Continuous tight frames for finite dimensional spaces

Abstract

The tight frame expansion for n equally spaced unit vectors in $\mathbb {R}^2$ is $f = {2\over n} \sum \limits _{j=1}^n \langle f, u_j\rangle u_j, \quad \forall f\in \mathbb {R}^2, \quad u_j:=\begin{pmatrix}\cos {2\pi j\over n}\\ \sin {2\pi j\over n}\end{pmatrix}.$

Appendices

Notes

Continuous tight frames were introduced and studied in detail in [AAG93]. A good account of zonal harmonics is given in [ABR01]. The section on the orthogonal polynomials for a radially symmetric weight was adapted from [Wal09]. I thank Tom Ter Elst and Joey Iverson for many useful discussions about this chapter.

Exercises

16.1.

Let $(f_j)_{j\in J}$ be a generalised frame with respect to $\mu$ for a Hilbert space ${\mathscr {H}}$ (possibly infinite dimensional).

(a) Show that

$$ g \mapsto \int _J \overline{\langle f,f_j\rangle }\langle g,f_j\rangle \,d\mu (j) =:\langle g, Sf\rangle $$

defines a bounded linear functional on ${\mathscr {H}}$ and denote its Riesz representer by

$$ Sf=\int _J \langle f,f_j\rangle f_j\, d\mu (j). $$

(b) Show this defines a linear map $S:{\mathscr {H}}\rightarrow {\mathscr {H}}$, with

$$ A\Vert f\Vert ^2\le \langle Sf, f\rangle \le B\Vert f\Vert ^2, \qquad \forall f\in {\mathscr {H}}. $$

(c) Show that S has a bounded inverse, and

$$\begin{aligned} f&= \int _J \langle f, S^{-1} f_j\rangle f_j\,d\mu (j) =\int _J \langle f, f_j\rangle S^{-1} f_j\,d\mu (j) \\&=\int _J \langle f, S^{-1/2} f_j\rangle S^{-1/2}f_j\, d\mu (j), \qquad \forall f\in {\mathscr {H}}. \end{aligned}$$

(d) Show that the synthesis and analysis operators (see §2.4) can be generalised to

$$ V:\ell _2(\mu )\rightarrow {\mathscr {H}}:a\mapsto \int _J a_jf_j\, d\mu (j), $$

$$ V^*: {\mathscr {H}}\rightarrow \ell _2(\mu ) :f\mapsto (\langle f, f_j\rangle )_{j\in J}, $$

where Va is defined as the Riesz representer of

$$ f\mapsto \int _J\langle f,a_jf_j\rangle \, d\mu (j) =:\langle f, Va\rangle . $$

Remark: Here $S=VV^*$, and one can define the Gramian as $V^*V:\ell _2(\mu )\rightarrow \ell _2(\mu )$.

16.2.

Let $(f_j)_{j\in J}$ be a generalised frame with respect to $\mu$ for a d-dimensional Hilbert space ${\mathscr {H}}$. Show that the variational characterisation (Theorem 6.1) extends in the obvious way (Proposition 16.2), i.e.,

$$ \int _J \int _J |\langle f_j, f_k\rangle |^2 \,d\mu (j)\, d\mu (k) \ge {1\over d} \Bigl (\int _J \Vert f_j\Vert ^2\, d\mu (j)\Bigl )^2, $$

with equality if and only if $(f_j)_{j\in J}$ is tight.

16.3.

The canonical Gramian of a generalised frame $\varPhi =(f_j)_{j\in J}$ with respect to $\mu$ is given by

$$ P_\varPhi = V^*S^{-1}V:L_2(\mu )\rightarrow L_2(\mu ), $$

where $S=VV^*$ is the frame operator and V is the synthesis operator of $\varPhi$.

(a) Show that $P_\varPhi$ is an orthogonal projection.

