Tight p-fusion frames

Introduction

In modern signal processing, basis-like systems are applied to derive stable and redundant signal representations. Frames are basis-like systems that span a vector space but allow for linear dependency, that can be used to reduce noise, find sparse representations, or obtain other desirable features unavailable with orthonormal bases . In fusion frame theory as introduced in , see also , the signal is projected onto a collection of linear subspaces that can represent, for instance, sensors in a network or nodes in a computing cluster . To obtain a signal reconstruction that is robust against noise and data loss, the subspaces are usually chosen in some redundant fashion and, as such, fusion frames are tightly connected to coding theory .

Tight fusion frames provide a direct reconstruction formula, and can be characterized as the minimizers of the fusion frame potential. The error caused from the loss of one or two subspaces within a tight fusion frame is minimized for equidimensional subspaces that satisfy the simplex bound with equality . The simplex bound, as derived in , is an extremal estimate on the maximum of the inner products $\langle P_{V},P_{W}\rangle:=\operatorname{trace}(P_{V}P_{W})$ between the projectors associated to equidimensional linear subspaces $V$ and $W$. Equality in this bound implies that the subspaces are equiangular, meaning that the inner products between distinct pairs take the same value.

We introduce the notions of $p$-fusion frames and tight $p$-fusion frames, where $p\geq 1$ is an integer. These notions generalize the notion of (tight) fusion frames corresponding to the case $p=1$. For subspaces of equal dimension, we apply methods from harmonic analysis on Grassmann spaces in order to analyse these objects. In particular we characterize tight $p$-fusion frames by the evaluation of certain multivariate Jacobi polynomials at the principal angles between subspaces. Moreover we relate them to cubature formulas in Grassmann space.

A general framework for cubatures in polynomial spaces is proposed in . Designs for the Grassmann space, i.e., cubatures with constant weights, have been introduced and studied in . We prove that cubatures of strength $2p$ can be characterized as the minimizers of the $p$-fusion frame potential. The notions of tight $p$-fusion frames and of cubatures of strength $2p$ coincide for $p=1$; however the latter is stronger than the former for $p\geq 2$.

The outline is as follows: In Section 2, we list the basic properties of fusion frames. We recall the simplex bound in Section 3.1 and derive the generalized simplex bound in Section 3.2. An upper bound on the number of equiangular subspaces is given in Section 3.3. We introduce $p$-fusion frames in Section 4. The relations between tight $p$-fusion frames and cubatures of strength $2p$ are investigated in Section 5. Constructions of tight $p$-fusion frames are discussed in Section 6. Section 7 contains a lower bound on the $p$-fusion frame potential that does not require the subspaces to be equidimensional.

Fusion frames

Let $\{V_{j}\}_{j=1}^{n}\subset\operatorname{\mathcal{G}}_{d}$ and let $\{\omega_{j}\}_{j=1}^{n}$ be a collection of positive weights. Then $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is called a fusion frame if there are positive constants $A$ and $B$ such that

If $A=B$, then $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is called a tight fusion frame.

where \big{(}\bigoplus_{j=1}^{n}V_{j}\big{)}_{\omega} is the space $\bigoplus_{j=1}^{n}V_{j}$ endowed with the inner product $\langle\{f_{j}\}_{j=1}^{n},\{g_{j}\}_{j=1}^{n}\rangle_{\omega}:=\sum_{j=1}^{n}\omega_{j}\langle f_{j},g_{j}\rangle$. Its adjoint is the synthesis operator

and the fusion frame operator is defined by

The simplex bound and equiangular fusion frames

Our goal in this section is to give lower bounds on $\max_{i\neq j}(\langle P_{V_{i}},P_{V_{j}}\rangle)$. We start to review the known results in the case of subspaces of equal dimension.

