha)\right\|\leq(b-\varepsilon)\|x\|,\quad\forall
x\in\mathcal{X},\forall 1\leq k\leq M.$

######  Conjecture 2.65.

(Continuous Akemann-Weaver conjecture for Banach spaces)  Let $\mathcal{X}$ be
a Banach space (finite or infinite dimensional). There exists a universal
constant $c$ such that the following holds. Let
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
be a continuous p-approximate Bessel family with Bessel bound 1 for
$\mathcal{X}$ satisfying

$\displaystyle\sup_{\alpha\in\Omega}\|f_{\alpha}\|^{q}\leq\varepsilon,\quad\sup_{\alpha\in\Omega}\|\tau_{\alpha}\|^{s}\leq\varepsilon,$

for some $\epsilon>0$, for some $q,s>0$. Let
$\\{r_{\alpha}\\}_{\alpha\in\Omega}$ be any family in [0,1]. Then there exists
a measurable subset $\Delta\subseteq\Omega$ and a $d>0$ such that

$\displaystyle\left\|\int_{\Delta}f_{\alpha}(x)\tau_{\alpha}\,d\mu(\alpha)-\int_{\Omega}r_{\alpha}f_{\alpha}(x)\tau_{\alpha}\,d\mu(\alpha)\right\|\leq
c\varepsilon^{\frac{1}{d}}\|x\|,\quad\forall x\in\mathcal{X},$

i.e.,

$\displaystyle\left\|\int_{\Delta}f_{\alpha}(\cdot)\tau_{\alpha}\,d\mu(\alpha)-\int_{\Omega}r_{\alpha}f_{\alpha}(\cdot)\tau_{\alpha}\,d\mu(\alpha)\right\|\leq
c\varepsilon^{\frac{1}{d}}.$

######  Conjecture 2.66.

(Continuous Akemann-Weaver conjecture for Banach spaces - strong form)  Let
$\mathcal{X}$ be a Banach space (finite or infinite dimensional). There exists
a universal constant $c$ such that the following holds. Let
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
be a continuous p-approximate Bessel family with Bessel bound 1 for
$\mathcal{X}$ satisfying

$\displaystyle\sup_{\alpha\in\Omega}\|f_{\alpha}\|^{q}\leq\varepsilon,\quad\sup_{\alpha\in\Omega}\|\tau_{\alpha}\|^{s}\leq\varepsilon,$

for some $\epsilon>0$, for some $q,s>0$ and the spectrum of the operator
$S_{f,\tau}:\mathcal{X}\ni
x\mapsto\int_{\Omega}f_{\alpha}(x)\tau_{\alpha}\,d\mu(\alpha)\in\mathcal{X}$
lies in $[0,\infty)$. Let $\\{r_{\alpha}\\}_{\alpha\in\Omega}$ be any family
in [0,1]. Then there exists a measurable subset $\Delta\subseteq\Omega$ and a
$d>0$ such that

$\displaystyle\left\|\int_{\Delta}f_{\alpha}(x)\tau_{\alpha}\,d\mu(\alpha)-\int_{\Omega}r_{\alpha}f_{\alpha}(x)\tau_{\alpha}\,d\mu(\alpha)\right\|\leq
c\varepsilon^{\frac{1}{d}}\|x\|,\quad\forall x\in\mathcal{X},$

i.e.,

$\displaystyle\left\|\int_{\Delta}f_{\alpha}(\cdot)\tau_{\alpha}\,d\mu(\alpha)-\int_{\Omega}r_{\alpha}f_{\alpha}(\cdot)\tau_{\alpha}\,d\mu(\alpha)\right\|\leq
c\varepsilon^{\frac{1}{d}}.$

######  Definition 2.67.

A continuous p-ASF
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
for a Banach space $\mathcal{X}$ is said to be p-scalable if there exist
families of scalars $\\{a_{\alpha}\\}_{\alpha\in\Omega}$,
$\\{b_{\alpha}\\}_{\alpha\in\Omega}$ such that

$\displaystyle(\\{a_{\alpha}f_{\alpha}\\}_{\alpha\in\Omega},\\{b_{\alpha}\tau_{\alpha}\\}_{\alpha\in\Omega})\text{
is a Parseval continuous p-ASF for }\mathcal{X}.$

######  Problem 2.68.

(Continuous scaling problem for Banach spaces) Classify continuous p-ASFs
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
for a Banach space $\mathcal{X}$ so that there are families of scalars
$\\{a_{\alpha}\\}_{\alpha\in\Omega}$, $\\{b_{\alpha}\\}_{\alpha\in\Omega}$
such that
$(\\{a_{\alpha}f_{\alpha}\\}_{\alpha\in\Omega},\\{b_{\alpha}\tau_{\alpha}\\}_{\alpha\in\Omega})$
is a Parseval continuous p-ASF for $\mathcal{X}$, i.e., continuous p-ASF
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
is p-scalable.

######  Problem 2.69.

(Continuous frame algorithm problems for Banach spaces) Whether there is a

  1. (i)

Duffin-Schaeffer algorithm for continuous p-ASFs?

  2. (ii)

Grochenig-Chebyshev algorithm for continuous p-ASFs?

  3. (iii)

Grochenig conjugate gradient algorithm for continuous p-ASFs?