(b) Show that the “matrix” $[P_\varPhi ]:=[\langle f_k, S^{-1}f_j\rangle ]_{j, k\in J}$ represents $P_\varPhi$ in the sense

$$ P_\varPhi a = [P_\varPhi ] \cdot _\mu a := \bigl ( \int _J [P_\varPhi ]_{jk} a_k \, d\mu (k)\bigr )_{j\in J} =\bigl (\int _J \langle f_k, S^{-1} f_j\rangle a_k\, d\mu (k)\bigr )_{j\in J}. $$

(c) Let $v_k:=(\langle f_k, S^{-1}f_j\rangle )_{j\in J}$ be the k-th “column” of $[P_\varPhi ]$. Show that $v_k\in L_2(\mu )$, and $(v_j)_{j\in J}$ gives a copy of the canonical tight frame $(S^{-1/2}f_j)_{j\in J}$, i.e.,

$$\langle v_r, v_s\rangle _{L_2(\mu )} = \langle f_r,f_s\rangle ,\qquad \forall r, s\in J. $$

16.4.

Suppose $(f_j)_{j\in J}$ is a unit-norm generalised tight frame for a d-dimensional space ${\mathscr {H}}$, i.e., $\Vert f_j\Vert =1$, $\forall j\in J$. Show that $\mu$ is finite, and

$$ f= {d\over \mu (J)} \int _J \langle f, f_j\rangle f_j\, d\mu (j), \qquad \forall f\in {\mathscr {H}}. $$

16.5.

Let ${\mathscr {H}}$ be a subspace of $\mathbb {F}^n$, and P be the orthogonal projection onto ${\mathscr {H}}$.

(a) Show that the tight frame $(K_j)_{j=1}^n$ corresponding to the reproducing kernel for ${\mathscr {H}}$ is given by $K_j=Pe_j$.

(b) Find this normalised tight frame explicitly for

$$ {\mathscr {H}}=\{x\in \mathbb {F}^n:x_1+\cdots +x_n=0\}. $$

16.6.

Suppose that ${\mathscr {H}}$ is a reproducing kernel Hilbert space.

(a) Show that any subspace of ${\mathscr {H}}$ is again a reproducing kernel Hilbert space.

(b) Suppose ${\mathscr {H}}=\bigoplus _j {\mathscr {H}}_j$, an orthogonal direct sum of subspaces. Show that the reproducing kernel of ${\mathscr {H}}$ is $K=\sum _j K_j$, where $K_j$ is the reproducing kernel of ${\mathscr {H}}_j$.

16.7.

Use the orthogonal decomposition $L_2(\mathbb {S})=\bigoplus _j{\mathscr {H}}_j$ of $L_2(\mathbb {S})$ into absolutely irreducible rotationally invariant subspaces to show that the rotationally invariant subspaces of $L_2(\mathbb {S})$ have the form

$$ {\mathscr {H}}=\bigoplus _{j\in J} {\mathscr {H}}_j, \qquad \hbox {for some}\;J\subset \mathbb {N}. $$

16.8.

The Poisson kernel for the unit ball $B=\{x\in \mathbb {R}^d:\Vert x\Vert <1\}$ is given by

$$ P(x,\xi ) = {1-\Vert x\Vert ^2\over \Vert x-\xi \Vert ^d} = {1-\Vert x\Vert ^2\over (1-2\langle x,\xi \rangle +\Vert x\Vert ^2)^{d\over 2}}, \qquad x\in B,\ \xi \in \mathbb {S}. $$

It has the property that for every u which is harmonic on the closed unit ball

$$ u(x) =\int _\mathbb {S}u(\xi )P(x,\xi )\, d\xi . $$

From this, and (16.8), (16.9), (16.7), it follows that

$$ P(x,\xi ) = \sum _{k=0}^\infty Z_\xi ^{(k)}(x), \qquad x\in B,\ \xi \in \mathbb {S}, $$

where the series converges absolutely and locally uniformally. Use the generating function for the Gegenbauer polynomials

$$ {1\over (1-2yt+t^2)^\lambda } = \sum _{k=0}^\infty C_k^{(\lambda )}(y)t^k $$

to expand the Poisson kernel in terms of the zonal harmonics, to obtain the formula

$$ Z_\xi ^{(k)}(x) = \Vert x\Vert ^k C_{k}^{d\over 2} \Bigl ({\langle x,\xi \rangle \over \Vert x\Vert }\Bigr ) - \Vert x\Vert ^k C_{k-2}^{d\over 2} \Bigl ({\langle x,\xi \rangle \over \Vert x\Vert }\Bigr ). $$

16.9.