The chordal distance $d_{c}(V,W)$, for $V,W\in\mathcal{G}_{k,d}$, was introduced in and is defined by

where $\theta_{1},\ldots,\theta_{k}$ are the principal angles between $V$ and $W$, cf. . If $\{V_{j}\}_{j=1}^{n}\subset\mathcal{G}_{k,d}$, then the simplex bound as derived in yields

and equality requires $n\leq\binom{d+1}{2}$. Of course, in view of (7), the above simplex bound is a lower bound for $\max_{i\neq j}(\langle P_{V_{i}},P_{V_{j}}\rangle)$. The following result is proven in :

If $\{V_{j}\}_{j=1}^{n}\subset\mathcal{G}_{k,d}$ is an equidistance fusion frame, i.e. if $d_{c}(V_{i},V_{j})$ is independent of $i\neq j$, then it is tight if and only if it satisfies the simplex bound (8) with equality.

We shall extend the simplex bound (8) and the above theorem to collections of weighted subspaces that do not all have the same dimension.

2 The simplex bound for subspaces of arbitrary dimension

When the subspaces do not have the same dimension, we replace the notion of subspaces being equidistant with the notion of equiangular subspaces:

Let $\{V_{j}\}_{j=1}^{n}$ be a collection in $\mathcal{G}_{d}$ and let $\{\omega_{j}\}_{j=1}^{n}$ be positive weights. We then call both, $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ and $\{V_{j}\}_{j=1}^{n}$, equiangular if $\langle P_{V_{i}},P_{V_{j}}\rangle$ does not depend on $i\neq j$.

If all the subspaces $V_{j}$ are of dimension $1$, then our definition of $\{V_{j}\}_{j=1}^{n}$ being equiangular coincides with the classical notion of equiangular lines. If all subspaces $V_{j}$ are of dimension $k$, being equiangular amounts to being equidistant with respect to the chordal distance. However, we remark that being equiangular in our sense does not mean that the $k$-tuples of principal angles between the pairs $(V_{i},V_{j})$ are the same (unless $k=1$).

The proof of the simplex bound (8) in heavily relies on the embedding of $\mathcal{G}_{k,d}$ into a higher dimensional sphere. For subspaces that are not equidimensional, we cannot use this embedding. Instead, we use the $p$-fusion frame potential.

The $p$-fusion frame potential of the collection of weighted subspaces $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is defined for $1\leq p<\infty$ by:

Note that the $1$-fusion frame potential $\operatorname{FFP}(\{(V_{j},\omega_{j}),1\}_{j=1}^{n})=\operatorname{trace}(S^{2})$, where $S$ is the fusion frame operator, has already been considered in .

We can now derive a new weighted simplex bound for collections of subspaces that are not necessarily equidimensional.

Given positive weights $\{\omega_{j}\}_{j=1}^{n}$, if $\{V_{j}\}_{j=1}^{n}\subset\mathcal{G}_{d}$ and $m=\sum_{j=1}^{n}\omega_{j}\dim(V_{j})$, then the following points hold:

If $p=1$, then equality holds if and only if $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is a tight fusion frame. If $1<p<\infty$, then equality holds if and only if $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is an equiangular tight fusion frame.

Equality holds if and only if $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is an equiangular tight fusion frame.

Part 1) for $p=1$ has already been derived in . For $1<p<\infty$, we take the $p$-th root and only consider the terms $i\neq j$. We then see that (10) is equivalent to

By applying the Hölder inequality, we obtain, for $1<p\leq\infty$ and $1=\frac{1}{p}+\frac{1}{q}$,

In the last step we have used the Cauchy-Schwartz inequality for $(\lambda_{k})_{k=1}^{d}$ and the constant sequence. The inequality (12) turns into an equality if and only if $\{V_{j}\}_{j=1}^{n}$ are equiangular. The Cauchy-Schwartz inequality turns into an equality if and only if $S$ is a multiple of the identity. ∎

By applying (7), we observe that (11) is equivalent to the simplex bound (8) if all the subspaces have the same dimension and the weights are constant.

If $\{V_{j}\}_{j=1}^{n}\subset\mathcal{G}_{k,d}$ and $\{\omega_{j}\}_{j=1}^{n}$ are positive weights, then $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is an equiangular tight fusion frame if and only if the weights are constant and $\langle P_{V_{i}},P_{V_{j}}\rangle=\frac{k(nk-d)}{(n-1)d}$, for all $i\neq j$.

