Equiangular Finite Unit-Norm Tight Systems in Complex \( \ell_\infty^2 \) and \( \ell_1^2 \)

1. Introduction

Equiangular tight frames in Hilbert spaces are closely connected with the Welch bound, optimal correlation, and Grassmannian packings [5,6]; related equiangularity questions for dual pairs of Hilbert-space frames were studied in [3]. In Banach spaces, the vector and its norming functional are separate objects, so the natural analogue is a system of pairs $({\tau }_j,f_j)\in X\times X^{*}$ whose frame operator is

$$S_{f,\tau }x={\displaystyle \sum _{j=1}^n}{f}_j\left(x\right){\tau }_j.$$

This point of view belongs to the theory of Schauder and approximate Schauder frames; see, for example, [1]. Chávez-Domínguez, Freeman, and Kornelson developed the frame potential for finite-dimensional Banach spaces and studied finite unit-norm tight frames in this setting [2]. In particular, their results imply that every complex Banach space with a normalized 1-unconditional basis admits finite unit-norm tight frames of every length at least its dimension. Thus ${\ell }_1^2$ and ${\ell }_{\infty }^2$ have tight systems of arbitrary length; the obstruction proved here comes specifically from equiangularity.

Krishna established Banach-space analogues of Welch bounds and posed the following existence problem [4]. Given a d-dimensional Banach space X and $\gamma >0$, determine the integers n for which there are vectors ${\tau }_j\in X$ and functionals $f_j\in X^{*}$ satisfying

$$\parallel {\tau }_j\parallel =\parallel f_j\parallel =f_j\left({\tau }_j\right)=1,x={\displaystyle \frac{d}{n}}{\displaystyle \sum _{j=1}^n}{f}_j\left(x\right){\tau }_j,\left|f_j\left({\tau }_k\right)\right|=\gamma (j\ne k).$$

(1)

We use the parameter convention of [4], so that $\gamma$ denotes the off-diagonal modulus $|f_j\left({\tau }_k\right)|$. In [4], equiangularity is instead parametrized by $|f_j\left({\tau }_k\right){|}^2$; under that convention, the interior parameter $1/2$ obtained below is recorded as $1/4$.

To the best of our knowledge, the purpose of this paper is to solve the specialization of (1) completely for $X={\ell }_{\infty }^2\left(\mathbb{C}\right)$ and, by duality, for $X={\ell }_1^2\left(\mathbb{C}\right)$. The answer is particularly rigid:

$$\begin{array}{|c|}\hline \mathrm{existence}\mathrm{occurs}\mathrm{exactly}\mathrm{for}(n,\gamma )=(3,1/2)\mathrm{or}\gamma =1\mathrm{and}n\ge 4\mathrm{even}.\\ \hline\end{array}$$

(2)

For $0<\gamma <1$, all systems form a single geometric family based on the third roots of unity. At $\gamma =1$, all norming functionals are supported on coordinate axes, and tightness becomes a pair of zero-sum conditions on unimodular numbers.

Throughout the paper all functionals are complex-linear. We identify ${\left({\ell }_{\infty }^2\right)}^{*}$ with ${\ell }_1^2$ by writing $f(z_1,z_2)=\alpha z_1+\beta z_2$, so that ${\parallel f\parallel }_1=\left|\alpha \right|+\left|\beta \right|$. We also write $\mathbb{T}=\{z\in \mathbb{C}:|z|=1\}$.

2. Normalized Equiangular Systems and Norming Functionals

Definition 1. 

Let X be a finite-dimensional Banach space. A family ${({\tau }_j,f_j)}_{j=1}^n\subset X\times X^{*}$ is anormalized $\gamma$-equiangular system if

$$\parallel {\tau }_j\parallel =\parallel f_j\parallel =f_j\left({\tau }_j\right)=1,\left|f_j\left({\tau }_k\right)\right|=\gamma (j\ne k).$$

(3)

It is tight if

$${\displaystyle \sum _{j=1}^n}{\tau }_j\otimes f_j={\displaystyle \frac{n}{dimX}}{I}_X,(\tau \otimes f)\left(x\right)=f\left(x\right)\tau .$$

(4)

The following elementary equality-case lemma drives the classification.