(Linear polynomials on the sphere). The spaces $\varPi _1(\mathbb {S})$ and $\varPi _1(\mathbb {R}^d)$ of linear polynomials on the sphere and on $\mathbb {R}^d$ have dimension $d+1$.

(a) Find the reproducing kernel tight frame $(Z_\xi )$ for these spaces, with the norms $\Vert \cdot \Vert _\mathbb {S}$ and $\Vert \cdot \Vert _w$, respectively.

(b) Let $\xi _1,\ldots ,\xi _4$ be four points on the sphere in $\mathbb {R}^3$. Show that the zonal functions $(Z_{\xi _j})_{j=1}^4$ are a basis for $\varPi _1(\mathbb {S})$ if and only if the points $\{\xi _j\}$ do not lie on a circle.

(c) Show that $(Z_{\xi _j})_{j=1}^4$ is an orthogonal basis for $\varPi _1(\mathbb {S})$ if and only if the points $\{\xi _j\}$ are the vertices of a regular tetrahedron.

16.10.

The ${\mathscr {U}}$-invariant subspaces of $L_2({\mathbb {S}_\mathbb {C}})$ are given by the subsums of (16.55). Thus the reproducing kernel of such a space is the sum of the reproducing kernels of its summands (cf Theorem 16.1).

(a) Show that the reproducing kernel of H(p, 0) is zonal.

(b) Find the reproducing kernel for the holomorphic polynomials of degree $\le n$.

(c) Show that the sum of the reproducing kernels of the homogeneous holomorphic functions of all degrees (these are orthogonal) is the the Szegö kernel

$$ S(z, w):={1\over (1-\langle z, w\rangle )^d}, \qquad z\in B, \quad w\in \mathbb {S}. $$

It has the property that for every f which is holomorphic on the closed unit ball

$$ f(z) = \int _{\mathbb {S}_\mathbb {C}}f(w) S(z,w)\, d\sigma (w). $$

(d) Expand the Poisson kernel

$$ P(z, w) = \sum _{p=0}^\infty \sum _{q=0}^\infty K_{pq}(z, w) = {1-\Vert z\Vert ^2\over \Vert w-z\Vert ^{2d}}, $$

to find a formula for the reproducing kernel for ${\mathscr {H}}_k({\mathbb {S}_\mathbb {C}})={\mathscr {H}}_k(\mathbb {C}^d)$.

16.11.

Let S be the bilateral shift on $\ell _2(\mathbb {Z})$, given by $Se_j:=e_{j+1}$. Then $j\cdot v=S^j v$ defines unitary continuous action of the noncompact group $G=\mathbb {Z}$ on ${\mathscr {H}}=\ell _2(\mathbb {Z})$.

(a) Let $v\in \ell _2(\mathbb {Z})$ be nonzero. Show that if $(S^j v)_{j\in \mathbb {Z}}$ is a tight frame for $\ell _2(\mathbb {Z})$, then the frame bound is $A=\Vert v\Vert ^2$, i.e., $(S^jv)$ is orthogonal basis.

(b) Let $v=v_a e_a+v_b e_b\ne 0$, $a\ne b$. Show

$$ \sum _j|\langle x, S^jv\rangle |^2 = \Vert v\Vert ^2\Vert x\Vert ^2 + 2\mathfrak {R}\bigl ( v_a\overline{v_b} \langle S^{a-b}x, x\rangle \bigr ), $$

and conclude that $(S^j v)_{j\in \mathbb {Z}}$ is a tight frame for $\ell _2(\mathbb {Z})$ if and only if $v_a=0$ or $v_b=0$.

(c) Find all finitely supported vectors v for which $(S^j v)_{j\in \mathbb {Z}}$ is a tight frame for $\ell _2(\mathbb {Z})$.