Without loss of generality, we can assume that $\sum_{j=1}^{n}\omega_{j}=1$. For $\dim(V_{j})=k$, $j=1,\ldots,n$, the right-hand side of (11) equals $k(\frac{\frac{k}{d}-1}{1-\sum_{j=1}^{n}\omega^{2}_{j}}+1)$ and is maximized if and only if $\omega_{j}=1/n$, $j=1,\ldots,n$. By applying Theorem 3.4, we can conclude the proof. ∎

3 The maximal number of equiangular subspaces

To match the generalized simplex bound of Theorem 3.4 with equality, the subspaces need to be equiangular. It is natural to ask how large a collection of equiangular subspaces can be. The classical Gerzon upper bound $n\leq\binom{d+1}{2}$ for equiangular lines was extended to equiangular subspaces of equal dimension $k$ in [3, Theorem 3.6]. In the present section, we prove that this upper bound extends further to equiangular subspaces of arbitrary dimensions.

If $\{V_{j}\}_{j=1}^{n}$ is a collection of equiangular pairwise distinct subspaces in $\mathcal{G}_{d}$, then $n\leq\binom{d+1}{2}$.

Let $\alpha=\langle P_{V_{i}},P_{V_{j}}\rangle$, for all $i\neq j$. We split the proof into two cases.

Case 1) Suppose that there exists $i$ such that $\alpha=\dim(V_{i})$. Without loss of generality, we assume that $i=1$. A short computation yields that $\langle P_{V_{1}},P_{V}\rangle\leq\dim(V_{1})$, for all $V\in\mathcal{G}_{d}$, and equality holds if and only if $V_{1}$ is contained in $V$. Thus, we have $V_{1}\subset V_{j}$, for all $j=2,\ldots,n$. Let $W_{j}$ be the orthogonal complement of $V_{1}$ in $V_{j}$, i.e., $V_{j}=V_{1}\oplus W_{j}$, for all $j=2,\ldots,n$. It can be checked that the equiangularity implies that the collection $\{W_{j}\}_{j=2}^{n}$ is pairwise orthogonal. Since $\{V_{j}\}_{j=1}^{n}$ are pairwise distinct, none of the $\{W_{j}\}_{j=2}^{n}$ can be zero. Thus, $n-1\leq d$ must hold, which implies $n\leq\binom{d+1}{2}$, for $d\geq 2$.

We can check by induction and elimination that $\operatorname{Gram}$ has full rank. Therefore, $\{P_{V_{j}}\}_{j=1}^{n}$ is linearly independent. Since the real vector space of self-adjoint matrices is $\binom{d+1}{2}$-dimensional, we must have $n\leq\binom{d+1}{2}$. ∎

The following examples form equiangular subspaces:

Let $d$ be a prime which is either $3$ or congruent to $-1$ modulo $8$. A collection of $\binom{d+1}{2}$ subspaces in $\mathcal{G}_{\frac{d-1}{2},d}$ satisfying the simplex bound was constructed in .

In , codes in Grassmann spaces were constructed from $2$-transitive groups. By construction, these codes are equiangular.

Tight p𝑝p-fusion frames

The notion of (tight) fusion frames generalizes in a natural way when squares are replaced by $2p$-powers for $p$ a positive integer:

If the weights are all equal to $1$, then we suppress them in our notation and simply write $\{V_{j}\}_{j=1}^{n}$ for the $p$-fusion frame. If $A=B$, then $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is called a tight $p$-fusion frame. If, in addition, all the subspaces are one-dimensional, then $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is simply called a tight $p$-frame.

Of course, tight $1$-fusion frames are tight fusion frames. Also, it is clear from the definition that the union of tight $p$-fusion frames is again a tight $p$-fusion frame.

The defining property of a $p$-fusion frame can be rephrased in the following way:

where $m_{k}=\sum_{\dim(V_{j})=k}\omega_{j}$. Equality holds for tight $p$-fusion frames. Since $T_{1,k,d}(1)=\frac{k}{d}$, cf. , the frame bounds of a fusion frame satisfy $A\leq\frac{m}{d}\leq B$, where $m=\sum_{j=1}^{n}\omega_{j}\dim(V_{j})$.

