A frame decomposition of the atmospheric tomography operator

In this section, we consider the periodic tomography operator (2.3) from [22], but now for the case that instead of only NGSs, we consider a setting using only LGSs. Since in this case c_l,g = c_l independent of the guide star direction g, the operator à can now be written in the form

$$\begin{equation}\begin{aligned}\hfill & \tilde{A}\enspace :\enspace \prod _{l=1}^{L}{L}_{2}\left({c}_{l}\enspace {{\Omega}}_{T}\right)\to {{L}_{2}\left({{\Omega}}_{T}\right)}^{G},\hfill \\ \hfill & \phi {\mapsto}{\varphi }_{g}={\left(\tilde{A}\phi \right)}_{g}\left(r\right){:=}\sum _{l=1}^{L}{\phi }_{l}\left({c}_{l}r+{\alpha }_{g}{h}_{l}\right),\quad g=1,\dots ,G,\hfill \end{aligned}\end{equation} \tag{ 4.1 }$$

where, as before, Ω_T = [−T, T], however now with T sufficiently large, such that

$$\begin{equation*}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}\subset {c}_{l}\enspace {{\Omega}}_{T},\quad g=1,\dots ,G,\quad l=1,\dots ,L.\end{equation*}$$

We now derive a singular-value-type decomposition of the adapted operator à using the ideas from [22]. First, due to the presence of the constants c_l , it makes sense to use, for ever layer l, a different orthonormal basis of L₂(c_l Ω_T ), namely the functions

$$\begin{equation}{w}_{jk,l}\left(x,y\right){:=}{c}_{l,g}^{-1}\enspace {w}_{jk}\left(\left(x,y\right)/{c}_{l,g}\right)\text{for}\;\text{only}\;\text{LGS}{=}\;{{c}_{l}}^{-1}\enspace {w}_{jk}\left(\left(x,y\right)/{c}_{l}\right).\end{equation} \tag{ 4.2 }$$

Any function ϕ_l ∈ L₂(c_l Ω_T ) can be written in the form

$$\begin{equation}{\phi }_{l}\left(x,y\right)=\sum _{jk\in \mathbb{Z}}{\phi }_{jk,l}\enspace {w}_{jk,l}\left(x,y\right),\quad {\phi }_{jk,l}{:=}{\left\langle \enspace {\phi }_{l},{w}_{jk,l}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)},\end{equation} \tag{ 4.3 }$$

and for $\phi ={\left({\phi }_{l}\right)}_{l=1}^{L}\in {\prod }_{l=1}^{L}{L}_{2}\left({c}_{l}\enspace {{\Omega}}_{T}\right)$ we collect the expansion coefficients ϕ_jk,l in the vectors ${\phi }_{jk}{:=}{\left({\phi }_{jk,l}\right)}_{l=1}^{L}\in {\mathbb{C}}^{L}$. With this, we now get

Proposition 4.1. Let à be defined as in (4.1) and let the G × L matrices Ã_jk be defined by

$$\begin{equation*}{\left({\tilde{A}}_{jk}\right)}_{gl}{:=}\left(2T\right)\enspace {w}_{jk,l}\left({\alpha }_{g}^{x}{h}_{l},{\alpha }_{g}^{y}{h}_{l}\right).\end{equation*}$$

Then there holds

$$\begin{equation}\left(\tilde{A}\phi \right)\left(x,y\right)=\sum _{jk\in \mathbb{Z}}\left({\tilde{A}}_{jk}{\phi }_{jk}\right){w}_{jk}\left(x,y\right).\end{equation} \tag{ 4.4 }$$

Proof. Using the definition (4.1) of à and the expansion (4.3), we get

$$\begin{equation*}\begin{aligned}\hfill {\left(\tilde{A}\phi \right)}_{g}\left(x,y\right)& =\sum _{l=1}^{L}{\phi }_{l}\left({c}_{l}x+{\alpha }_{g}^{x}{h}_{l},{c}_{l}y+{\alpha }_{g}^{y}{h}_{l}\right)\hfill \\ \hfill & =\sum _{l=1}^{L}\sum _{jk\in \mathbb{Z}}{\phi }_{jk,l}\enspace {w}_{jk,l}\left({c}_{l}x+{\alpha }_{g}^{x}{h}_{l},{c}_{l}y+{\alpha }_{g}^{y}{h}_{l}\right)\hfill \\ \hfill & =\sum _{l=1}^{L}\sum _{jk\in \mathbb{Z}}{\phi }_{jk,l}\left(2T\right){w}_{jk}\left(x,y\right){w}_{jk,l}\left({\alpha }_{g}^{x}{h}_{l},{\alpha }_{g}^{y}{h}_{l}\right),\hfill \end{aligned}\end{equation*}$$

from which the assertion immediately follows after interchanging the series, which is allowed since the norm of Ã_jk is bounded independently of j, k.□

As in [22] we consider the singular system for each of the matrices Ã_jk , i.e., the vectors ${v}_{jk,n}\in {\mathbb{C}}^{L}$, ${u}_{jk,n}\in {\mathbb{C}}^{G}$ and numbers σ_jk,n , n = 1, ..., r_jk ⩽ min {G, L} such that

$$\begin{equation}\begin{aligned}\hfill {\tilde{A}}_{jk}{\phi }_{jk}=& \sum _{n=1}^{{r}_{jk}}{\sigma }_{jk,n}\left({v}_{jk,n}^{H}\enspace {\phi }_{jk}\right){u}_{jk,n},\hfill \\ \hfill {v}_{jk,m}^{H}{v}_{jk,n}=& {\delta }_{mn},\quad {u}_{jk,m}^{H}{u}_{jk,n}={\delta }_{mn},\hfill \\ \hfill {\sigma }_{jk,1}& {\geqslant}\cdots {\geqslant}{\sigma }_{jk,{r}_{jk}}{ >}0,\hfill \end{aligned}\end{equation} \tag{ 4.5 }$$

where the superscript H denotes the Hermitian adjoint (i.e. the complex transpose), r_jk is the rank of the matrix Ã_jk , and the ${\sigma }_{jk,n}^{2}$ are the positive eigenvalues of the matrices ${\tilde{A}}_{jk}^{H}{\tilde{A}}_{jk}$ and ${\tilde{A}}_{jk}{\tilde{A}}_{jk}^{H}$, respectively.

Hence, the decomposition of à in terms of the functions w_jk is given by

$$\begin{equation}\left(\tilde{A}\phi \right)\left(x,y\right)=\sum _{jk\in \mathbb{Z}}\left(\sum _{n=1}^{{r}_{jk}}{\sigma }_{jk,n}\left({v}_{jk,n}^{H}\enspace {\phi }_{jk}\right){u}_{jk,n}\right){w}_{jk}\left(x,y\right),\end{equation} \tag{ 4.6 }$$

which is completely analogous to [22, equation (10)], however with a different singular systems (σ_jk,n , u_jk,n , v_jk,n ). Thus, for incoming wavefronts

$$\begin{equation*}\varphi =\sum _{jk\in \mathbb{Z}}{\varphi }_{jk}{w}_{jk}\in {{L}_{2}\left({{\Omega}}_{T}\right)}^{G},\quad {\varphi }_{jk}\in {\mathbb{C}}^{G},\quad \end{equation*}$$

the best-approximate solution of the equation

$$\begin{equation*}\tilde{A}\phi =\varphi \end{equation*}$$

is given by

$$\begin{equation*}\left({\tilde{A}}^{{\dagger}}\varphi \right)\left(x,y\right){:=}\sum _{jk\in \mathbb{Z}}\left(\sum _{n=1}^{{r}_{jk}}\frac{\left({u}_{jk,n}^{H}\enspace {\varphi }_{jk}\right)}{{\sigma }_{jk,n}}{v}_{jk,n}\right){w}_{jk,l}\left(x,y\right),\end{equation*}$$

where Ã^† denotes the Moore–Penrose generalized inverse of à (see for example [10, 22]), which is well-defined if and only if the following Picard-condition holds

$$\begin{equation*}\sum _{jk\in \mathbb{Z}}\sum _{n=1}^{{r}_{jk}}\frac{{\left\vert {u}_{jk,n}^{H}\enspace {\varphi }_{jk}\right\vert }^{2}}{{\sigma }_{jk,n}^{2}}{< }\infty .\end{equation*}$$

In this section, we derive a frame decomposition for the general atmospheric tomography operator A as defined in (2.1), i.e., for the case of arbitrary aperture domains Ω_A and both NGSs and LGSs.