(d) Let $v=(\ldots ,-{1\over 5}, 0,-{1\over 3}, 0,-1,0,1,0,{1\over 3}, 0,{1\over 5}, 0,{1\over 7},\ldots )$. Show that $(S^j v)_{j\in \mathbb {Z}}$ is a tight frame for $\ell _2(\mathbb {Z})$.

(e) Determine all vectors v for which $(S^j v)_{j\in \mathbb {Z}}$ is a tight frame for $\ell _2(\mathbb {Z})$.

16.12.

Let ${\mathbb {S}_\mathbb {C}}:=\{z\in \mathbb {C}^d:\Vert z\Vert =1\}$ be the complex unit sphere in $\mathbb {C}^d\approx \mathbb {R}^{2d}$, and $\sigma$ be Lebesgue surface-area measure on ${\mathbb {S}_\mathbb {C}}$ viewed as a unit sphere in $\mathbb {R}^{2d}$. Deduce the analogue of (16.15), i.e.,

$$ z={d\over \mathrm{area}({\mathbb {S}_\mathbb {C}})}\int _{{\mathbb {S}_\mathbb {C}}} \langle z,\xi \rangle \xi \, d\sigma (\xi ), \qquad \forall z\in \mathbb {C}^d. $$

16.13.

Show that the zonal harmonic $Z_\xi ^{(k)}$ is localised at $\xi \in \mathbb {S}$ in the following sense.

(a) $|Z_\xi ^{(k)}(x)|<Z_\xi ^{(k)}(\xi )$, $\forall x\ne \xi$, $x\in \mathbb {S}$.

(b) $Z_\xi ^{(k)}(\xi )=\Vert Z_\xi ^{(k)}\Vert _\mathbb {S}^2=\dim ({\mathscr {H}}_k)$.

(c) The maximum

$$ \max \{ p(\xi ):\Vert p\Vert _\mathbb {S}=1,p\in {\mathscr {H}}_k(\mathbb {S}^{d-1})\} = \sqrt{\dim ({\mathscr {H}}_k)} =\mathrm{O}(k^{d\over 2}), \qquad k\rightarrow \infty $$

is attained if and only if $p={ 1 \over \sqrt{\dim ({\mathscr {H}}_k)}}Z_\xi ^{(k)}$.

16.14.

The variational condition (6.5) for $(f_j)$ to be a finite tight frame for ${F_d}$ is

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

Show that the following analogous condition holds

$$\int _\mathbb {S}\int _\mathbb {S}|\langle x, y\rangle |^2 \,d\sigma (x)\, d\sigma (y) = {1\over d} \Bigl (\int _\mathbb {S}\langle x,x\rangle \, d\sigma (x) \Bigr )^2. $$

Remark: This can be interpreted as saying that although a generic frame of n unit vectors for ${F_d}$ is not tight, for $n\gg d$ it is close to being tight (also see Exer. 6.3).

16.15.

Let $\langle \cdot ,\cdot \rangle _{\circ , k}$ denote the apolar inner product of (6.19) on H(k, 0) (the space of holomorphic homogeneous polynomials of degree k).

(a) Use (6.21), to show that

$$ \langle f,g\rangle _{\circ , k}= \left( {\begin{array}{c}k+d-1\\ d-1\end{array}}\right) \langle f, g\rangle _{\mathbb {S}_\mathbb {C}}. $$

(b) Express the reproducing kernel tight frame given by (16.57) in terms of $\langle \cdot ,\cdot \rangle _{\circ , k}$.

(c) Use the formula

$$ \int _{\mathbb {S}_\mathbb {C}}f(\langle \xi ,\eta \rangle )\, d\sigma (\xi ) = {d-1\over \pi } \int _0^{2\pi }\int _0^1 (1-r^2)^{d-2} f(re^{i\theta })\, r\,dr\, d\theta , \quad \eta \in {\mathbb {S}_\mathbb {C}}, $$

to calculate $\Vert \langle \cdot ,\eta \rangle ^k\Vert _{\mathbb {S}_\mathbb {C}}$.