It should also be mentioned that reweighting of a tight $p$-fusion frame leads to tight $p^{\prime}$-fusion frames for the entire range $1\leq p^{\prime}\leq p$:

We introduce the Laplace operator $\Delta=\sum_{i=1}^{d}\frac{\partial^{2}}{\partial x_{i}^{2}}$. In spherical coordinates, for $x\neq 0$, we use the parametrization $x=r\varphi$, for $r>0$ and $\varphi\in S^{d-1}$, so that the function $f(x)=\|x\|^{2p}$ is constant in $\varphi$. Thus, we have $\Delta f=r^{1-d}\partial_{r}(r^{d-1}\partial_{r}f)$, which yields

More generally, for a subspace $V$, we obtain

Applying $\Delta$ to both sides of the identity $\sum_{j=1}^{n}\omega_{j}\|P_{V_{j}}(x)\|^{2p}=A\|x\|^{2p}$, we obtain

proving that $\{(V_{j},\omega_{j}(p-1+\dim(V_{j})/2))\}_{j=1}^{n}$ is a $(p-1)$-tight fusion frame. ∎

Iteration of Theorem 4.2 yields that if $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is a tight $p$-fusion frame, then $\{(V_{j},\omega^{\prime}_{j})\}_{j=1}^{n}$ is a tight fusion frame, where $\omega^{\prime}_{j}=\omega_{j}\prod_{l=1}^{p-1}(l+\dim(V_{j})/2)$.

Equidimensional tight p𝑝p-fusion frames, cubature formulas and the p𝑝p-fusion frame potential

In this section, we assume that the subspaces $V_{j}$ have the same dimension $k$. Using tools from harmonic analysis on the Grassmann manifold ${\mathcal{G}}_{k,d}$, the tight $p$-fusion frames can be characterized in terms of the principal angles of the pairs of subspaces. The same holds for minimizers of the $p$-fusion frame potential, where we minimize over all collections of $k$-dimensional subspaces whose weights add up to one. We shall recognize in these minimizers the cubatures for the Grassmann space, also called Grassmann designs in the case of constant weights. It will turn out that the minimizers of the $p$-fusion frame potential are tight $p$-fusion frames, while the converse holds only in the cases $p=1$ or $k=1$.

The use of harmonic analysis, namely the irreducible decomposition of $L^{2}$ and the associated zonal spherical functions, is standard in the study of designs in homogeneous spaces. The unit sphere of Euclidean space served as a model for many other spaces . We refer to for a general framework for cubature formulas in polynomial spaces and to for the notion of designs in Grassmann spaces (see also ).

The next proposition shows that, after possibly a change from $\{V_{j}\}$ to $\{V_{j}^{\perp}\}$, the condition $k\leq d/2$ can be fulfilled. The assumption that $k\leq d/2$ will be conveniently followed in the remaining of this section.

If $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is a $p$-tight fusion frame of equal dimension $k$, then:

$\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is a $p^{\prime}$-tight fusion frame for all $1\leq p^{\prime}\leq p$.

$\{(V_{j}^{\perp},\omega_{j})\}_{j=1}^{n}$ is also a $p$-tight fusion frame.

Part (1) follows from Theorem 4.2 by putting $(p-1+k/2)^{-1}$ into the fusion frame constant. For (2), we observe that $\|x\|^{2}=\|P_{V_{j}}(x)\|^{2}+\|P_{{V_{j}}^{\perp}}(x)\|^{2}$, so

where the second last equality, follows from the property that $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is a $k$-tight fusion frame for all $1\leq k\leq p$, and insures the existence of some constants $A_{k}>0$ such that $\sum_{j=1}^{n}\omega_{j}\|P_{V_{j}}(x)\|^{2k}=A_{k}\|x\|^{2k}$. Since $V_{1}^{\perp}$ is not empty, $\sum_{k=0}^{p}(-1)^{k}\binom{p}{k}A_{k}>0$, so that $\{(V^{\perp}_{j},\omega_{j})\}_{j=1}^{n}$ is a tight $p$-fusion frame. ∎

It follows from (16) that the constant $A_{p}$ in the characteristic property of tight $p$-fusion frames $\sum_{j=1}^{n}\omega_{j}\|P_{V_{j}}(x)\|^{2p}=A_{p}\|x\|^{2p}$ equals $A_{p}=T_{1,k,d}(p)\sum_{j=1}^{n}\omega_{j}$. Applying the Laplace operator $p$ times leads to another, more explicit, formula:

where we employ the standard notation $(a)_{p}=a(a+1)\cdots(a+p-1)$.