The main idea of [22], where some of the ideas of the upcoming analysis are taken from, was to use the special properties of the functions w_jk (2.4), in particular that they form an orthogonal basis of L₂(Ω_A ). Unfortunately, for general domains Ω_A , the functions w_jk do not necessarily form a basis of L₂(Ω_A ). However, they do form a (tight) frame in the sense of definition 3.1, as we see in the following

Lemma 4.2. Let w_jk be defined as in (2.4). If T is large enough such that Ω_A ⊂ Ω_T , then the system ${\left\{{w}_{jk}\right\}}_{j,k\in \mathbb{Z}}$ forms a tight frame over L₂(Ω_A ) with frame bound 1, i.e.,

$$\begin{equation*}{\Vert}\psi {{\Vert}}_{{L}_{2}\left({{\Omega}}_{A}\right)}^{2}=\sum _{j,k\in \mathbb{Z}}{\left\vert {\left\langle \enspace \psi ,{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}\right\vert }^{2},\quad \forall \enspace \psi \in {L}_{2}\left({{\Omega}}_{A}\right).\end{equation*}$$

Proof. We start by defining $\tilde {\psi }{:=}\psi {I}_{{{\Omega}}_{A}}$ where ${I}_{{{\Omega}}_{A}}$ denotes the indicator function of Ω_A . Now since $\tilde {\psi }\in {L}_{2}\left({{\Omega}}_{T}\right)$ and $\tilde {\psi }=\psi$ on Ω_A and $\tilde {\psi }=0$ on Ω_T \Ω_A , we get

$$\begin{equation*}{\Vert}\psi {{\Vert}}_{{L}_{2}\left({{\Omega}}_{A}\right)}^{2}={\Vert}\tilde {\psi }{{\Vert}}_{{L}_{2}\left({{\Omega}}_{T}\right)}^{2}=\sum _{jk\in \mathbb{Z}}{\left\vert {\left\langle \enspace \tilde {\psi },{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{T}\right)}\right\vert }^{2}=\sum _{jk\in \mathbb{Z}}{\left\vert {\left\langle \enspace \psi ,{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}\right\vert }^{2},\end{equation*}$$

where we have used that since the w_jk form an orthonormal basis of L₂(Ω_T ), they are also a tight frame with frame bounds B₁ = B₂ = 1. This proves the assertion.□

Now, since ${\left\{{w}_{jk}\right\}}_{jk\in \mathbb{Z}}$ forms a tight frame with bound 1, it is also its own dual frame. Hence, (3.4) implies that any function ψ ∈ L₂(Ω_A ) can be written in the form

$$\begin{equation}\psi \left(x,y\right)=\sum _{j,k\in \mathbb{Z}}{\left\langle \enspace \psi ,{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}{w}_{jk}\left(x,y\right).\end{equation} \tag{ 4.7 }$$

In particular, we have that the incoming wavefronts φ can be written in the form

$$\begin{equation}\varphi =\sum _{jk\in \mathbb{Z}}{\varphi }_{jk}{w}_{jk},\quad {\varphi }_{jk}{:=}{\left({\varphi }_{jk,g}\right)}_{g=1}^{G}{:=}{\left({\left\langle \enspace {\varphi }_{g},{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}\right)}_{g=1}^{G}\in {\mathbb{C}}^{G}.\end{equation} \tag{ 4.8 }$$

We also want to expand functions on L₂(Ω_l ) in terms of frames. It is not difficult to find frames for L₂(Ω_l ); for example, for large enough T, the sets ${\left\{{w}_{jk}\right\}}_{jk\in \mathbb{Z}}$ or ${\left\{{w}_{jk,l}\right\}}_{jk\in \mathbb{Z}}$ already form frames over L₂(Ω_l ). However, for the upcoming analysis, those frames do not satisfy the specific needs of the problem under consideration. Hence, we use a different, problem tailored frame, which we build from the functions

$$\begin{equation}{w}_{jk,lg}\left(x,y\right){:=}\enspace {c}_{l,g}^{-1}{w}_{jk}\left(\left(x,y\right)/{c}_{l,g}\right)\enspace {I}_{{c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}}\left(x,y\right),\end{equation} \tag{ 4.9 }$$

with w_jk defined as in (2.4), where we choose T large enough such that

$$\begin{equation}{{\Omega}}_{A}\subset {{\Omega}}_{T}\quad \text{and}\quad {{\Omega}}_{l}\subset {c}_{l,g}{{\Omega}}_{T}\quad \forall \enspace l\in \left\{1,\dots ,L\right\},\quad g\in \left\{1,\dots ,G\right\},\end{equation} \tag{ 4.10 }$$

which we assume to hold from now on. That these functions can indeed form frames can be seen from the following

Proposition 4.3. Let w_jk,lg be defined as in (4.9) and let T be large enough such that (4.10) holds. Then, for fixed l, the set ${\left\{{w}_{jk,lg}\right\}}_{jk\in \mathbb{Z},\enspace g=1,\dots ,G}$ forms a frame over L₂(Ω_l ) with frame bounds 1 ⩽ B₁ ⩽ B₂ ⩽ G.

Proof. Let ψ ∈ L₂(Ω_l ) be arbitrary but fixed and start by looking at

$$\begin{align}\hfill & \sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\left\vert {\left\langle \enspace \psi ,{w}_{jk,lg}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)}\right\vert }^{2}\hfill \\ \hfill & \quad =\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\left\vert {\left\langle \enspace \psi ,{c}_{l,g}^{-1}{w}_{jk}\left(\left(\cdot ,\cdot \right)/{c}_{l,g}\right)\enspace {I}_{{c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)}\right\vert }^{2}\hfill \\ \hfill & \quad =\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\left\vert {\left\langle \enspace \psi \enspace {I}_{{c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}},{c}_{l,g}^{-1}{w}_{jk}\left(\left(\cdot ,\cdot \right)/{c}_{l,g}\right)\enspace \right\rangle }_{{L}_{2}\left({c}_{l,g}\enspace {{\Omega}}_{T}\right)}\right\vert }^{2},\hfill \end{align} \tag{ 4.11 }$$

where we have used that c_l,g Ω_A + α_g h_l ⊂ Ω_l ⊂ c_l,g Ω_T . Now since for each (fixed) g the functions ${c}_{l,g}^{-1}{w}_{jk}\left(\left(x,y\right)/{c}_{l,g}\right)$ form an orthonormal basis of L₂(c_l,g Ω_T ) (and thus a tight frame with bound 1), we get that

$$\begin{align*}\hfill & \sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\left\vert {\left\langle \enspace \psi \enspace {I}_{{c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}},{c}_{l,g}^{-1}{w}_{jk}\left(\left(\cdot ,\cdot \right)/{c}_{l,g}\right)\enspace \right\rangle }_{{L}_{2}\left({c}_{l,g}\enspace {{\Omega}}_{T}\right)}\right\vert }^{2}\hfill \\ \hfill & \quad =\sum _{g=1}^{G}{\Vert}\psi \enspace {I}_{{c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}}{{\Vert}}_{{L}_{2}\left({c}_{l,g}\enspace {{\Omega}}_{T}\right)}^{2}\hfill \\ \hfill & \quad =\sum _{g=1}^{G}{\int }_{{c}_{l,g}{{\Omega}}_{T}}{\left\vert \psi \enspace {I}_{{c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}}\right\vert }^{2}=\sum _{g=1}^{G}{\int }_{{c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}}{\left\vert \psi \right\vert }^{2},\hfill \end{align*}$$

which, together with (4.11), yields

$$\begin{equation}\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\left\vert {\left\langle \enspace \psi ,{w}_{jk,lg}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)}\right\vert }^{2}=\sum _{g=1}^{G}{\int }_{{c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}}{\left\vert \psi \right\vert }^{2}.\end{equation} \tag{ 4.12 }$$

Now since ${{\Omega}}_{l}={\bigcup }_{g=1}^{G}\left({c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}\right)$, we get that

$$\begin{equation*}{\Vert}\psi {{\Vert}}_{{L}_{2}\left({{\Omega}}_{l}\right)}^{2}={\int }_{{{\Omega}}_{l}}{\left\vert \psi \right\vert }^{2}\enspace \enspace {\leqslant}\sum _{g=1}^{G}{\int }_{{c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}}{\left\vert \psi \right\vert }^{2}\enspace \enspace {\leqslant}\sum _{g=1}^{G}\underset{{{\Omega}}_{l}}{\int }{\left\vert \psi \right\vert }^{2}\enspace \enspace =G{\Vert}\psi {{\Vert}}_{{L}_{2}\left({{\Omega}}_{l}\right)}^{2}.\end{equation*}$$