Lemma 1 

(Norming functionals of the bidisc). Let $\tau =(u,v)\in {\ell }_{\infty }^2\left(\mathbb{C}\right)$ and $f(z_1,z_2)=\alpha z_1+\beta z_2\in {\ell }_1^2\left(\mathbb{C}\right)$ satisfy ${\parallel \tau \parallel }_{\infty }={\parallel f\parallel }_1=f\left(\tau \right)=1$.

(i)

If $\left|u\right|=1>\left|v\right|$, then $f(z_1,z_2)=\overline{u}{z}_1$.

(ii)

If $\left|v\right|=1>\left|u\right|$, then $f(z_1,z_2)=\overline{v}{z}_2$.

(iii)

If $\left|u\right|=\left|v\right|=1$, then there is an $a\in [0,1]$ such that

$$f(z_1,z_2)=a\overline{u}{z}_1+(1-a)\overline{v}{z}_2.$$

Proof. 

The chain

$$1=f\left(\tau \right)\le \left|\alpha \right|\left|u\right|+\left|\beta \right|\left|v\right|\le \left|\alpha \right|+\left|\beta \right|=1$$

is an equality throughout. If one coordinate has modulus strictly less than one, the coefficient of that coordinate must vanish, which proves (i) and (ii). In the remaining case, equality in the triangle inequality implies that $\alpha u$ and $\beta v$ are nonnegative real numbers. Taking $a=\left|\alpha \right|$ proves (iii). □

3. A Cardinality Bound below the Endpoint

The next theorem requires no reconstruction or invertibility assumption.

Theorem 1. 

Let $0\le \gamma <1$. Every normalized γ-equiangular system in ${\ell }_{\infty }^2\left(\mathbb{C}\right)$ has length $n\le 3$.

Proof. 

Suppose, for contradiction, that $n\ge 4$, and write ${\tau }_j=(u_j,v_j)$.

First suppose that some ${\tau }_i$ has a unique coordinate of maximal modulus. After interchanging the coordinates, assume $|u_i|=1>|v_i|$. Lemma 1 gives $f_i(z_1,z_2)=\overline{u_i}{z}_1$. Hence $|u_k|=|f_i\left({\tau }_k\right)|=\gamma <1$ for every $k\ne i$, and therefore $|v_k|=1$. Lemma 1 now gives $f_k(z_1,z_2)=\overline{v_k}{z}_2$ for every $k\ne i$. Taking two distinct indices $k,l\ne i$ yields $|f_k\left({\tau }_l\right)|=1$, a contradiction.

Consequently

$$|u_j|=|v_j|=1(1\le j\le n).$$

(5)

By Lemma 1,

$$f_j(z_1,z_2)=a_j\overline{u_j}{z}_1+(1-a_j)\overline{v_j}{z}_2$$

(6)

for some $a_j\in [0,1]$. Since $n\ge 2$ and $\gamma <1$, neither endpoint $a_j=0$ nor $a_j=1$ is possible. Indeed, if $a_j=1$, then for every $k\ne j$ we have $|u_k|=|f_j\left({\tau }_k\right)|=\gamma$, contradicting (5); the case $a_j=0$ is identical with the coordinates interchanged. Thus $0<a_j<1$.

Set $r_j=u_j/v_j\in \mathbb{T}$. The ratios are distinct: if $r_k=r_l$ for $k\ne l$, then ${\tau }_l$ is a unimodular multiple of ${\tau }_k$, and $|f_k\left({\tau }_l\right)|=1$. For fixed j and every $k\ne j$,

$$\begin{array}{cc}\hfill {\gamma }^2& ={\left|(1-a_j)+a_j\overline{r_j}{r}_k\right|}^2\hfill \\ \hfill & =a_j^2+{(1-a_j)}^2+2a_j(1-a_j)Re\left(\overline{r_j}{r}_k\right).\hfill \end{array}$$

(7)

Because $0<a_j<1$, the real part in (7) is independent of $k\ne j$. A line of prescribed real part meets the unit circle in at most two points. Thus there are at most two choices for $r_k$ with $k\ne j$, contradicting $n-1\ge 3$. □

Remark 1. 

The bound is sharp: the tight systems classified in Theorem 2 have three members. The proof also shows that if one vector has a unique maximal coordinate, then $n\le 2$.

4. The Tight Classification for $0<\gamma <1$

In dimension two, tightness means

$$S:={\displaystyle \sum _{j=1}^n}{\tau }_j\otimes f_j={\displaystyle \frac{n}{2}}I.$$

(8)

Theorem 2. 

