Variational characterisations of tight frames

Abstract

If $(f_j)_{j\in J}$ is a finite tight frame for $\mathscr {H}$, then (see Proposition 2.1). $\sum _{j\in J}\sum _{k\in J} |\langle f_j, f_k\rangle |^2 = {1\over d} \Bigl (\sum _{j\in J}\langle f_j, f_j\rangle \Bigr )^2, \qquad d=\dim (\mathscr {H}).$

Appendices

Notes

Neil Sloane has a webpage of putatively optimal real spherical t-designs

http://neilsloane.com/sphdesigns/ (see [HS96])

and the author has a similar list for real and complex spherical (t, t)-designs.

There is recent interest in complex spherical (t, t)-designs (weighted complex projective t-designs), see, e.g., [KR05], [RS07], [RS14]. Our unified treatment of real spherical half-designs of order 2t and complex (t, t)-designs in Theorem 6.7 was adapted from [Kön99], [Wal16] (also see [DHC12], [BH15]). There are other equivalences. Those involving the evaluation of Gegenbauer polynomials can be used to estimate the minimum number of vectors in a (t, t)-design (see [DGS77], [Hog89] for spherical t-designs). Some (t, t)-designs meeting these bounds (for spherical 2t-designs) are termed tight (this is not related to being a tight frame).

Thanks to Aidan Roy, Andreas Klappenecker and Wei–Hsuan Yu for insightful discussions about this chapter.

Exercises

6.1.

Let $A\in \mathbb {C}^{n\times n}$ be Hermitian, $A\ne 0$. Show that $\mathrm{{rank}}(A)\ge {\mathrm{{trace}}(A)^2\over \mathrm{{trace}}(A^2)}$ with equality if and only if $A=cUU^*$, $U=[u_1,\ldots , u_r]\in \mathbb {C}^{n\times r}$, with orthonormal columns.

Hint. If $\lambda _1,\ldots ,\lambda _r$ are the nonzero eigenvalues of A, then Cauchy–Schwarz gives

$$ (\mathrm{{trace}}(A))^2 =(\sum _{j=1}^n\lambda _j)^2 \le r\sum _{j=1}^r \lambda _j^2=r\;\mathrm{{trace}}(A^2). $$

6.2.

Use the generalised Welch bound (6.4) to prove Example 6.2, i.e., that

$$ \max _{j\ne k} |\langle f_j, f_k\rangle |^2 \ge { (\sum _j\Vert f_j\Vert )^2/d-\sum _j\Vert f_j\Vert ^4 \over n^2-n} > 0. $$

6.3.

$^\mathtt{m}$Do a numerical investigation of the variational inequality (6.4)? Does one get close to equality for large numbers of random (unit or otherwise) vectors?

6.4.

We may write the variational characterisation (6.4) for tight frames $(v_j)_{j\in J}$ as

$$ \sum _{\{j, k\}\subset J} \left( 2|\langle v_j, v_k\rangle |^2 +{\Vert v_j\Vert ^4+\Vert v_k\Vert ^4\over n-1}\right) = {1\over d} \sum _{\{j, k\}\subset J} \left( 2\Vert v_j\Vert ^2\Vert v_k\Vert ^2 +{\Vert v_j\Vert ^4+\Vert v_k\Vert ^4\over n-1}\right) . $$

We say that a frame $(v_j)_{j=1}^n$ for $\mathbb {C}^d$ is perfectly tight if equality holds for all pairs.

(a) Show that for $d=1$, every tight frame is perfectly tight.

(b) Show that for $d\ge 2$, every perfectly tight frame has nonzero vectors.

(c) Describe the equal-norm perfectly tight frames.

(d) Do there exist perfectly tight frames for $d\ge 2$ which do not have equal norms?

6.5.

Let $(f_j)$ be a sequence of $n\le d$ unit vectors in a $\mathscr {H}$, where $\dim (\mathscr {H})=d$. Show that

$$ \mathrm{{FP}}(f_1,\ldots ,f_n) := \sum _j \sum _k |\langle f_j, f_k\rangle |^2 $$

has a minimum value of n, which is attained if and only if $(f_j)$ is orthogonal.

6.6.