2 Characterization of tight p𝑝p-fusion frames by means of principal angles

Here $2\mu=(2\mu_{1},\dots,2\mu_{d})$ runs over the partitions with even parts. The subspace

coincides with the space of polynomial functions on ${\mathcal{G}}_{k,d}$ of degree bounded by $2p$. We also introduce the subspace

so that the orthogonal complement of $\operatorname{Pol}^{1}_{\leq 2p}({\mathcal{G}}_{k,d})$ in $\operatorname{Pol}_{\leq 2p}({\mathcal{G}}_{k,d})$ is the direct sum of all $H_{k,d}^{2\mu}$, such that $2\leq l(\mu)\leq k$ and $\deg(\mu)\leq p$.

To every subspace $H_{k,d}^{2\mu}$ is associated a polynomial $P_{\mu}(y_{1},\dots,y_{k})$ which is symmetric in the variables $y_{i}$, of degree equal to $\deg(\mu)$, satisfying $P_{\mu}(1,\dots,1)=1$, and such that $V\mapsto P_{\mu}(y_{1}(V,W),\dots,y_{k}(V,W))$ belongs to $H_{k,d}^{2\mu}$. In fact, these two last properties uniquely determine $P_{\mu}$. For example, $P_{(0)}=1$ and $P_{(1)}=(y_{1}+\dots+y_{k})-k^{2}/d$ up to a multiplicative constant. These polynomials are called the zonal spherical polynomials of the Grassmann manifold. They were calculated in , where it is shown that they belong to the family of multivariate Jacobi polynomials. They do depend on the parameters $k$ and $d$, although those parameters are not involved in our notation, see also .

Moreover, the functions $(V,W)\mapsto P_{\mu}(y_{1}(V,W),\dots,y_{k}(V,W))$ are positive definite functions on ${\mathcal{G}}_{k,d}$, meaning that, for all $n\geq 1$ and all $\{V_{j}\}_{j=1}^{n}\subset{\mathcal{G}}_{k,d}$, the matrix $(P_{\mu}(y_{1}(V_{i},V_{j}),\dots,y_{k}(V_{i},V_{j})))_{1\leq i,j\leq n}$ is positive semidefinite. As a consequence, we have:

Taking $\mu=(1)$, the inequality (21) becomes

so we already see here a connection with Theorem 3.4. Now we are in the position to characterize the tight $p$-fusion frames.

The following properties are equivalent for $\{(V_{j},\omega_{j})\}_{j=1}^{n}$, where $\{V_{j}\}_{j=1}^{n}\subset\mathcal{G}_{k,d}$ and $\sum_{j=1}^{n}\omega_{j}=1$:

$\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is a tight $p$-fusion frame.

For all $f\in\operatorname{Pol}_{\leq 2p}^{1}({\mathcal{G}}_{k,d})$,

Theorem 5.3(4) is the characterization of tight $p$-fusion frames we were aiming at, involving only the principal angles of the pairs $(V_{i},V_{j})$.

3 Cubature formulas as minimizers of the p𝑝p-fusion frame potential

In this section, we define cubature formulas on the Grassmann space and discuss their relations to the $p$-fusion frame potential.

Let $\{V_{j}\}_{j=1}^{n}$ be a finite subset of ${\mathcal{G}}_{k,d}$ and let $\{\omega_{j}\}_{j=1}^{n}$ be a collection of positive weights, with $\sum_{j=1}^{n}\omega_{j}=1$. Then $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is called a cubature formula of strength $2p$ (or for short a cubature of strength $2p$) if:

We say that $\{V_{j}\}_{j=1}^{n}$ is a design of strength $2p$ or a $2p$-design if $\{(V_{j},1/n)\}_{j=1}^{n}$ is a cubature of strength $2p$.