Combing this together with (4.12), we get that

$$\begin{equation*}{\Vert}\psi {{\Vert}}_{{L}_{2}\left({{\Omega}}_{l}\right)}^{2}{\leqslant}\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\left\vert {\left\langle \enspace \psi ,{w}_{jk,lg}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)}\right\vert }^{2}{\leqslant}G{\Vert}\psi {{\Vert}}_{{L}_{2}\left({{\Omega}}_{l}\right)}^{2},\end{equation*}$$

which proves the assertion.□

Now, denoting for fixed l the dual frame of ${\left\{{w}_{jk,lg}\right\}}_{jk\in \mathbb{Z},\enspace g=1,\dots ,G}$ by ${\left\{{\tilde {w}}_{jk,lg}\right\}}_{jk\in \mathbb{Z},\enspace g=1,\dots ,G}$, the above proposition together with (3.4) implies that any function ϕ_l ∈ L₂(Ω_l ) can be written in the form

$$\begin{equation}{\phi }_{l}\left(x,y\right)=\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\phi }_{jk,lg}{\tilde {w}}_{jk,lg}\left(x,y\right),\quad {\phi }_{jk,lg}={\left\langle \enspace {\phi }_{l},{w}_{jk,lg}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)}.\end{equation} \tag{ 4.13 }$$

Next, we want to find an expansion of Aϕ in terms of the frame ${\left\{{w}_{jk}\right\}}_{jk\in \mathbb{Z}}$. Due to (4.7) and since Aϕ ∈ L₂(Ω_A ), such an expansion is given by

$$\begin{equation}A\phi ={\left({\left(A\phi \right)}_{g}\right)}_{g=1}^{G}=\sum _{jk\in \mathbb{Z}}{\left({\left\langle \enspace {\left(A\phi \right)}_{g},{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}\right)}_{g=1}^{G}{w}_{jk},\end{equation} \tag{ 4.14 }$$

As we already know, even though this might not be the only possible expansion, it is the most economical one in the sense of proposition 3.1. Furthermore, due to (3.9), this expansion allows us to express ∥Aϕ − φ∥ in terms of the expansion coefficients of Aϕ and φ, which is important for determining an (approximate) solution of Aϕ = φ. We now derive an explicit expression for the coefficients ${\left\langle \enspace {\left(A\phi \right)}_{g},{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}$ in terms of the coefficients ϕ_jk,lg in the following

Proposition 4.4. Let $\phi ={\left({\phi }_{l}\right)}_{l=1,\dots ,L}$ with ϕ_l ∈ L₂(Ω_l ) for all l = 1, ..., L, and let A, w_jk , and ϕ_jk,lg be defined as in (2.1), (2.4), and (4.13), respectively. Then there holds

$$\begin{equation}{\left\langle \enspace {\left(A\phi \right)}_{g},{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}=\left(2T\right)\sum _{l=1}^{L}{c}_{l,g}^{-1}{w}_{jk}\left({\alpha }_{g}{h}_{l}/{c}_{l,g}\right){\phi }_{jk,lg}.\end{equation} \tag{ 4.15 }$$

Proof. Due to the definition of A, and setting r = (x, y) we have

$$\begin{align*}\hfill {\left\langle \enspace {\left(A\phi \right)}_{g},{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}& =\sum _{l=1}^{L}{\left\langle \enspace {\phi }_{l}\left({c}_{l,g}\cdot +{\alpha }_{g}{h}_{l}\right),{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}\hfill \\ \hfill & =\sum _{l=1}^{L}{\int }_{{{\Omega}}_{A}}{\phi }_{l}\left({c}_{l,g}r+{\alpha }_{g}{h}_{l}\right)\overline{{w}_{jk}\left(r\right)}\enspace \mathrm{d}r.\hfill \end{align*}$$

Using substitution in the integral yields

$$\begin{align*}\hfill & {\int }_{{{\Omega}}_{A}}{\phi }_{l}\left({c}_{l,g}r+{\alpha }_{g}{h}_{l}\right)\overline{{w}_{jk}\left(r\right)}\enspace \mathrm{d}r\hfill \\ \hfill & \quad ={c}_{l,g}^{-2}{\int }_{{c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}}{\phi }_{l}\left(r\right)\overline{{w}_{jk}\left(\left(r-{\alpha }_{g}{h}_{l}\right)/{c}_{l,g}\right)}\enspace \mathrm{d}r\hfill \\ \hfill & \quad ={c}_{l,g}^{-2}\left(2T\right){w}_{jk}\left({\alpha }_{g}{h}_{l}/{c}_{l,g}\right){\int }_{{c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}}{\phi }_{l}\left(r\right)\overline{{w}_{jk}\left(r/{c}_{l,g}\right)}\enspace \mathrm{d}r.\hfill \end{align*}$$

Since c_l,g Ω_A + α_g h_l ⊂ Ω_l , due to (4.9) and (4.13), we get

$$\begin{align*}\hfill & {c}_{l,g}^{-1}{\int }_{{c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}}{\phi }_{l}\left(r\right)\overline{{w}_{jk}\left(r/{c}_{l,g}\right)}\enspace \mathrm{d}r\hfill \\ \hfill & \quad ={c}_{l,g}^{-1}{\int }_{{{\Omega}}_{l}}{\phi }_{l}\left(r\right)\overline{{w}_{jk}\left(r/{c}_{l,g}\right){I}_{{c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}}\left(r\right)}\enspace \mathrm{d}r={\phi }_{jk,lg}.\hfill \end{align*}$$

Combining the above results, we get

$$\begin{equation*}\begin{aligned}\hfill {\left\langle \enspace {\left(A\phi \right)}_{g},{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}& =\sum _{l=1}^{L}{\int }_{{{\Omega}}_{A}}{\phi }_{l}\left({c}_{l,g}r+{\alpha }_{g}{h}_{l}\right)\overline{{w}_{jk}\left(r\right)}\enspace \mathrm{d}r\hfill \\ \hfill & =\left(2T\right)\sum _{l=1}^{L}{c}_{l,g}^{-1}{w}_{jk}\left({\alpha }_{g}{h}_{l}/{c}_{l,g}\right){\phi }_{jk,lg}.\hfill \end{aligned}\end{equation*}$$

which directly yields the assertion.□

Hence, combining (4.14) and the above proposition, we get that

$$\begin{equation}\left(A\phi \right)\left(x,y\right)=\sum _{jk\in \mathbb{Z}}{\left(\left(2T\right)\sum _{l=1}^{L}{c}_{l,g}^{-1}{w}_{jk}\left({\alpha }_{g}{h}_{l}/{c}_{l,g}\right){\phi }_{jk,lg}\right)}_{g=1}^{G}{w}_{jk}\left(x,y\right),\end{equation} \tag{ 4.16 }$$

Thus, if we define (with a slight abuse of notation), the vectors

$$\begin{equation}{\phi }_{jk}{:=}\left({\left({\phi }_{jk,l1}\right)}_{l=1}^{L},\dots ,{\left({\phi }_{jk,lG}\right)}_{l=1}^{L}\right)\in {\mathbb{C}}^{L\enspace \cdot G},\end{equation} \tag{ 4.17 }$$

and the ${\mathbb{C}}^{G{\times}\left(L\cdot G\right)}$ matrices

$$\begin{equation}{A}_{jk}{:=}\left(2T\right)\left(\begin{matrix}\hfill {\left({c}_{l,1}^{-1}{w}_{jk}\left(\frac{{\alpha }_{1}{h}_{l}}{{c}_{l,1}}\right)\right)}_{l=1}^{L}\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\left({c}_{l,2}^{-1}{w}_{jk}\left(\frac{{\alpha }_{2}{h}_{l}}{{c}_{l,2}}\right)\right)}_{l=1}^{L}\hfill & \hfill \ddots \hfill & \hfill {\vdots}\hfill \\ \hfill {\vdots}\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill {\left({c}_{l,G}^{-1}{w}_{jk}\left(\frac{{\alpha }_{G}{h}_{l}}{{c}_{l,G}}\right)\right)}_{l=1}^{L}\hfill \end{matrix}\right),\end{equation} \tag{ 4.18 }$$

then we get the following expansion of the tomography operator

$$\begin{equation}\left(A\phi \right)\left(x,y\right)=\sum _{jk\in \mathbb{Z}}\left({A}_{jk}{\phi }_{jk}\right){w}_{jk}\left(x,y\right).\end{equation} \tag{ 4.19 }$$