Show the normalised frame potential satisfies (6.7), i.e.,

$$ {1\over d} \le \hat{\mathrm{{FP}}}(f_1,\ldots , f_n)\le 1, $$

and that

(a) $\hat{\mathrm{{FP}}}(f_1,\ldots , f_n)$ equals ${1\over d}$ if and only if $(f_j)$ is a tight frame.

(b) $\hat{\mathrm{{FP}}}(f_1,\ldots , f_n)$ equals 1 if and only if $\mathrm{span}(f_j)$ is 1-dimensional.

Remark: For unit-norm vectors $\mathrm{{FP}}= n^2 \hat{\mathrm{{FP}}} |_{\mathbb {S}^n}$, and so this extends Theorem 6.2.

6.7.

Real spherical 2-designs. Let $\varPhi =\{\phi _1,\ldots ,\phi _n\}$ be unit vectors in $\mathbb {R}^d$.

(a) Suppose that $\varPhi$ is a real spherical 2-design. For $y\in \mathbb {R}^d$, let $p_y\in \varPi _2^\circ (\mathbb {R}^d)$ be given by

$$p_y (x) := |\langle y, x\rangle |^2 = (\langle y, x\rangle )^2. $$

Show that the integral of $p_y$ over the unit sphere $\mathbb {S}$ is $c\Vert y\Vert ^2$, with $c>0$ independent of y, and hence conclude that $\varPhi$ is a tight frame.

(b) Now suppose that $\varPhi$ is a tight frame. By considering the integral of $p_y$ above, or otherwise, show that it is a real spherical 2-design.

Remark: This is a special case of the key arguments of §6.8.

6.8.

Let $d\ge 2$. Show that there exists a real spherical 2-design of n points for $\mathbb {R}^d$ unless $n\le d$ or $n=d+2$ and n is odd (the existence part requires a construction). In particular, there is no real spherical 2-design of five points for $\mathbb {R}^3$.

Hint: For the construction use harmonic frames (see Chapter 11).

6.9.

Real spherical designs and Waring type formulas.

Here we consider the case of equality in Theorem 6.7 for $\mathscr {H}=\mathbb {R}^d$.

(a) Show that $c_t(d,\mathbb {R})\ge c_t(d,\mathbb {C})$, with strict inequality when $t>1$, $d>1$.

(b) A sequence $(f_j)$ of unit vectors in $\mathbb {R}^d$ satisfying (6.33) is, by definition, a real spherical half-design of order 2t. By substituting t for 2t, write down the equivalent conditions for being a real spherical half-design of order t (for t even) given by equality in (6.30), (6.31) and (6.35).

(c) Show that if $(f_j)$ is centrally symmetric, i.e., of the form $(\pm f_j)_{j=1}^{n/2}$, then the cubature rule (6.33) holds for all odd polynomials, i.e., $p\in \varPi _1^\circ \oplus \varPi _3^\circ \oplus \varPi _5^\circ \oplus \cdots$.

6.10.

The n equally spaced (unit) vectors in $\mathbb {R}^2$ are

$$ \varPhi =(v_j) = \bigl \{\bigl (\cos {2\pi \over n}j,\sin {2\pi \over n}j\bigr ) :j=0,\ldots , n-1\bigr \},$$

and the n equally spaced lines in $\mathbb {R}^2$ are

$$ \varPsi =(w_j)=\bigl \{\bigl (\cos {\pi \over n}j,\sin {\pi \over n}j\bigr ) :j=0,\ldots , n-1\bigr \}.$$

(a) Show that the n equally spaced vectors in $\mathbb {R}^2$ are a spherical $(n-1)$-design.

(b) Show that the $n=t+1$ equally spaced lines in $\mathbb {R}^2$ are a spherical half-design of order 2t, i.e., a (t, t)-design.

Hint: Use the integrals of (6.25) and (6.26).

6.11.

A SIC consists of $d^2$ equiangular unit vectors in $\mathbb {C}^d$, with a common angle $|\langle f_j, f_k\rangle |^2={1\over \sqrt{d+1}}$, $j\ne k$, and m MUBs are m orthonormal bases for $\mathbb {C}^d$, with the property that $|\langle f, g\rangle |= {1\over \sqrt{d}}$ for f and g from different bases (see §2.11).