If $n=\binom{d+1}{2}$ holds in Theorem 3.6 and all subspaces have the same dimension, then it follows from [3, Theorem 3.6] that $\{V_{j}\}_{j=1}^{n}$ is a $4$-design.

Cubatures can be characterized in a similar way as tight $p$-fusion frames with the help of the zonal spherical polynomials of the Grassmann manifold, and they also match lower bounds on the weighted $p$-potential. These results extend straightforwardly similar characterizations of designs on the unit sphere and in Grassmann spaces, see . For preparation and extending (14), we define, for $1\leq k,l\leq d-1$,

To shorten notation, let $T_{k,d}(p):=T_{k,k,d}(p)$.

Let $\{V_{j}\}_{j=1}^{n}\subset\mathcal{G}_{k,d}$ and $\sum_{j=1}^{n}\omega_{j}=1$. We then have

Moreover, the following properties are equivalent:

$\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is a cubature of strength $2p$ in ${\mathcal{G}}_{k,d}$.

For all $\mu$, $1\leq\deg(\mu)\leq p$, for all $f\in H_{k,d}^{2\mu}$, $\sum_{j=1}^{n}\omega_{j}f(V_{j})=0$.

For all $\mu$, $1\leq\deg(\mu)\leq p$, $\sum_{i,j=1}^{n}\omega_{i}\omega_{j}P_{\mu}(y_{1}(V_{i},V_{j}),\dots,y_{k}(V_{i},V_{j}))=0$.

$\operatorname{FFP}(\{(V_{j},\omega_{j})\}_{j=1}^{n},p)=T_{k,d}(p)$.

There is a constant $A>0$ such that $\sum_{j=1}^{n}\omega_{j}\langle P_{W},P_{V_{j}}\rangle^{p}=A$, for all $W\in\mathcal{G}_{k,d}$.

There are constants $A_{l}>0$, $l=1,\ldots,k$, such that $\sum_{j=1}^{n}\omega_{j}\langle P_{W_{l}},P_{V_{j}}\rangle^{p}=A_{l}$, for all $W_{l}\in\mathcal{G}_{l,d}$.

where $\lambda_{\mu}\geq 0$ for all $\mu$. This important result goes back to . Since $(V,W)\mapsto P_{\mu}(y_{1}(V,W),\ldots,y_{k}(V,W))$ is positive definite, $F-\lambda_{0}$ is positive definite too, so

for all $\{V_{j}\}_{j=1}^{n}\subset{\mathcal{G}}_{k,d}$. For $F=s^{p}$, we have $\lambda_{0}=T_{k,d}(p)$.

The equivalences between (1)-(4) have already been proven in for constant weights. Incorporating weights is straightforward so we omit it here.

(1)$\Rightarrow$(5): The mapping $V\mapsto\langle P_{W},P_{V}\rangle^{p}$ is an element in $\operatorname{Pol}_{\leq 2p}({\mathcal{G}}_{k,d})$, for all $W\in\mathcal{G}_{l,d}$ and $l=1,\ldots,k$. For $W\in\mathcal{G}_{k,d}$, the property (24) implies

The implication (1)$\Rightarrow$(6) follows in the same way using $A_{l}=T_{l,k,d}(p)$. Since (6)$\Rightarrow$(5) is obvious, we only need to verify (5)$\Rightarrow$(1): As for (16), we can compute $A=T_{k,d}(p)$. Therefore, we derive $\sum_{i,j}\omega_{i}\omega_{j}\langle P_{V_{i}},P_{V_{j}}\rangle^{p}=T_{k,d}(p)$, which implies (1).

Since $\operatorname{Pol}^{1}_{\leq 2}(\mathcal{G}_{k,d})\subset\operatorname{Pol}_{\leq 2}(\mathcal{G}_{k,d})$, every cubature of strength $2p$ is a tight $p$-fusion frame according to Theorem 5.3. In particular, the designs of strength $2p$ in Grassmann spaces provide an interesting subclass of tight $p$-fusion frames.

We have already seen that $T_{k,d}(1)=k^{2}/d$. In [1, Remark 6.4], an explicit expression of $T_{k,d}(p)$ is given for $p=2,3$. In general, $T_{k,d}(p)$ can be calculated from the expression of $(y_{1}+\dots+y_{k})^{p}$ as a linear combination of the zonal polynomials $P_{\mu}$, cf. [1, Lemma 6.2].