Let $0<\gamma <1$. A normalized γ-equiangular tight system in ${\ell }_{\infty }^2\left(\mathbb{C}\right)$ exists if and only if

$$n=3,\gamma ={\displaystyle \frac{1}{2}}.$$

More precisely, after a permutation of the indices, every such system has the following form. Choose $\rho ,v_0,v_1,v_2\in \mathbb{T}$, let $\omega =e^{2\pi i/3}$, and set

$${\tau }_j=v_j(\rho {\omega }^j,1),f_j(z_1,z_2)={\displaystyle \frac{\overline{v_j}}{2}}\left(\overline{\rho }{\omega }^{-j}{z}_1+z_2\right),j=0,1,2.$$

(9)

Conversely, every choice of the four unimodular parameters in (9) gives such a system.

Proof. 

The rank of the left-hand side of (8) is at most n, so $n\ne 1$. If $n=2$, define the synthesis and analysis operators by

$$T(c_1,c_2)=c_1{\tau }_1+c_2{\tau }_2,Fx=(f_1\left(x\right),f_2\left(x\right)).$$

Tightness gives $TF=I_X$. Both spaces have dimension two, hence $F=T^{-1}$ and $FT=I_{{\mathbb{C}}^2}$. It follows that $f_j\left({\tau }_k\right)={\delta }_{jk}$, contradicting $\gamma >0$. Therefore Theorem 1 gives $n=3$.

The unique-maximal-coordinate case in the proof of Theorem 1 has length at most two, so all three vectors satisfy (5). Retain the notation $r_j=u_j/v_j\in \mathbb{T}$. For each fixed j, equation (7) says that the other two ratios are reflections of one another across the radial line through 0 and $r_j$.

Rotate the three ratios so that $r_0=1$. Reflection at $r_0$ allows us to write the other two as $e^{i\theta }$ and $e^{-i\theta }$ with $0<\theta <\pi$. Applying (7) at $e^{i\theta }$ gives

$$cos\theta =cos\left(2\theta \right).$$

Thus $2{cos}^2\theta -cos\theta -1=0$. Distinctness excludes $cos\theta =1$, so $cos\theta =-1/2$. After a permutation and undoing the rotation,

$$r_j=\rho {\omega }^j,j=0,1,2,$$

(10)

for some $\rho \in \mathbb{T}$.

Write ${\tau }_j=v_j(r_j,1)$. Formula (6) becomes

$$f_j(z_1,z_2)=\overline{v_j}\left(a_j\overline{r_j}{z}_1+(1-a_j)z_2\right),0<a_j<1.$$

Relative to the standard basis,

$${\tau }_j\otimes f_j=\left(\begin{array}{cc}{a}_j& (1-a_j)r_j\\ a_j\overline{r_j}& 1-a_j\end{array}\right).$$

(11)

The upper-left and lower-left entries of (8) yield

$${\displaystyle \sum _{j=0}^2}{a}_j={\displaystyle \frac{3}{2}},{\displaystyle \sum _{j=0}^2}{a}_j\overline{r_j}=0.$$

(12)

Using (10), the second equality is ${\sum }_j{a}_j{\omega }^{-j}=0$. Since the $a_j$ are real, its real and imaginary parts imply $a_0=a_1=a_2$. The first equality in (12) then gives $a_j=1/2$ for every j. Consequently, for $j\ne k$,

$$|f_j\left({\tau }_k\right)|={\displaystyle \frac{1}{2}}\left|1+{\omega }^{k-j}\right|={\displaystyle \frac{1}{2}}.$$

This proves necessity and gives (9).

Conversely, direct substitution gives $\parallel {\tau }_j{\parallel }_{\infty }={\parallel f_j\parallel }_1=f_j\left({\tau }_j\right)=1$ and common off-diagonal modulus $1/2$. Finally, $1+\omega +{\omega }^2=0$ and (11) give ${\sum }_j{\tau }_j\otimes f_j=(3/2)I$. □

5. Complete Classification of the Endpoint $\gamma =1$

At the endpoint, arbitrarily long systems become possible, but tightness still imposes a complete and elementary classification. Here and throughout, “equiangular” is used in the precise sense of (3). In particular, at the degenerate endpoint $\gamma =1$, distinct vectors or distinct one-dimensional subspaces are not required, in accordance with [4].