The above expansion is obviously similar to (4.4) for the pure LGS case, however now with different (and slightly larger) matrices A_jk . Hence, we can again consider the singular value decomposition of each of the matrices A_jk , i.e., with another small abuse of notation, the vectors ${v}_{jk,n}\in {\mathbb{C}}^{L\cdot G}$, ${u}_{jk,n}\in {\mathbb{C}}^{G}$ and numbers σ_jk,n , n = 1, ..., r_jk ⩽ G such that

$$\begin{equation}\begin{aligned}\hfill {A}_{jk}{\phi }_{jk}=& \;\sum _{n=1}^{{r}_{jk}}{\sigma }_{jk,n}\left({v}_{jk,n}^{H}\enspace {\phi }_{jk}\right){u}_{jk,n},\hfill \\ \hfill {v}_{jk,m}^{H}{v}_{jk,n}=& \;{\delta }_{mn},\quad {u}_{jk,m}^{H}{u}_{jk,n}={\delta }_{mn},\hfill \\ \hfill {\sigma }_{jk,1}& {\geqslant}\cdots {\geqslant}{\sigma }_{jk,{r}_{jk}}{ >}0,\hfill \end{aligned}\end{equation} \tag{ 4.20 }$$

to get, in complete analogy to (4.6), the following frame decomposition of the atmospheric tomography operator A defined in (2.1):

$$\begin{equation}\left(A\phi \right)\left(x,y\right)=\sum _{jk\in \mathbb{Z}}\left(\sum _{n=1}^{{r}_{jk}}{\sigma }_{jk,n}\left({v}_{jk,n}^{H}\enspace {\phi }_{jk}\right){u}_{jk,n}\right){w}_{jk}\left(x,y\right).\end{equation} \tag{ 4.21 }$$

Using this decomposition, we are now able to define the operator

$$\begin{equation}\left(\mathcal{A}\varphi \right)\left(x,y\right){:=}\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\left(\left(\sum _{n=1}^{{r}_{jk}}\frac{\left({u}_{jk,n}^{H}\enspace {\varphi }_{jk}\right)}{{\sigma }_{jk,n}}{\left({v}_{jk,n}\right)}_{l+\left(g-1\right)L}\right){\tilde {w}}_{jk,lg}\left(x,y\right)\right)}_{l=1}^{L},\end{equation} \tag{ 4.22 }$$

which we use to find a solution to the equation Aϕ = φ in theorem 4.8 below. Before we proceed to that though, we observe that the structure of the matrices A_jk allows to compute their singular-value-decomposition explicitly, which leads to the following

Proposition 4.5. Let ${\left({\sigma }_{jk,n},{u}_{jk,n},{v}_{jk,n}\right)}_{n=1}^{{r}_{jk}}$, for $j,k\in \mathbb{Z}$ be the singular systems of the matrices A_jk as defined in (4.20). Then there holds

$$\begin{equation}{r}_{jk}=G,\quad {\sigma }_{jk,n}=\sqrt{\sum _{l=1}^{L}{c}_{l,n}^{-2}},\quad {u}_{jk,n}={ \overrightarrow {e}}_{n},\end{equation} \tag{ 4.23 }$$

$$\begin{equation*}{\left({v}_{jk,n}\right)}_{i}=\left(2T\right){\sigma }_{jk,n}^{-1}\begin{cases}_{l,n}^{-1}{w}_{jk}\left(\frac{-{\alpha }_{n}{h}_{l}}{{c}_{l,n}}\right)\quad \hfill & \exists \enspace 1{\leqslant}l{\leqslant}L:i=l+\left(n-1\right)L,\hfill \\ 0\quad \hfill & \text{else},\hfill \end{cases}\end{equation*}$$

where ${ \overrightarrow {e}}_{n}$ denotes the nth unit vector in ${\mathbb{C}}^{G}$.

Proof. Due to the structure of A_jk , we have that

$$\begin{equation*}{A}_{jk}{A}_{jk}^{H}={\left(2T\right)}^{2}\enspace \mathrm{diag}{\left({\Vert}{\left({c}_{l,g}^{-1}{w}_{jk}\left({\alpha }_{g}{h}_{l}/{c}_{l,g}\right)\right)}_{l=1}^{L}{{\Vert}}_{{\ell }_{2}}^{2}\right)}_{g=1,\dots ,G}=\mathrm{diag}{\left(\sum _{l=1}^{L}{c}_{l,g}^{-2}\right)}_{g=1,\dots ,G},\end{equation*}$$

from which the formula for σ_jk,n , ${u}_{jk,n}={ \overrightarrow {e}}_{n}$ and r_jk = G immediately follow. Furthermore, since there holds

$$\begin{equation*}{v}_{jk,n}=\left(1/{\sigma }_{jk,n}\right)\enspace {A}_{jk}^{H}\enspace {u}_{jk,n},\end{equation*}$$

the expression for (v_jk,n ) immediately follows, which concludes the proof.□

Using the explicit representation derived above, we get the following

Corollary 4.6. The operator $\mathcal{A}$ defined in (4.22) can be written in the form

$$\begin{equation}\left(\mathcal{A}\varphi \right)\left(x,y\right){:=}\left(2T\right)\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\left(\frac{{c}_{l,g}^{-1}}{{\sigma }_{jk,g}^{2}}{w}_{jk}\left(-\frac{{\alpha }_{g}{h}_{l}}{{c}_{l,g}}\right){\left({\varphi }_{jk}\right)}_{g}{\tilde {w}}_{jk,lg}\left(x,y\right)\right)}_{l=1}^{L},\end{equation} \tag{ 4.24 }$$

with the explicit form of the singular-values σ_jk,g as derived in proposition 4.5.

Proof. Note that by proposition 4.5, it follows that

$$\begin{equation*}{\left({v}_{jk,n}\right)}_{l+\left(g-1\right)L}=\left(2T\right){\sigma }_{jk,n}^{-1}\begin{cases}_{l,n}^{-1}{w}_{jk}\left(\frac{-{\alpha }_{n}{h}_{l}}{{c}_{l,n}}\right)\quad \hfill & g=n,\hfill \\ 0\quad \hfill & \text{else},\hfill \end{cases}\end{equation*}$$

from which, together with the expression for σ_jk,n from proposition 4.5, there follows

$$\begin{equation}\sum _{n=1}^{{r}_{jk}}\frac{\left({u}_{jk,n}^{H}\enspace {\varphi }_{jk}\right)}{{\sigma }_{jk,n}}{\left({v}_{jk,n}\right)}_{l+\left(g-1\right)L}=\left(2T\right)\frac{\left({u}_{jk,g}^{H}\enspace {\varphi }_{jk}\right)}{{\sigma }_{jk,g}^{2}}{c}_{l,g}^{-1}{w}_{jk}\left(\frac{-{\alpha }_{g}{h}_{l}}{{c}_{l,g}}\right).\end{equation} \tag{ 4.25 }$$

Furthermore, again due to proposition 4.5, we get that

$$\begin{equation}{u}_{jk,g}^{H}\enspace {\varphi }_{jk}={\left({ \overrightarrow {e}}_{n}\right)}^{H}\enspace {\varphi }_{jk}={\left({\varphi }_{jk}\right)}_{g},\end{equation} \tag{ 4.26 }$$

which, together with (4.25) and the definition of $\mathcal{A}$ immediately yields the assertion.□

Concerning the well-definedness of $\mathcal{A}$, we can now derive the following

Lemma 4.7. Let $\varphi \in {{L}_{2}\left({{\Omega}}_{A}\right)}^{G}$ and let $\mathcal{A}$ be defined as in (4.22). Then $\mathcal{A}\varphi$ is well-defined and there holds

$$\begin{equation}{\Vert}\mathcal{A}{\Vert}{\leqslant}{\left(\underset{g=1,\dots ,G}{\mathrm{min}}\left\{\sum _{l=1}^{L}{c}_{l,g}^{-2}\right\}\right)}^{-\frac{1}{2}}{\leqslant}1/\sqrt{L}.\end{equation} \tag{ 4.27 }$$