For $p=1$, Theorems 5.7 and 3.4 show that the tight fusion frames of equal dimension $k$ are exactly the cubatures of strength $2$ of ${\mathcal{G}}_{k,d}$.

It is natural to ask for the existence of the objects discussed in this section, namely tight $p$-fusion frames and cubatures, and beyond existence, it is also desirable to discuss the size $n$ of these objects as a function of $p$ and $d$. In these directions, the following results are borrowed from :

There exists a tight $p$-fusion frame $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ with $n\leq\dim(\operatorname{Pol}_{\leq 2p}^{1}({\mathcal{G}}_{k,d}))-1=\binom{2p+d-1}{d-1}-1$.

There exists a cubature $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ of strength $2p$ such that the inequality $n\leq\dim(\operatorname{Pol}_{\leq 2p}(\mathcal{G}_{k,d}))-1$ holds.

If $\{(V_{j},\omega_{j})\}_{j=1}^{n}$ is a cubature of strength $4p$, then $n\geq\dim(\operatorname{Pol}_{\leq 2p}(\mathcal{G}_{k,d}))$.

It should be noted that the existence statements above are non constructive by nature. In the next section, some explicit constructions are discussed.

We aim to minimize the $p$-fusion frame potential among all collections of $k$-dimensional linear subspaces whose weights add up to one. Proposition 5.9 and Theorem 5.7 ensure that there exists a minimizer of cardinality less than $\dim(\operatorname{Pol}_{\leq 2p}(\mathcal{G}_{k,d}))$.

Some constructions of tight p𝑝p-fusion frames

We remark first that, if an orbit $G.V:=\{g(V):g\in G\}$ is a tight $p$-fusion frame, then it satisfies the property (13) with equal weights. To see this, one has to sum up the conditions (13) for $x=g(y)$, when $g$ runs in $G$.

For all $1\leq k<d$ and all $V\in{\mathcal{G}}_{k,d}$, the collection $G.V:=\{g(V):g\in G\}$ is a tight $p$-fusion frame.

Obviously $(X_{1}^{2}+\dots+X_{d}^{2})^{p}$ is invariant by the orthogonal group so the condition (2) means that $G$ does not afford other invariant polynomials than the ones which are invariant by the full orthogonal group.

It is a straightforward adaptation of the proof of [1, Theorem 4.1]. In view of (2) in Proposition 5.1 we can assume $k\leq d/2$. We recall that

2 Tight p𝑝p-fusion frames from p𝑝p-designs in projective spaces

for all functions $f\in\operatorname{Pol}_{\leq p}({\mathcal{P}}(K^{d}))$, where $\sigma$ denotes the normalized Lebesgue measure on ${\mathcal{P}}(K^{d})$ and $\operatorname{Pol}_{\leq p}({\mathcal{P}}(K^{d}))$ is a subspace of functions on ${\mathcal{P}}(K^{d})$, which are polynomial of degree bounded by $p$ in some reasonable sense.

Many examples of projective $p$-designs for $p\leq 5$ are described in . Most of them are related to complex or quaternionic reflection groups.

3 Extension and refinement of tight p𝑝p-fusion frames

If $\operatorname{\mathcal{F}}_{0}$ and $\operatorname{\mathcal{F}}_{1}$ are tight $p$-fusion frames, then $\operatorname{\mathcal{F}}$ is also a tight $p$-fusion frame.