(a) Show that a SIC is a (2, 2)-design.

(b) Show that m MUBs in $\mathbb {C}^d$ form a (2, 2)-design if and only if $m=d+1$.

(c) Show that $d+1$ MUBs in $\mathbb {C}^d$ form a (3, 3)-design if and only if $d=2$.

6.12.

There is a highly symmetric tight frame of 240 vectors for $\mathbb {C}^4$ given as an orbit of the Shephard–Todd group 32 (see §13.8), which gives 40 lines, since the group contains scalar multiplication by the sixth roots of unity. Take a vector from each line. This set $\varPhi$ of 40 vectors has the property that each is orthogonal to 12 others and makes an angle ${1\over \sqrt{3}}$ with 27 others. Show that $\varPhi$ is a (3, 3)-design for $\mathbb {C}^4$.

6.13.

Show that an equiangular tight frame $\varPhi =(v_j)$ of n unit vectors for $\mathbb {R}^d$ is a spherical (2, 2)-design for $\mathbb {R}^d$ if and only if $n={1\over 2}d(d+1)$.

Remark: Such equiangular lines are known to exist for $d=2,3,7,23$ (see §12.1).

6.14.

Let $\varPhi$ be the set of 240 vectors $v\in \mathbb {R}^8$ of with $\Vert v\Vert ^2=2$, and the form

$$\begin{aligned} \text{ type } \text{1: }&\qquad v_j\in \{\pm {1\over 2}\}\, \hbox {and}\, v\,\text {has}\,\text {an}\,\text {even}\,\text {number}\,\text {of}\,\text {positive entries}, \\\hbox {type 2:}&\qquad v_j\in \{0,\pm 1\} \quad \hbox {(such}\, v\,\hbox {have two nonzero entries)}. \end{aligned}$$

Since $\varPhi$ is centrally symmetric, it can be written $\varPhi =\varPhi _0\cup -\varPhi _0$.

(a) Show that $\varPhi _0$ is a spherical (3, 3)-design of 120 vectors for $\mathbb {R}^8$.

(b) Show that $\varPhi$ is a spherical 7-design of 240 vectors $\mathbb {R}^8$.

Remark: This $\varPhi$ (minimal vectors of the Korkin–Zolotarev lattice) is due to [KP11].

6.15.

Show that equality in (6.22) is equivalent to

(a) The generalised Plancherel identity

$$\langle x, y\rangle ^t = {{d+t-1\atopwithdelims ()t}\over \sum _{\ell =1}^n\Vert f_\ell \Vert ^{2t}} \sum _{j=1}^n \langle x,f_j\rangle ^t \langle f_j,y\rangle ^t, \qquad \forall x, y\in \mathscr {H}. $$

(b) The generalised Bessel identity

$$ \Vert x\Vert ^{2t} = {{d+t-1\atopwithdelims ()t}\over \sum _{\ell =1}^n\Vert f_\ell \Vert ^{2t}} \sum _{j=1}^n |\langle x, f_j\rangle |^{2t}, \qquad \forall x\in \mathscr {H}. $$

6.16.

Define $\xi \in \mathrm{{Sym}}^t(\mathscr {H})\otimes \mathrm{{Sym}}^t(\overline{\mathscr {H}})$ and $Q:\mathrm{{Sym}}^t(\mathscr {H})\rightarrow \mathrm{{Sym}}^t(\mathscr {H})$ by

$$\begin{aligned} \xi&:= \int _\mathbb {S}x^{\otimes t}\otimes \overline{x}^{\otimes t}\, d\sigma (x) -{1\over C} \sum _{j=1}^n f_j^{\otimes t}\otimes \overline{f_j}^{\otimes t}, \\ Q&:= \int _\mathbb {S}\langle \cdot , x^{\otimes t}\rangle x^{\otimes t}\, d\sigma (x) -{1\over C} \sum _{j=1}^n \langle \cdot , f_j^{\otimes t}\rangle f_j^{\otimes t}, \end{aligned}$$

where $C:=\sum _\ell \Vert f_\ell \Vert ^{2t}$. Show that

$$ \langle \xi ,\xi \rangle _\circ =\langle Q, Q\rangle _F={1\over C^2}\sum _j\sum _k |\langle f_j, f_k\rangle |^{2t}-c_t(d,\mathbb {F}), $$

where the apolar and Frobenius inner products are used, respectively.