Theorem 3. 

A normalized 1-equiangular tight system ${({\tau }_j,f_j)}_{j=1}^n$ exists in ${\ell }_{\infty }^2\left(\mathbb{C}\right)$ if and only if $n\ge 4$ is even.

More precisely, every such system is obtained as follows. Partition the index set into sets A and B, choose $u_j,v_j\in \mathbb{T}$, and put

$$\begin{array}{ccc}j\in A:\hfill & {\tau }_j=(u_j,v_j),\hfill & f_j(z_1,z_2)=\overline{u_j}{z}_1,\hfill \\ j\in B:\hfill & {\tau }_j=(u_j,v_j),\hfill & f_j(z_1,z_2)=\overline{v_j}{z}_2.\hfill \end{array}$$

(13)

The necessary and sufficient constraints are

$$\left|A\right|=\left|B\right|={\displaystyle \frac{n}{2}},\sum _{j\in A}{v}_j\overline{u_j}=0,\sum _{j\in B}{u}_j\overline{v_j}=0.$$

(14)

Proof. 

Write ${\tau }_k=(u_k,v_k)$ and $f_j(z_1,z_2)={\alpha }_j{z}_1+{\beta }_j{z}_2$. We first show that every $f_j$ is supported on a single coordinate. Suppose instead that ${\alpha }_i{\beta }_i\ne 0$ for some i. For every k, equality in

$$1=|f_i\left({\tau }_k\right)|\le |{\alpha }_i\left|\right|u_k|+|{\beta }_i\left|\right|v_k|\le |{\alpha }_i|+|{\beta }_i|=1$$

forces $|u_k|=|v_k|=1$ and forces ${\alpha }_i{u}_k$ and ${\beta }_i{v}_k$ to have the same argument. Hence the ratio $u_k/v_k$ is independent of k. All the vectors ${\tau }_k$ are then collinear, so the range of ${\sum }_k{\tau }_k\otimes f_k$ is one-dimensional, contradicting tightness.

Thus every $f_j$ is supported on the first or second coordinate. Both types must occur, since otherwise the frame operator cannot have rank two. Choose i of the first type and l of the second type. Normalization gives $|u_i|=1$ and $|v_l|=1$, while the off-diagonal identities $|f_i\left({\tau }_k\right)|=1$ for $k\ne i$ and $|f_l\left({\tau }_k\right)|=1$ for $k\ne l$ give, respectively, $|u_k|=1$ for $k\ne i$ and $|v_k|=1$ for $k\ne l$. Consequently $|u_k|=|v_k|=1$ for every k, and normalization gives precisely the two forms in (13).

Summing the associated rank-one matrices gives

$${\displaystyle \sum _{j=1}^n}{\tau }_j\otimes f_j=\left(\begin{array}{cc}\left|A\right|& {\displaystyle \sum _{j\in B}{u}_j\overline{v_j}}\\ {\displaystyle \sum _{j\in A}{v}_j\overline{u_j}}& \left|B\right|\end{array}\right).$$

This matrix equals $(n/2)I$ exactly when (14) holds. In particular n is even. If $n=2$, each zero sum would contain only one unimodular term, which is impossible. Hence $n\ge 4$.

Conversely, (13) and (14) immediately give normalization, common off-diagonal modulus one, and tightness. □

Corollary 1 

(Explicit endpoint construction). Let $n=2m$ with $m\ge 2$, and set $\zeta =e^{2\pi i/m}$. For $k=0,\dots ,m-1$, define

$$\begin{array}{cccc}\hfill {\tau }_k& =(1,{\zeta }^k),\hfill & \hfill f_k(z_1,z_2)& =z_1,\hfill \\ \hfill {\tau }_{m+k}& =({\zeta }^k,1),\hfill & \hfill f_{m+k}(z_1,z_2)& =z_2.\hfill \end{array}$$

Then the system is normalized and 1-equiangular, and

$${\displaystyle \sum _{j=0}^{2m-1}}{\tau }_j\otimes f_j=mI={\displaystyle \frac{n}{2}}I.$$

Proof. 