Also $\operatorname{\mathcal{F}}_{1}$ is a tight $p$-fusion frame so, for some $A_{\operatorname{\mathcal{F}}_{1}}$,

Let $x_{i}:=P_{W_{i}}(x)$. Then, $P_{V_{i,j}}(x)=P_{V_{i,j}}(x_{i})$. So,

An unrestricted lower bound for the p𝑝p-fusion frame potential

Given positive weights $\{\omega_{j}\}_{j=1}^{n}$, let $\{V_{j}\}_{j=1}^{n}\subset\mathcal{G}_{d}$ and $p\geq 1$ be an integer. Define $m_{k}:=\sum_{\dim(V_{i})=k}\omega_{i}$. If $M=(m_{1},\ldots,m_{d-1})$, then

We recall that a function $F\in L^{2}(\mathcal{G}_{d}\times\mathcal{G}_{d})$ is said to be positive definite if, for all $f\in L^{2}(\mathcal{G}_{d})$,

Let $s^{p}(V,W):=\langle P_{V},P_{W}\rangle^{p}$. Then, $s^{p}=F_{0}+F_{1}$, with $F_{0}\in L\otimes L$, $F_{1}\in L^{\perp}\otimes L^{\perp}$, and $F_{0}$, $F_{1}$ are positive definite functions.

for some constant $\gamma_{k}$. Because $f_{1}\in L^{\perp}$, the right hand side is zero. Therefore, we have $s^{p}=F_{0}+F_{1}\in(L\otimes L)\oplus(L^{\perp}\otimes L^{\perp})$.

The functions $s^{p}(V,W)$ are positive definite on $\operatorname{\mathcal{G}}_{d}$. So, for all $f=f_{0}+f_{1}\in L\oplus L^{\perp}$, we obtain

Hence, $F_{0}$ is positive definite. A similar argument shows that $F_{1}$ is also positive definite. ∎

Since the set of functions $\{\operatorname{\mathbf{1}}_{\operatorname{\mathcal{G}}_{k,d}}(V)\operatorname{\mathbf{1}}_{\operatorname{\mathcal{G}}_{l,d}}(W),1\leq k,l\leq d-1\}$ is an orthonormal basis of $L\otimes L$, we have

Since $s^{p}-F_{0}$ is positive definite, we obtain $\sum_{i,j}\omega_{i}(s^{p}(V_{i},V_{j})-F_{0}(V_{i},V_{j}))\omega_{j}\geq 0$, and this yields

The measures $d\sigma_{l}d\sigma_{k}$ induce measures $d\lambda_{l,k}$ on $^{l}$ for the variables $y_{i}=\cos^{2}(\theta_{i}(V,W))$, which are computed in . Up to a multiplicative constant, one has, for $l\leq k\leq d/2$,

Note that $\lambda_{1,k}$ has already occurred in the proof of Theorem 5.3. The zonal spherical intertwining polynomials for the Grassmann spaces $\operatorname{\mathcal{G}}_{l,d}$ and $\operatorname{\mathcal{G}}_{k,d}$, denoted $P_{\mu}^{l,k}(y_{1},\dots,y_{k})$, are symmetric polynomials, and are orthogonal for the measure $\lambda_{l,k}$ (). They are indexed by the partitions $\mu$ of length at most $l$. These polynomials already occurred in Section 5 for $(l,k)=(k,k)$ and for $(l,k)=(1,k)$.

Since $\langle P_{V},P_{W}\rangle^{p}=(y_{1}+\dots+y_{k})^{p}$, the number $T_{l,k,d}(p)$ corresponds to the constant term in the expression of $(y_{1}+\dots+y_{k})^{p}$ as a linear combination of these polynomials. For example, $P_{(1)}^{l,k}=(y_{1}+\dots+y_{l})-lk/d$ (up to a multiplicative factor) and thus $T_{l,k,d}(1)=lk/d$. Knowledge of these polynomials allows to give an explicit expression for $T_{l,k,d}(p)$. We observe that, because these polynomials have rational coefficients, $T_{l,k,d}(p)$ is a rational function of $l,k,p$.

Thus, we recover the lower bound for $p=1$ in Theorem 3.4.

For $p\geq 1$, if $\{V_{j}\}_{j=1}^{n}\subset{\mathcal{G}}_{k,d}$, then $M=(0,\dots,m_{k}=\sum_{j=1}^{n}\omega_{j},0,\dots)$ and MT(p)M^{\top}=\big{(}\sum_{j=1}^{n}\omega_{j}\big{)}^{2}T_{k,d}(p) so that we recover (26).

Acknowledgements

The authors would like to thank the referees for their suggestions that improved the presentation of this paper. M. E. is supported by the NIH/DFG Research Career Transition Awards Program (EH 405/1-1/575910).

References