6.17.

We consider the vector space $\varPi _{t, r}^\circ (\mathbb {C}^d)$ of polynomials $\mathbb {C}^d\rightarrow \mathbb {C}$ which are homogeneous of degree t in z and of degree r in $\overline{z}$, i.e.,

$$\begin{aligned} \varPi _{t, r}^\circ (\mathbb {C}^d):= \mathrm{span}\{z\mapsto z^\alpha \overline{z}^\beta :|\alpha |=t,|\beta |=r\}. \end{aligned}$$

(6.68)

This absolutely irreducible $\mathscr {U}$-invariant space is denoted by H(t, r) in §16.7. There is a natural identification $\mathrm{{Sym}}^t(\mathscr {H}^*)\otimes \mathrm{{Sym}}^r(\overline{\mathscr {H}}^*)\rightarrow \varPi _{t, r}^\circ (\mathscr {H})$ given by

$$\langle \cdot ,v\rangle ^{\otimes t}\otimes \langle \overline{\cdot },\overline{x}\rangle ^{\otimes r} \mapsto \langle \cdot ,v\rangle ^t\langle \overline{\cdot },\overline{x}\rangle ^r =\langle \cdot , v\rangle ^t\langle x,\cdot \rangle ^r. $$

For polynomials $p:\mathbb {C}^d\rightarrow \mathbb {C}$, we define an associated differential operator $p(\partial )$ by

$$\begin{aligned} p(\partial ):=\sum _{(\alpha ,\beta )} c_{\alpha \beta } \partial ^\alpha \overline{\partial }^\beta , \qquad \hbox {where}\ p(z)=\sum _{(\alpha ,\beta )} c_{\alpha \beta } z^\alpha \overline{z}^\beta , \end{aligned}$$

(6.69)

where $\partial$ and $\overline{\partial }$ are the Wirtinger complex differential operators given by

$$ \partial _j = {\partial \over \partial z_j} ={1\over 2}\left( {\partial \over \partial x_j}-i{\partial \over \partial y_j}\right) , \qquad \overline{\partial }_j = {\partial \over \partial \overline{z_j}} ={1\over 2}\left( {\partial \over \partial x_j}+i{\partial \over \partial y_j}\right) . $$

(a) Show that the monomials $z\mapsto z^\alpha \overline{z}^\beta$ in (6.68) are linearly independent, and so

$$\begin{aligned} \dim (\varPi _{t, r}^\circ (\mathbb {F}^d)) = {d+t-1\atopwithdelims ()t}{d+r-1\atopwithdelims ()r}. \end{aligned}$$

(6.70)

(b) By taking the apolar inner product on $\mathrm{{Sym}}^t(\mathscr {H}^*)\otimes \mathrm{{Sym}}^r(\overline{\mathscr {H}}^*)$, show that

$$\begin{aligned} \langle \langle \cdot ,v\rangle ^t\langle x,\cdot \rangle ^r, \langle \cdot ,w\rangle ^t\langle y,\cdot \rangle ^r \rangle _\circ := \langle w,v\rangle ^t \langle x, y\rangle ^r \end{aligned}$$

(6.71)

defines an inner product on $\varPi _{t, r}^\circ (\mathbb {C}^d)$.

(c) Show the Riesz representer of point evaluation at w is $\langle \cdot , w\rangle ^t\langle w,\cdot \rangle ^r$, i.e.,

$$\langle p,\langle \cdot ,w\rangle ^t\langle w,\cdot \rangle ^r\rangle _\circ = p(w), \qquad \forall p\in \varPi _{t, r}^\circ (\mathbb {C}^d),\quad \forall w\in \mathbb {C}^d.$$

(d) Use this to conclude that $\varPi _{t, r}^\circ (\mathbb {C}^d)$ is spanned by ridge functions, i.e.,

$$\begin{aligned} \varPi _{t,r}^\circ (\mathbb {C}^d) = P\,{:=}\,\mathrm{span}\{z \mapsto \langle z,v\rangle ^t\langle v, z\rangle ^r:v\in \mathbb {C}^d\}. \end{aligned}$$