All coordinates are unimodular, so every off-diagonal value has modulus one. The tightness identity follows from ${\sum }_{k=0}^{m-1}{\zeta }^k=0$. □

6. Duality and the Complete Answer in ${\ell }_1^2$

Proposition 1 

(Duality principle). Let X be finite-dimensional and let ${({\tau }_j,f_j)}_{j=1}^n\subset X\times X^{*}$ be normalized and γ-equiangular. Under the canonical embedding $J:X\to X^{**}$, the swapped system

$${(f_j,J{\tau }_j)}_{j=1}^n\subset X^{*}\times X^{**}$$

is also normalized and γ-equiangular. Its frame operator is the adjoint of the original frame operator. In particular, tightness is preserved.

Proof. 

The diagonal pairing is $\left(J{\tau }_j\right)\left(f_j\right)=f_j\left({\tau }_j\right)=1$, while the off-diagonal values are $\left(J{\tau }_j\right)\left(f_k\right)=f_k\left({\tau }_j\right)$, which transposes the modulus matrix. Moreover,

$$\sum _j{f}_j\otimes J{\tau }_j={\left(\sum _j{\tau }_j\otimes f_j\right)}^{*}.$$

□

Since ${\left({\ell }_1^2\right)}^{*}={\ell }_{\infty }^2$, the preceding principle transfers all the classifications above and is involutive in finite dimensions.

Corollary 2. 

Every normalized γ-equiangular tight system in ${\ell }_1^2\left(\mathbb{C}\right)$ with $0<\gamma <1$ has $n=3$ and $\gamma =1/2$. Explicitly, after a permutation, all such systems are

$${\tau }_j={\displaystyle \frac{\overline{v_j}}{2}}(\overline{\rho }{\omega }^{-j},1),f_j(z_1,z_2)=v_j(\rho {\omega }^j{z}_1+z_2),j=0,1,2,$$

(15)

where $\rho ,v_0,v_1,v_2\in \mathbb{T}$ and $\omega =e^{2\pi i/3}$.

Corollary 3. 

Every normalized 1-equiangular tight system in ${\ell }_1^2\left(\mathbb{C}\right)$ is the dual of a system in Theorem 3. Equivalently, with the notation of that theorem, its pairs have the form

$$\begin{array}{ccc}j\in A:\hfill & {\tau }_j=(\overline{u_j},0),\hfill & f_j(z_1,z_2)=u_j{z}_1+v_j{z}_2,\hfill \\ j\in B:\hfill & {\tau }_j=(0,\overline{v_j}),\hfill & f_j(z_1,z_2)=u_j{z}_1+v_j{z}_2,\hfill \end{array}$$

where $u_j,v_j\in \mathbb{T}$ and (14) holds. Such a system exists exactly for even $n\ge 4$.

Corollary 4 

(Complete solution for the two spaces). Let X be either ${\ell }_{\infty }^2\left(\mathbb{C}\right)$ or ${\ell }_1^2\left(\mathbb{C}\right)$, and let $\gamma >0$. A system satisfying Krishna’s conditions (1) exists if and only if exactly one of the following holds:

(i)

$n=3$ and $\gamma =1/2$;

(ii)

$n\ge 4$ is even and $\gamma =1$.

Proof. 

Normalization gives $|f_j\left({\tau }_k\right)|\le \parallel f_j\parallel \parallel {\tau }_k\parallel =1$, so $\gamma >1$ is impossible. The interval $0<\gamma <1$ is settled by Theorem 2 and Corollary 2; the endpoint follows from Theorem 3, Corollary 1, and Corollary 3. □

Remark 2. 

Problem 3 of [4] assumes $\gamma >0$. If $\gamma =0$ is admitted, the normalized coordinate Auerbach basis gives the additional tight example $n=2$; this case is therefore outside the scope of Corollary 4.

7. Concluding Remarks

The argument separates two geometric regimes. Below the endpoint, a norming functional with two nonzero coordinates converts equiangularity into a line-circle intersection condition, which bounds the length by three. Tightness then selects the equilateral triple and fixes its weights. At the endpoint, equality in the duality estimate forces coordinate support, and the classification reduces to two balanced zero-sum conditions. This contrast explains why ordinary finite unit-norm tight frames can have arbitrary length in these spaces while equiangular tight systems have only the lengths listed in (2).

To the best of our knowledge, the corresponding classification for other two-dimensional polygonal norms, and for higher-dimensional ${\ell }_1^d$ and ${\ell }_{\infty }^d$, remains open.