######  Problem 2.70.

(Continuous dynamical sampling problem for Banach spaces) Let $\mathcal{X}$ be
a Banach space (finite or infinite dimensional). Let
$\mathbb{M}\subseteq\mathbb{N}$ and $\Delta$ be a measurable subset of
$\Omega$, $U:\mathcal{X}\to\mathcal{X}$, $V:\mathcal{X}\to\mathcal{X}$ be
bounded linear operators, $\\{\tau_{\alpha}\\}_{\alpha\in\Delta}$ be a family
in $\mathcal{X}$ and $\\{f_{\alpha}\\}_{\alpha\in\Delta}$ be a family in
$\mathcal{X}^{*}$. Find conditions on $\Delta,\mathbb{M}$, $U$ and $V$ such
that

  1. (i)

$(\\{f_{\alpha}U^{m}\\}_{\alpha\in\Delta,m\in\mathbb{M}},\\{V^{m}\tau_{\alpha}\\}_{\alpha\in\Delta,m\in\mathbb{M}})$
is a continuous p-approximate Riesz basis for $\mathcal{X}$.

  2. (ii)

$(\\{f_{\alpha}U^{m}\\}_{\alpha\in\Delta,m\in\mathbb{M}},\\{V^{m}\tau_{\alpha}\\}_{\alpha\in\Delta,m\in\mathbb{M}})$
is a continuous p-approximate Riesz family for $\mathcal{X}$.

  3. (iii)

$(\\{f_{\alpha}U^{m}\\}_{\alpha\in\Delta,m\in\mathbb{M}},\\{V^{m}\tau_{\alpha}\\}_{\alpha\in\Delta,m\in\mathbb{M}})$
is a continuous p-approximate Schauder frame for $\mathcal{X}$.

  4. (iv)

$(\\{f_{\alpha}U^{m}\\}_{\alpha\in\Delta,m\in\mathbb{M}},\\{V^{m}\tau_{\alpha}\\}_{\alpha\in\Delta,m\in\mathbb{M}})$
is a Parseval continuous p-approximate Schauder frame for $\mathcal{X}$.

  5. (v)

$(\\{f_{\alpha}U^{m}\\}_{\alpha\in\Delta,m\in\mathbb{M}},\\{V^{m}\tau_{\alpha}\\}_{\alpha\in\Delta,m\in\mathbb{M}})$
is a continuous p-approximate Bessel family for $\mathcal{X}$.

The set $\mathbb{M}$ can be replaced by subsets $\Sigma\subseteq\mathbb{C}$
whenever $U^{\alpha}$ and $V^{\alpha}$ make sense for $\alpha\in\Sigma$.

######  Definition 2.71.

  1. (i)

Let $\mathcal{X}$ be a Banach space (finite or infinite dimensional). A
continuous p-approximate Schauder frame
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
for $\mathcal{X}$ does continous phase retrieval for $\mathcal{X}$ if $x,y$
are any two elements of $\mathcal{X}$ and $\phi,\psi$ are any two elements of
$\mathcal{X}^{*}$ satisfying

$\displaystyle|\phi(\tau_{\alpha})|=|\psi(\tau_{\alpha})|\quad\text{and}\quad|f_{\alpha}(x)|=|f_{\alpha}(y)|,\quad\forall\alpha\in\Omega,$

then $y=\alpha x$, for some $|\alpha|=1$ and $\psi=\beta\phi$, for some
$|\beta|=1$.

  2. (ii)

Let $\mathcal{X}$ be a Banach space (finite or infinite dimensional). A
continuous p-approximate Schauder frame
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
for $\mathcal{X}$ does continuous norm retrieval for $\mathcal{X}$ if $x,y$
are any two elements of $\mathcal{X}$ and $\phi,\psi$ are any two elements of
$\mathcal{X}^{*}$ satisfying

$\displaystyle|\phi(\tau_{\alpha})|=|\psi(\tau_{\alpha})|\quad\text{and}\quad|f_{\alpha}(x)|=|f_{\alpha}(y)|,\quad\forall\alpha\in\Omega,$

then $\|y\|=\|x\|$ and $\|\phi\|=\|\psi\|$.

######  Problem 2.72.

  1. (i)

(Continuous phase retrieval problem for Banach spaces) For a given Banach
space $\mathcal{X}$, classify continuous p-approximate Schauder frames
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
for $\mathcal{X}$ which does continuous phase retrieval for $\mathcal{X}$.

  2. (ii)

(Continuous norm retrieval problem for Banach spaces) For a given Banach space
$\mathcal{X}$, classify continous p-approximate Schauder frames
$(\\{f_{\alpha}\\}_{\alpha\in\Omega},\\{\tau_{\alpha}\\}_{\alpha\in\Omega})$
for $\mathcal{X}$ which does continuous norm retrieval for $\mathcal{X}$.

We end by asking the following interesting question.

######  Problem 2.73.

If any of the Conjectures 2.13, 2.14, 2.15, 2.16, 2.19, 2.21, 2.22, 2.23,
2.24, 2.25, 2.26, 2.27, 2.28, 2.29, 2.30, 2.34, 2.35, 2.36, 2.38, 2.40 fails
in a given dimension, classify the spaces and/or p-approximate Schauder frames
and/or p-approximate Bessel sequences in which the corresponding conjecture
holds.

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