(6.72)

In particular, $\varPi _{t, t}^\circ (\mathbb {C}^d)$ is spanned by ridge functions (plane waves), i.e.,

$$ \varPi _{t,t}^\circ (\Omega ) =\mathrm{span}\{z\mapsto |\langle z, v\rangle |^{2t}:v\in \Omega \}, \quad \hbox {where}\;\,\Omega =\mathbb {C}^d\hbox {or}\,\,\mathbb {S}_\mathbb {C}.$$

(e) With $p(\partial )$ given by (6.69) and $\tilde{q}(z)\,{:=}\,\overline{q(\overline{z})}$, show that

$$ \langle p, q\rangle _\circ = {1\over t!r!}p(\partial )\tilde{q}(0), \qquad \forall p,q\in \varPi _{t, r}^\circ (\mathbb {C}^d). $$

In particular, the monomials $z\mapsto z^\alpha \overline{z}^\beta$ in (6.68) form an orthogonal basis.

Remark: It follows from (6.25), that for $\varPi _{t, 0}^\circ (\mathbb {C}^d)$ and $\varPi _{0,t}^\circ (\mathbb {C}^d)$, one has

$$ \langle p, q\rangle _\circ = {t+d-1\atopwithdelims ()t} \int _{\mathbb {S}(\mathbb {C}^d)} p(z)\overline{q(z)}\, d\sigma (z). $$

Cubature rules which integrate $\varPi _{t, r}^\circ (\mathbb {C}^d)$, $(t, r)\in \mathscr {T}$ for some set of indices $\mathscr {T}$ are studied in [RS14], where they are called spherical $\mathscr {T}$-designs.

6.18.

Make the substitution (6.41) in Theorem 6.7 to obtain the weighted versions of the conditions (a)–(e).

6.19.

Let $\varDelta$ be the Laplacian for functions $\mathbb {F}^d\rightarrow \mathbb {F}$, i.e., for $\mathbb {F}$ equal $\mathbb {R}$ and $\mathbb {C}$

$$ \varDelta =\sum _{j=1}^d\Bigl ({\partial \over \partial x_j}\Bigr )^2, \qquad \varDelta =\sum _{j=1}^d\Bigl ({\partial \over \partial x_j}\Bigr )^2 +\sum _{j=1}^d\Bigl ({\partial \over \partial y_j}\Bigr )^2 = 4 \sum _{j=1}^d \partial _j \overline{\partial _j}. $$

(a) Take the Laplacian with respect to $x\in \mathbb {R}^d$ to get

$$ \varDelta (\Vert x\Vert ^{2t}) = 2t(d+2t-2)\Vert x\Vert ^{2t-2}, \qquad \varDelta (\langle x, v\rangle ^{2t}) = 2t(2t-1) \langle x, v\rangle ^{2t-2}\Vert v\Vert ^2. $$

(b) Take the Laplacian with respect to $z\in \mathbb {R}^d$ to get

$$ \varDelta (\Vert z\Vert ^{2t}) = 4t(d+t-1) \Vert z\Vert ^{2t-2}, \qquad \varDelta (|\langle z, v\rangle |^{2t}) = 4t^2|\langle z, v\rangle |^{2(t-1)} \Vert v\Vert ^{2}. $$

(c) Using (a) and (b), apply the Laplacian to the Bessel identity (6.31).

6.20.

Show that if $(v_j)$ and $(w_k)$ are spherical (t, t)-designs for $\mathbb {F}^d$, with

$$ \sum _j \Vert v_j\Vert ^{2t} = \sum _k \Vert w_k\Vert ^{2t}, $$

then their union $(v_j)\cup (w_k)$ is a spherical (t, t)-design for $\mathbb {F}^d$.

6.21.

Use the generalised Bessel identity (6.31) to show that the minimal number n of vectors in a weighted (t, t)-design satisfies

$$ n \le \dim (\varPi _{t, t}^\circ (\mathbb {F}^d)) = {\left\{ \begin{array}{ll} {d+t-1\atopwithdelims ()t}^2, &{} \mathbb {F}=\mathbb {C}\\ {d+2t-1\atopwithdelims ()2t}, &{} \mathbb {F}=\mathbb {R}\\ \end{array}\right. } =O(d^{2t}),\qquad d\rightarrow \infty . $$

6.22.