Proof. Let ${\tilde {F}}_{l}$ be the frame operator [compare with (3.2)] corresponding to the dual frame ${\left\{{\tilde {w}}_{jk,lg}\right\}}_{jk\in \mathbb{Z},g=1,\dots ,G}$ of ${\left\{{w}_{jk,lg}\right\}}_{jk\in \mathbb{Z},g=1,\dots ,G}$ and let ${\tilde {F}}_{l}^{{\ast}}$ be its adjoint. Since the dual frame is also a frame but with inverse frame bounds, it follows from proposition 4.3 together with (3.3) that

$$\begin{equation*}1/\sqrt{G}{\leqslant}{\Vert}{\tilde {F}}_{l}{\Vert}={\Vert}{\tilde {F}}_{l}^{{\ast}}{\Vert}{\leqslant}1.\end{equation*}$$

Hence, for any sequence ${a}_{l}={\left\{{a}_{jk,\mathrm{lg}}\right\}}_{jk\in \mathbb{Z},g=1,\dots ,G}\in {\ell }_{2}$ there holds

$$\begin{equation}{\Vert}{\tilde {F}}_{l}^{{\ast}}{a}_{l}{{\Vert}}_{{L}_{2}\left({{\Omega}}_{l}\right)}^{2}{\leqslant}{\Vert}{a}_{l}{{\Vert}}_{{\ell }_{2}}^{2}.\end{equation} \tag{ 4.28 }$$

which we now use for the choice

$$\begin{equation}{a}_{l}{:=}{\left\{\left(2T\right)\frac{{c}_{l,g}^{-1}}{{\sigma }_{jk,g}^{2}}{w}_{jk}\left(-\frac{{\alpha }_{g}{h}_{l}}{{c}_{l,g}}\right){\left({\varphi }_{jk}\right)}_{g}\right\}}_{jk\in \mathbb{Z},g=1,\dots ,G},\end{equation} \tag{ 4.29 }$$

where ${\left\{{\varphi }_{jk}\right\}}_{jk\in \mathbb{Z}}$ are the expansion coefficients of φ defined in (4.8). Since due to (4.24) and the choice of a_l there holds

$$\begin{equation}{\left(\mathcal{A}\varphi \right)}_{l}={\tilde {F}}_{l}^{{\ast}}{a}_{l},\end{equation} \tag{ 4.30 }$$

we now obtain from (4.28) that

$$\begin{equation*}\begin{aligned}\hfill {\Vert}{\left(\mathcal{A}\varphi \right)}_{l}{{\Vert}}_{{L}_{2}\left({{\Omega}}_{l}\right)}^{2}& {\leqslant}\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\left\vert \left(2T\right)\frac{{c}_{l,g}^{-1}}{{\sigma }_{jk,g}^{2}}{w}_{jk}\left(-\frac{{\alpha }_{g}{h}_{l}}{{c}_{l,g}}\right){\left({\varphi }_{jk}\right)}_{g}\right\vert }^{2}\hfill \\ \hfill & ={\left(2T\right)}^{2}\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}\frac{{c}_{l,g}^{-2}}{{\sigma }_{jk,g}^{4}}{\left\vert {\left({\varphi }_{jk}\right)}_{g}\right\vert }^{2}.\hfill \end{aligned}\end{equation*}$$

Now summing over l we get that

$$\begin{equation*}\sum _{l=1}^{L}{\Vert}{\left(\mathcal{A}\varphi \right)}_{l}{{\Vert}}_{{L}_{2}\left({{\Omega}}_{l}\right)}^{2}{\leqslant}{\left(2T\right)}^{2}\sum _{l=1}^{L}\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}\frac{{c}_{l,g}^{-2}}{{\sigma }_{jk,g}^{4}}{\left\vert {\left({\varphi }_{jk}\right)}_{g}\right\vert }^{2}\;\left(4.23\right){=}\;\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}\frac{{\left\vert {\left({\varphi }_{jk}\right)}_{g}\right\vert }^{2}}{{\sigma }_{jk,g}^{2}},\end{equation*}$$

and therefore,

$$\begin{equation*}\sum _{l=1}^{L}{\Vert}{\left(\mathcal{A}\varphi \right)}_{l}{{\Vert}}_{{L}_{2}\left({{\Omega}}_{l}\right)}^{2}{\leqslant}\underset{\underset{g=1,\dots ,G}{jk\in \mathbb{Z}}}{\mathrm{max}}\left\{{\sigma }_{jk,g}^{-2}\right\}\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\left\vert {\left({\varphi }_{jk}\right)}_{g}\right\vert }^{2}=\underset{\underset{g=1,\dots ,G}{jk\in \mathbb{Z}}}{\mathrm{max}}\left\{{\sigma }_{jk,g}^{-2}\right\}{\Vert}\varphi {{\Vert}}_{{{L}_{2}\left({{\Omega}}_{A}\right)}^{G}}^{2},\end{equation*}$$

where the last equality follows from the definition (4.8) of the coefficients φ_jk together with lemma 4.2. Due to proposition 4.5 and since 0 < c_l,g ⩽ 1 the singular values σ_jk,g are independent of j, k and bounded away from 0. Hence, since $\varphi \in {{L}_{2}\left({{\Omega}}_{A}\right)}^{G}$, it follows that $\mathcal{A}\varphi$ is well-defined. Furthermore, the above estimate together with the explicit expression for σ_jk,g from proposition 4.5 implies that

$$\begin{equation*}{\Vert}\mathcal{A}{\Vert}{\leqslant}\underset{\underset{g=1,\dots ,G}{jk\in \mathbb{Z}}}{\mathrm{max}}\left\{{\sigma }_{jk,g}^{-1}\right\}={\left(\underset{g=1,\dots ,G}{\mathrm{min}}\left\{{\sigma }_{jk,g}\right\}\right)}^{-1}={\left(\underset{g=1,\dots ,G}{\mathrm{min}}\left\{\sqrt{\sum _{l=1}^{L}{c}_{l,g}^{-2}}\right\}\right)}^{-1},\end{equation*}$$

which, together with 0 < c_l,g ⩽ 1, immediately yields the assertion.□

With this, we are now able to prove the main theorem of this section.

Theorem 4.8. Let $\varphi \in {{L}_{2}\left({{\Omega}}_{A}\right)}^{G}$ be given and let ${\left\{{\varphi }_{jk}\right\}}_{jk\in \mathbb{Z}}$ be its expansion coefficients defined as in (4.8). Furthermore, let ${\left({\sigma }_{jk,n},{u}_{jk,n},{v}_{jk,n}\right)}_{n=1}^{{r}_{jk}}$ for $j,k\in \mathbb{Z}$ be the singular systems of the matrices A_jk defined in (4.20). Moreover, for all $l\in \left\{1,\dots ,L\right\}$ let a_l be defined as in (4.29) and satisfy a_l ∈ R(F_l ), where F_l denotes the frame operator corresponding to ${\left\{{w}_{jk,lg}\right\}}_{jk\in \mathbb{Z},g=1,\dots ,G}$. Then the function $\mathcal{A}\varphi$ is a solution of the equation Aϕ = φ. Furthermore, among all other solutions $\psi \in \mathcal{D}\left(A\right)$ of Aϕ = φ there holds

$$\begin{equation}\sum _{g,l=1}^{G,L}\sum _{jk\in \mathbb{Z}}{\left\vert {\left\langle \enspace {\left(\mathcal{A}\varphi \right)}_{l},{w}_{jk,lg}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)}\right\vert }^{2}{\leqslant}\sum _{g,l=1}^{G,L}\sum _{jk\in \mathbb{Z}}{\left\vert {\left\langle \enspace {\psi }_{l},{w}_{jk,lg}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)}\right\vert }^{2},\end{equation} \tag{ 4.31 }$$

Moreover, among all other possible expansions of $\mathcal{A}\varphi$ in terms of the functions ${\tilde {w}}_{jk,lg}$, (4.22) is the most economical one in the sense of proposition 3.1.

Proof. Let $\varphi \in {{L}_{2}\left({{\Omega}}_{A}\right)}^{G}$ and let ${\varphi }_{jk,g}={\left\langle \enspace {\varphi }_{g},{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}$ be its canonical expansion coefficients, collected in the vectors φ_jk ; compare with (4.8). Furthermore, let $\phi \in \mathcal{D}\left(A\right)$ and let ${\phi }_{jk,lg}={\left\langle \enspace {\phi }_{l},{w}_{jk,lg}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)}$ be its canonical expansion coefficients, collected in the vectors ϕ_jk ; compare with (4.13) and (4.17). Due to proposition 4.4 there holds,

$$\begin{equation}{\left\langle \enspace {\left(A\phi \right)}_{g},{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}\;\left(4.15\right){=}\;\left(2T\right)\sum _{l=1}^{L}{c}_{l,g}^{-1}{w}_{jk}\left({\alpha }_{g}{h}_{l}/{c}_{l,g}\right){\phi }_{jk,lg}\;\left(4.18\right){=}\;{\left({A}_{jk}{\phi }_{jk}\right)}_{g},\end{equation} \tag{ 4.32 }$$

where the matrices A_jk are as in (4.18). Since due to lemma 4.2 the set ${\left\{{w}_{jk}\right\}}_{jk\in \mathbb{Z}}$ forms a tight frame over L₂(Ω_A ) with frame bound 1, it follows with (3.9) that

$$\begin{align*}\hfill {\Vert}A\phi -\varphi {{\Vert}}_{{L}_{2}\left({{\Omega}}_{A}\right)}^{2}& =\sum _{g=1}^{G}{\Vert}{\left(A\phi \right)}_{g}-{\varphi }_{g}{{\Vert}}_{{L}_{2}\left({{\Omega}}_{A}\right)}^{2}\hfill \\ \hfill & \left(3.9\right){=}\;\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\left\vert {\left\langle \enspace {\left(A\phi \right)}_{g}-{\varphi }_{g},{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}\right\vert }^{2}\hfill \\ \hfill & \left(4.32\right){=}\;\sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\left\vert {\left({A}_{jk}{\phi }_{jk}\right)}_{g}-{\varphi }_{jk,g}\right\vert }^{2}.\hfill \end{align*}$$

Hence, together with

$$\begin{align*}\hfill \sum _{g=1}^{G}\sum _{jk\in \mathbb{Z}}{\left\vert {\left({A}_{jk}{\phi }_{jk}\right)}_{g}-{\varphi }_{jk,g}\right\vert }^{2}& =\sum _{jk\in \mathbb{Z}}\sum _{g=1}^{G}{\left\vert {\left({A}_{jk}{\phi }_{jk}\right)}_{g}-{\varphi }_{jk,g}\right\vert }^{2}\hfill \\ \hfill & =\sum _{jk\in \mathbb{Z}}{\Vert}{A}_{jk}{\phi }_{jk}-{\varphi }_{jk}{{\Vert}}_{{\ell }_{2}}^{2},\hfill \end{align*}$$

we obtain

$$\begin{equation}{\Vert}A\phi -\varphi {{\Vert}}_{{L}_{2}\left({{\Omega}}_{A}\right)}^{2}=\sum _{jk\in \mathbb{Z}}{\Vert}{A}_{jk}{\phi }_{jk}-{\varphi }_{jk}{{\Vert}}_{{\ell }_{2}}^{2}.\end{equation} \tag{ 4.33 }$$

Thus, ϕ is a solution of Aϕ = φ if and only if its expansion coefficients ϕ_jk satisfy

$$\begin{equation}{A}_{jk}{\phi }_{jk}={\varphi }_{jk},\quad \forall \enspace jk\in \mathbb{Z}.\end{equation} \tag{ 4.34 }$$

The solutions of these matrix-vector systems can be characterized via the singular systems ${\left({\sigma }_{jk,n},{u}_{jk,n},{v}_{jk,n}\right)}_{n=1}^{{r}_{jk}}$ of the matrices A_jk . Since in proposition 4.5 we showed that r_jk = G and ${u}_{jk,n}={ \overrightarrow {e}}_{n}$, it follows that at least one solution of (4.34) exists. Hence, by the properties of the SVD it follows that the vectors

$$\begin{equation}{A}_{jk}^{{\dagger}}{\varphi }_{jk}{:=}\sum _{n=1}^{{r}_{jk}}\frac{\left({u}_{jk,n}^{H}\enspace {\varphi }_{jk}\right)}{{\sigma }_{jk,n}}{v}_{jk,n},\end{equation} \tag{ 4.35 }$$

are the unique solution of (4.34) with minimal ℓ₂ norm. Hence, if ϕ can be found such that ${\phi }_{jk}={A}_{jk}^{{\dagger}}{\varphi }_{jk}$, then for all other solutions ψ of Aϕ = φ there holds

$$\begin{equation}\sum _{g,l=1}^{G,L}{\left\vert {\left\langle \enspace {\phi }_{l},{w}_{jk,lg}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)}\right\vert }^{2}={\Vert}{\phi }_{jk}{{\Vert}}_{{\ell }_{2}}{\leqslant}{\Vert}{\psi }_{jk}{{\Vert}}_{{\ell }_{2}}=\sum _{g,l=1}^{G,L}{\left\vert {\left\langle \enspace {\psi }_{l},{w}_{jk,lg}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)}\right\vert }^{2}.\end{equation} \tag{ 4.36 }$$

We now show that the choice $\phi {:=}\mathcal{A}\varphi$ is exactly such that

$$\begin{equation}{\phi }_{jk}={\left(\mathcal{A}\varphi \right)}_{jk}={A}_{jk}^{{\dagger}}{\varphi }_{jk},\quad \forall \enspace jk\in \mathbb{Z}.\end{equation} \tag{ 4.37 }$$

Since due to (4.17) there holds ${\phi }_{jk,lg}={\left({\phi }_{jk}\right)}_{l+\left(g-1\right)L}$, this is equivalent to showing

$$\begin{equation*}{\left({\phi }_{jk}\right)}_{l+\left(g-1\right)L}={\left({A}_{jk}^{{\dagger}}{\varphi }_{jk}\right)}_{l+\left(g-1\right)L},\quad \forall \enspace jk\in \mathbb{Z},\enspace 1{\leqslant}l{\leqslant}L,\enspace 1{\leqslant}g{\leqslant}G.\end{equation*}$$

For this, note first that as in the proof of corollary 4.6 we have that

$$\begin{align*}\hfill {\left({A}_{jk}^{{\dagger}}{\varphi }_{jk}\right)}_{l+\left(g-1\right)L}& \left(4.35\right){=}\;\sum _{n=1}^{{r}_{jk}}\frac{\left({u}_{jk,n}^{H}\enspace {\varphi }_{jk}\right)}{{\sigma }_{jk,n}}{\left({v}_{jk,n}\right)}_{l+\left(g-1\right)L}\hfill \\ \hfill & \underset{\left(4.26\right)}{\left(4.25\right){=}}\;\left(2T\right)\frac{{c}_{l,g}^{-1}}{{\sigma }_{jk,g}^{2}}{\left({\varphi }_{jk}\right)}_{g}{w}_{jk}\left(\frac{-{\alpha }_{g}{h}_{l}}{{c}_{l,g}}\right),\hfill \end{align*}$$

and thus together with the definition of a_l in (4.29) there holds

$$\begin{equation}{\left({A}_{jk}^{{\dagger}}{\varphi }_{jk}\right)}_{l+\left(g-1\right)L}={\left({a}_{l}\right)}_{jk,g}.\end{equation} \tag{ 4.38 }$$

Hence, we now have to show that

$$\begin{equation*}{\phi }_{jk,lg}={\left({a}_{l}\right)}_{jk,g},\quad \forall \enspace jk\in \mathbb{Z},\enspace 1{\leqslant}l{\leqslant}L,\enspace 1{\leqslant}g{\leqslant}G.\end{equation*}$$

To do so, note that in (4.30) in the proof of lemma 4.7 we saw that

$$\begin{equation*}{\phi }_{l}={\left(\mathcal{A}\varphi \right)}_{l}={\tilde {F}}_{l}^{{\ast}}{a}_{l},\end{equation*}$$

where ${\tilde {F}}_{l}$ is the frame operator of the dual frame ${\left\{{\tilde {w}}_{jk,lg}\right\}}_{jk\in \mathbb{Z},g=1,\dots ,G}$. Hence, we obtain

$$\begin{equation*}{F}_{l}{\phi }_{l}={F}_{l}{\left(\mathcal{A}\varphi \right)}_{l}={F}_{l}{\tilde {F}}_{l}^{{\ast}}{a}_{l}={P}_{l}{a}_{l}={a}_{l},\end{equation*}$$

where we used the fact (see [3]) that ${F}_{l}{\tilde {F}}_{l}^{{\ast}}={P}_{l}$, with P_l denoting the orthogonal projector in ℓ₂ onto $R\left({\tilde {F}}_{l}\right)=R\left({F}_{l}\right)$, and that by assumption a_l ∈ R(F_l ). However, since this implies that

$$\begin{equation*}{\phi }_{jk,lg}={\left\langle \enspace {\phi }_{l},{w}_{jk,lg}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)}={\left({F}_{l}{\phi }_{l}\right)}_{jk,g}={\left({a}_{l}\right)}_{jk,g},\end{equation*}$$

it now follows that (4.37) holds, and thus $\mathcal{A}\varphi$ is a solution of Aϕ = φ. Together with (4.36), this also implies the coefficient inequality (4.31). Note that due to lemma 4.7, $\mathcal{A}\varphi$ is well-defined since $\varphi \in {{L}_{2}\left({{\Omega}}_{A}\right)}^{G}$. Finally, since ${\phi }_{jk,lg}={\left\langle \enspace {\phi }_{l},{w}_{jk,lg}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)}={\left({a}_{l}\right)}_{jk,g}$, it follows that the expansion (4.22) is the most economical one in the sense of proposition 3.1, which concludes the proof.□