Use the generalised Plancherel identity (6.32) to show that the number n of vectors in a weighted (t, t)-design for $\mathbb {F}^d$ satisfies

$$ n \ge \dim \bigl (\varPi _t^\circ (\mathbb {F}^d)\bigr ) = {t+d-1\atopwithdelims ()d-1} = O(d^t), \qquad t\rightarrow \infty . $$

6.23.

Suppose that $(f_j)$ is a tight frame for $\mathbb {F}^d$, i.e., is a (1, 1)-design. Show the condition which ensures it comes from a (t, t)-design, as per Proposition 6.2 is that

$$ \sum _{j=1}^n \sum _{k=1}^n { |\langle f_j, f_k\rangle |^{2t} \over \Vert f_j\Vert ^{2t-2} \Vert f_k\Vert ^{2t-2}} = c_t(d,\mathbb {F}) \Bigl (\sum _{\ell =1}^n \Vert f_\ell \Vert ^2\Bigr )^2. $$

6.24.

The lines $\{\mathbb {F}x:x\ne 0\}$ in $\mathscr {H}=\mathbb {F}^d$ are in 1–1 correspondence with the rank-one orthogonal projections, i.e., points in the projective space $\mathbb {F}P^{d-1}$, via

$$ \mathbb {F}x \longleftrightarrow P_x:={\langle \cdot ,x\rangle \over \langle x, x\rangle }x. $$

(a) Show that the Frobenius inner product between orthogonal projections given by unit vectors is

$$ \langle P_x,P_y\rangle =|\langle x, y\rangle |^2 = \langle P_y, P_x\rangle . $$

(b) Show that the metric on $\mathbb {F}P^{d-1}$ given by the Frobenius inner product is

$$ \rho (P, Q) = \sqrt{2}\sqrt{1-\langle P,Q\rangle }, \qquad P, Q\in \mathbb {F}P^{d-1}. $$

(c) Show that in terms of lines this metric is

$$ \rho (\mathbb {F}x,\mathbb {F}y) = \sqrt{2} \sqrt{1-\Bigl |\langle {x\over \Vert x\Vert },{y\over \Vert y\Vert }\rangle \Bigr |^2 }. $$

(d) Show that the set of lines can also be embedded into the real vector space of traceless Hermitian matrices (with the Frobenius norm), via

$$ \mathbb {F}x \mapsto P_x-{1\over d}I. $$

6.25.

Polynomials on projective spaces.

Determine the vector space of polynomials $p:\mathbb {F}^d\rightarrow \mathbb {F}$ whose value at each $z\ne 0$ depends only on the 1-dimensional subspace given by z, i.e.,

$$ p(z)=p(a z), \qquad \forall z\in \mathbb {F}^d,\quad \forall a\in \mathbb {F},\ |a|=1. $$

6.26.

Let $F_{d+1}$ be the field of order $d+1$, where $d+1$ is a prime power. Suppose that y is a generator for the multiplicative group $F_{d+1}^*$. Show that $f:\mathbb {Z}_d\rightarrow \mathbb {Z}_{d+1}:y\mapsto y^j$ is a 1-uniform function.

6.27.

Let $f:G\rightarrow H$ be a map between finite abelian groups, with $|G|=d$. Show that if f is 1-uniform, then

$$ w+x-y-z=0,\qquad f(w)+f(x)-f(y)-f(z) = 0 $$

has exactly $d(2d+1)$ solutions in $(w,x,y, z)\in G^4$.

6.28.

Equilibrium with respect to the frame force in $\mathbb {R}^d$ and $\mathbb {C}^d$.

For unit vectors $a, b\in \mathbb {C}^d$, the frame force of b on a can be extended

$$ \mathrm{FF}(a,b):=\langle a, b\rangle (a-b), $$

though this is no longer a central force. The frame force between orthogonal vectors and between coincident vectors is zero. The effective frame force $\mathrm{{EFF}}(a, b)$ of b on a is the component of the frame force $v=\mathrm{FF}(a, b)$ which is orthogonal to a.