Remark. In the above theorem we saw that $\mathcal{A}\varphi$ is a solution of the atmospheric tomography problem given that a_l ∈ R(F_l ), which can be interpreted as a consistency condition. On the other hand, if instead we only assume that Aϕ = φ is solvable, then it follows from (4.33) that for any solution ϕ there holds A_jk ϕ_jk = φ_jk , and thus that

$$\begin{equation*}{P}_{N{\left({A}_{jk}\right)}^{\perp }}{\phi }_{jk}={A}_{jk}^{{\dagger}}{\varphi }_{jk}.\end{equation*}$$

Since the complete set of solutions is given by ϕ^† + N(A), where ϕ^† ∈ N(A)^⊥ denotes the minimum-norm solution of Aϕ = φ, and since by (4.33) for any $\tilde {\phi }\in N\left(A\right)$ there holds ${\tilde {\phi }}_{jk}\in N\left({A}_{jk}\right)$, it follows that

$$\begin{equation*}{A}_{jk}^{{\dagger}}{\varphi }_{jk}={P}_{N{\left({A}_{jk}\right)}^{\perp }}{\phi }_{jk}^{{\dagger}}.\end{equation*}$$

Noting that this can be rewritten as

$$\begin{equation}{A}_{jk}^{{\dagger}}{\varphi }_{jk}={\phi }_{jk}^{{\dagger}}+\left({P}_{N{\left({A}_{jk}\right)}^{\perp }}-I\right){\phi }_{jk}^{{\dagger}}={\phi }_{jk}^{{\dagger}}-{P}_{N\left({A}_{jk}\right)}{\phi }_{jk}^{{\dagger}},\end{equation} \tag{ 4.39 }$$

and since due to (4.30) and (4.38) there holds

$$\begin{equation*}{\left(\mathcal{A}\varphi \right)}_{l}={\tilde {F}}_{l}^{{\ast}}{\left\{{\left({A}_{jk}^{{\dagger}}{\varphi }_{jk}\right)}_{l+\left(g-1\right)L}\right\}}_{jk\in \mathbb{Z},g=1,\dots ,G},\end{equation*}$$

it now follows from (4.39) that

$$\begin{equation*}\mathcal{A}\varphi ={\phi }^{{\dagger}}+{\left({\tilde {F}}_{l}^{{\ast}}{b}_{l}\right)}_{l=1}^{L},\quad \text{where}\enspace {b}_{l}{:=}{\left\{{\left({P}_{N\left({A}_{jk}\right)}{\phi }_{jk}^{{\dagger}}\right)}_{l+\left(g-1\right)L}\right\}}_{jk\in \mathbb{Z},g=1,\dots ,G},\end{equation*}$$

which implies that if a_l ∉ R(F_l ) one can still consider $\mathcal{A}\varphi$ as an approximate solution.

Remark. The explicit representation of $\mathcal{A}$ given in (4.24) can be used as the basis of an efficient numerical routine for (approximately) solving the atmospheric tomography problem Aϕ = φ. Replacing the infinite sum over the indices j, k by a finite sum, all one needs for implementation are the vectors φ_jk and the evaluation of the functions ${\tilde {w}}_{jk,lg}\left(x,y\right)$ on a predetermined grid. For this, note that since the functions ${\left\{{\tilde {w}}_{jk,lg}\right\}}_{jk\in \mathbb{Z},g=1,\dots ,G}$ form the dual frame of the functions ${\left\{{w}_{jk,lg}\right\}}_{jk\in \mathbb{Z},g=1,\dots ,G}$, they can be efficiently numerically approximated via the approximation formula (3.8). Furthermore, the coefficients φ_jk can be efficiently computed via (4.8) using the Fourier transform. Thus, since apart from φ_jk all other quantities can be precomputed independently of the input data φ, the computation of $\mathcal{A}\varphi$ via formula (4.24) can be efficiently numerically implemented.

Remark. In lemma 4.7, we have derived that ${\Vert}\mathcal{A}{\Vert}{\leqslant}1/L$, i.e., that $\mathcal{A}$ is bounded. In particular, for any noisy measurement φ^δ of the incoming wavefronts φ this implies

$$\begin{equation*}{\Vert}\mathcal{A}\varphi -\mathcal{A}{\varphi }^{\delta }{{\Vert}}_{\mathcal{D}\left(A\right)}{\leqslant}1/L{\Vert}\varphi -{\varphi }^{\delta }{{\Vert}}_{{{L}_{2}\left({{\Omega}}_{A}\right)}^{G}},\end{equation*}$$

which shows that the (approximate) solution of the atmospheric tomography problem via the application of the operator $\mathcal{A}$ is stable with respect to noise in the data.

At first, this result seems counter-intuitive, since the atmospheric tomography operator is derived from the Radon transform under the assumption of a layered atmosphere. Hence, since inverting the Radon transform is known to be an ill-posed problem, one expects $\mathcal{A}$ to become unbounded with an increasing number of layers L. However, note that on the one hand $\mathcal{A}\varphi$ is only an approximate solution for a_l ∉ R(F_l ), and on the other hand, for L going to infinity, the atmospheric tomography operator A as defined in (2.1) does not tend towards the (limited-angle) Radon transform.

To see this, note that the sum in the definition of A stems from the discretization of the line integrals of the Radon transform via the simple quadrature rule

$$\begin{equation*}\underset{0}{\overset{1}{\int }}f\left(x\right)\enspace \mathrm{d}x\approx \sum _{k=1}^{L}f\left({x}_{k}\right)\left({x}_{k}-{x}_{k-1}\right),\end{equation*}$$

where the weights (x_k − x_k−1) were dropped. Assuming that (x_k − x_k−1) = 1/L, which would correspond to equidistant atmospheric layers, the above quadrature rule becomes

$$\begin{equation*}\underset{0}{\overset{1}{\int }}f\left(x\right)\enspace \mathrm{d}x\approx \frac{1}{L}\sum _{k=1}^{L}f\left({x}_{k}\right),\end{equation*}$$

which indicates that unless the atmospheric tomography operator is multiplied by 1/L in this case, it does not converge to the desired limit as the number of layers tends to infinity. Incidentally, if A is replaced by (1/L)A, then $\mathcal{A}$ has to be replaced by $L\mathcal{A}$, and it then follows from (4.27) that

$$\begin{equation}{\Vert}L\mathcal{A}{\Vert}=L{\Vert}\mathcal{A}{\Vert}{\leqslant}\sqrt{L},\end{equation} \tag{ 4.40 }$$

resulting in an upper bound for the solution operator $L\mathcal{A}$ which tends towards infinity for an increasing number of layers L, as one would expect.

Note that further adaptations to the tomography operator A would have to be made in order to ensure its convergence to the Radon transform in the limit for L → ∞. These correspond to the integration weights having to be adapted to different guide star directions. However, since this is practically not relevant as L is generally fixed, we do not go into further details here. In summary, we want to emphasize that the upper bound on $\mathcal{A}$ does not contradict the ill-posedness of the general tomography problem.

Remark. Equation (4.31) implies that $\mathcal{A}\varphi$ can be seen as the minimum-coefficient solution of Aϕ = φ. Since the sets ${\left\{{w}_{jk,lg}\right\}}_{jk\in \mathbb{Z},g=1,\dots ,G}$ do not form tight frames over L₂(Ω_l ), $\mathcal{A}\varphi$ is not necessarily also the minimum-norm solution of the equation.