(a) Calculate $\mathrm{{EFF}}(a, b)$ for $a, b\in \mathbb {S}$.

(b) Show that if $(a_j)_{j=1}^n$ is a minimiser of the frame potential, then the total effective frame force on each $a_j$ is zero, i.e.,

$$ \sum _k \mathrm{{EFF}}(a_j,a_k)= \sum _{k\ne j} \mathrm{{EFF}}(a_j, a_k) = 0. $$

6.29.

Let $V=[v_{\alpha \beta }]=[v_1,\ldots , v_n]$ and $p, g:\mathbb {F}^{d\times n}\rightarrow \mathbb {R}$ be given by

$$ p(V) := \sum _j\sum _k |\langle v_j, v_k\rangle |^{2t}, \qquad g(V):= \sum _\ell \Vert v_\ell \Vert ^{2t}. $$

With $\nabla f$ given by (6.59) when $\mathbb {F}=\mathbb {C}$, show the $\beta$-columns of $\nabla p(V)$ and $\nabla g(V)$ are

$$ 4 t\sum _j |\langle v_j, v_\beta \rangle |^{2(t-1)} \langle v_\beta , v_j\rangle v_j, \qquad 2t \Vert v_\beta \Vert ^{2(t-1)} v_{\beta }. $$

6.30.

Here we calculate the Hessian matrix $H_f$ of f of the functions $p, g:\mathbb {F}^{d\times n}\rightarrow \mathbb {R}$ given by (6.65). Let X be the real variables with some ordering, i.e.,

$$ X=\{x_{\alpha \beta }\}\cup \{y_{\alpha \beta }\}\quad \hbox {for}\ \mathbb {F}=\mathbb {C}, \qquad X=\{x_{\alpha \beta }\} \quad \hbox {for} \ \mathbb {F}=\mathbb {R}. $$

Then $H_f(V)$ is the $X\times X$ real symmetric matrix with (r, s)-entry given by

$$ H_f(V)_{rs} = {\partial ^2 f\over \partial r\partial s}(V). $$

Find the Hessian matrix of p, g, and hence the function f given by (6.66).

6.31.

A sequence $(f_j)_{j=1}^n$ is a finite tight frame for $\mathbb {F}^d$ if and only if

$$ g(x) := {d\over \sum _k\Vert f_k\Vert ^2} \sum _{j=1}^n |\langle x, f_j\rangle |^2 {1\over \Vert x\Vert ^2} = 1, \qquad \forall x\ne 0. $$

(a) Show for a general frame that g can take values which are $>1$ and $<1$. Thus the obvious generalisation of Bessel’s inequality does not hold. There are various generalisations in the literature. We now develop a few.

(b) By Cauchy–Schwarz, $|\sum _j\overline{c_j}\langle x, f_j\rangle |=|\langle x,\sum _j {c_j} f_j\rangle |^2 \le \Vert x\Vert ^2 \Vert \sum _j {c_j} f_j\Vert ^2$. Use the triangle and Cauchy–Schwarz inequalities to show Pečarić’s inequality

$$ \Bigl | \sum _{j=1}^n \overline{c_j} \langle x, f_j\rangle \Bigr |^2 \le \Vert x\Vert ^2 \sum _{j=1}^n|c_j|^2 \sum _{k=1}^n|\langle f_j, f_k\rangle |. $$

(c) From this deduce Selberg’s inequality

$$ \sum _{j=1}^n{|\langle x, f_j\rangle |^2\over \sum _{\ell =1}^n|\langle f_\ell , f_j\rangle |} \le \Vert x\Vert ^2. $$

(d) From Selberg’s inequality, deduce Bombieri’s inequality

$$\sum _{j=1}^n|\langle x, f_j\rangle |^2\le \Vert x\Vert ^2 \max _{1\le \ell \le n} \sum _{j=1}^n |\langle f_\ell , f_j\rangle |.$$

(e) Vary the argument of (b) to show the inequality.

$$ \sum _{j=1}^n|\langle x, f_j\rangle |^2\le \Vert x\Vert ^2 \Bigl ( \sum _{j=1}^n\sum _{k=1}^n |\langle f_j, f_k\rangle |^2\Bigr )^{1\over 2}. $$