One way to obtain a minimum-norm solution is to take M_l piecewise disjoint domains Ω_l,m ⊂ Ω_l satisfying

$$\begin{equation*}{{\Omega}}_{l}={\bigcup }_{m=1}^{{M}_{l}}{{\Omega}}_{l,m},\quad \text{and}\quad \forall \enspace l,g\enspace \enspace \enspace \exists \enspace {N}_{l,g}\subset \left\{1,\dots ,{M}_{l}\right\}:\enspace {c}_{l,g}{{\Omega}}_{A}+{\alpha }_{g}{h}_{l}={\bigcup }_{m\in {N}_{l,g}}{{\Omega}}_{l,m},\end{equation*}$$

and to use, instead of ${\left\{{w}_{jk,lg}\right\}}_{jk\in \mathbb{Z},g=1,\dots ,G}$, the set of functions

$$\begin{equation*}{\left\{{w}_{jk,lgm}\right\}}_{jk\in \mathbb{Z},\enspace g=1,\dots ,G,\enspace m=1,\dots ,{M}_{l}},\end{equation*}$$

where

$$\begin{equation*}{w}_{jk,lgm}\left(x,y\right){:=}{c}_{l,g}^{-1}{w}_{jk}\left(\left(x,y\right)/{c}_{l,g}\right){I}_{{{\Omega}}_{l,m}}\left(x,y\right).\end{equation*}$$

Similarly as in the proof of proposition 4.3, it can be shown that for fixed l, these functions form a tight frame with frame bound G for L₂(Ω_l ). Hence, in complete analogy to (4.13), it is possible to decompose ϕ_l as follows:

$$\begin{equation}\begin{aligned}\hfill {\phi }_{l}\left(x,y\right)& =\frac{1}{G}\sum _{g=1}^{G}\sum _{m=1}^{{M}_{l}}\sum _{jk\in \mathbb{Z}}{\phi }_{jk,lgm}{w}_{jk,lgm}\left(x,y\right),\hfill \\ \hfill {\phi }_{jk,lgm}& ={\left\langle \enspace {\phi }_{l},{w}_{jk,lgm}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)}.\hfill \end{aligned}\end{equation} \tag{ 4.41 }$$

Following the same steps as in the proof of proposition 4.4, one finds that

$$\begin{equation*}{\left\langle \enspace {\left(A\phi \right)}_{g},{w}_{jk}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{A}\right)}=\left(2T\right)\sum _{l=1}^{L}{c}_{l,g}^{-1}{w}_{jk}\left({\alpha }_{g}{h}_{l}/{c}_{l,g}\right)\sum _{m\in {N}_{l,g}}{\phi }_{jk,lgm},\end{equation*}$$

from which it follows that

$$\begin{equation*}{\left(A\phi \right)}_{g}\left(x,y\right)=\sum _{jk\in \mathbb{Z}}\left(\left(2T\right)\sum _{l=1}^{L}{c}_{l,g}^{-1}{w}_{jk}\left({\alpha }_{g}{h}_{l}/{c}_{l,g}\right)\sum _{m\in {N}_{l,g}}{\phi }_{jk,lgm}\right){w}_{jk}\left(x,y\right).\end{equation*}$$

Defining (again with a small abuse of notation) the vectors

$$\begin{equation}\begin{aligned}\hfill {\phi }_{jk}& {:=}{\left({\phi }_{jk,g}\right)}_{g=1,\dots ,G}\in {\mathbb{C}}^{G\cdot L\enspace \cdot {M}_{1}\cdot ,\dots ,\cdot {M}_{L}},\hfill \\ \hfill {\phi }_{jk,g}& {:=}\left({\left({\phi }_{jk,1gm}\right)}_{m=1}^{{M}_{1}},\dots ,{\left({\phi }_{jk,Lgm}\right)}_{m=1}^{{M}_{L}}\right)\in {\mathbb{C}}^{L\enspace \cdot {M}_{1}\cdot ,\dots ,\cdot {M}_{L}},\hfill \end{aligned}\end{equation} \tag{ 4.42 }$$

we get, in analogy to (4.19), that A can be written as

$$\begin{equation}\left(A\phi \right)\left(x,y\right)=\sum _{jk\in \mathbb{Z}}\left({A}_{jk}{\phi }_{jk}\right){w}_{jk}\left(x,y\right),\end{equation} \tag{ 4.43 }$$

but now with

$$\begin{equation*}{A}_{jk}=\left(\begin{matrix}\hfill {A}_{jk,1}\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {A}_{jk,2}\hfill & \hfill \ddots \hfill & \hfill {\vdots}\hfill \\ \hfill {\vdots}\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill {A}_{jk,G}\hfill \end{matrix}\right)\in {\mathbb{C}}^{G{\times}G\enspace \cdot L\enspace \cdot {M}_{1}\cdot ,\dots ,\cdot {M}_{L}},\end{equation*}$$

where

$$\begin{equation*}\begin{aligned}\hfill \left({A}_{jk,g}\right)=\left(2T\right)\left(\begin{matrix}\hfill {c}_{1,g}^{-1}{w}_{jk}\left({\alpha }_{g}{h}_{1}/{c}_{1,g}\right){\left({I}_{m\in {N}_{1,g}}\right)}_{m=1}^{{M}_{1}}\hfill & \hfill \cdots \hfill & \hfill {c}_{L,g}^{-1}{w}_{jk}\left({\alpha }_{g}{h}_{L}/{c}_{L,g}\right){\left({I}_{m\in {N}_{1,g}}\right)}_{m=1}^{{M}_{L}}\hfill \end{matrix}\right)\\ \hfill \in {\mathbb{C}}^{1{\times}L\enspace \cdot {M}_{1}\cdot ,\dots ,\cdot {M}_{L}},\end{aligned}\end{equation*}$$

and

$$\begin{equation*}{I}_{m\in {N}_{l,g}}{:=}\begin{cases}1\quad m\in {N}_{l,g},\quad \hfill \\ 0\quad \text{else}.\quad \hfill \end{cases}\end{equation*}$$

Denoting again by ${\left({\sigma }_{jk,n},{u}_{jk,n},{v}_{jk,n}\right)}_{n=1}^{{r}_{jk}}$ the singular value decompositions of the matrices A_jk , we arrive at the same decomposition of A as in (4.21), but obviously with the new singular values and functions. All results derived above hold analogously for this new decomposition, with the difference that, due to the tightness of the frame, $\mathcal{A}\varphi$ is then not only the minimum-coefficient solution of Aϕ = φ, but also the minimum-norm solution. Furthermore, one can again derive the singular values of A_jk explicitly, namely

$$\begin{equation*}{\sigma }_{jk,n}=\sqrt{\sum _{l=1}^{L}{c}_{l,n}^{-2}\sum _{m=1}^{{M}_{l}}{I}_{m\in {N}_{l,n}}}=\sqrt{\sum _{l=1}^{L}{c}_{l,n}^{-2}\left\vert {N}_{l,n}\right\vert },\end{equation*}$$

In addition, as in (4.7) one can again show that $\mathcal{A}\varphi$ is well-defined and that

$$\begin{equation*}{\Vert}\mathcal{A}{\Vert}{\leqslant}{\left(\underset{g=1,\dots ,G}{\mathrm{min}}\left\{{\sigma }_{jk,g}\right\}\right)}^{-1}={\left(\underset{g=1,\dots ,G}{\mathrm{min}}\left\{\sum _{l=1}^{L}{c}_{l,n}^{-2}\left\vert {N}_{l,n}\right\vert \right\}\right)}^{-\frac{1}{2}}.\end{equation*}$$

However, note that using this approach, the number of subdomains Ω_l,m can increase strongly with the number of guide stars G. Furthermore, note that the number of frames per layer considered in the two approaches presented in this section are directly related to G and M_l , respectively. Hence, since the used frames have discontinuities, using the approach based on the frames {w_jk,lgm } potentially introduces more discontinuities into the solution in a numerical implementation than the approach based on the frames {w_jk,lg }, which is not necessarily desirable in practice.

Remark. In some situations, it is desirable to, instead of the standard L₂ inner product (2.2), work with the weighted inner product

$$\begin{equation}\left\langle \enspace \phi ,\psi \enspace \right\rangle {:=}\sum _{l=1}^{L}\frac{1}{{\gamma }_{l}}{\left\langle \enspace {\phi }_{l},{\psi }_{l}\enspace \right\rangle }_{{L}_{2}\left({{\Omega}}_{l}\right)},\end{equation} \tag{ 4.44 }$$

where ${\left\{{\gamma }_{l}\right\}}_{l=1}^{L}$ denotes a normalized, non-zero sequence of weights. Although we do not focus on this issue in our current investigation, it should be noted that adapting the frame decomposition to include this weighted inner product should be straightforward.