Lorentz-covariant sampling theory for fields

The typical focus of sampling theory is the reconstruction of some particular class of functions on a space from a discrete set of sample points. The original sampling theorem (1) applies to functions on $\mathbb{R}$ with frequency support in a bounded interval, but the basic idea of reconstructing functions from sample values has been extended to many other contexts in communication theory (see, e.g., [119, 120, 121, 122]). For example, similar results apply to bandlimited functions whose Fourier transforms have support on more general subsets of finite measure. One can also consider analogues of sampling in higher dimensions and on manifolds different from $\mathbb{R}^{N}$. Going in this direction, the first task is to determine an appropriate generalization of bandlimitation. For example, a natural analogue of bandlimitation in higher-dimensional Euclidean space as well as Minkowski spacetime is a restriction in Fourier space to some subset of momenta, $\vec{k}$, or four-momenta, $k^{\mu}$, respectively.

Here we will be particularly interested in restrictions which preserve symmetries. In the one-dimensional scenario, where we restrict $|k|<\Omega$ and impose Dirichlet boundary conditions, we obtain a bandlimited subspace which is invariant under translations (we will discuss this in more detail in Section 5). In higher-dimensional Euclidean space, a simple bandlimited subspace which is invariant under both translations and rotations consists of functions whose Fourier transforms have support in the region $\vec{k}^{2}<\Omega^{2}$, for some fixed $\Omega>0$. Similarly, in Minkowski spacetime, a bandlimited subspace invariant under Poincaré transformations consists of wavepackets of the form:

where $k^{2}\equiv k_{\mu}k^{\mu}$ and $k\cdot x\equiv k_{\mu}x^{\mu}$, with metric signature $(+,-,-,-)$. This latter extension of the notion of bandlimitation to Minkowski spacetime was first proposed by Kempf in [85]. Here the momentum space cutoff is considered as a physical Lorentz-invariant cutoff. Note that this is different from introducing a formal Euclidean regulator, $k_{E}^{2}<\Omega^{2}$, after a Wick rotation. In particular, a Euclidean cutoff will fully regulate momentum space integrals by restricting the integration region to a finite volume. The Minkowski spacetime bandlimit generally will not.

In the settings of curved spaces and spacetimes, a generalization of plane waves are the eigenfunctions of the covariant Laplacian or d’Alembertian operator associated with a metric (namely, the operator $\Box_{g}=|g|^{-\frac{1}{2}}\partial_{\mu}(|g|^{\frac{1}{2}}g^{\mu\nu}\partial_{\nu}\cdot)$ for the metric $g_{\mu\nu}$). Bandlimitation can then be defined as a projection onto the closure of some subset of these eigenfunctions. In the flat space and spacetime cases, the symmetry-preserving restrictions $\vec{k}^{2}<\Omega^{2}$ and $|k^{2}|<\Omega^{2}$ can be thought of as a projection onto the eigenspaces corresponding to the eigenvalues of $\vec{\nabla}^{2}$ and $\Box=\partial_{t}^{2}-\vec{\nabla}^{2}$ (respectively) with magnitudes below some threshold $\Omega>0$. This kind of restriction can be generalized to manifolds with curvature, where one projects onto the eigenspaces of the covariant Laplacian or d’Alembertian with corresponding eigenvalues of magnitude below a cutoff $\Omega>0$. Because the spectra of these covariant differential operators are invariant under general coordinate transformations, so are the corresponding bandlimited subspaces. Therefore, this kind of restriction gives us a general coordinate-invariant notion of bandlimitation. This idea was introduced for the case of Riemannian manifolds by Pesenson in [123, 124] (see also developments in [85, 125, 126]), and for pseudo-Riemannian manifolds by Kempf in [85]. In this paper, however, we will be primarily focused on the case of flat Minkowski spacetime.

Once a bandlimited subspace has been fixed, one aims to find an appropriate set of sample points, $\{x_{n}\}_{n\in\mathbb{Z}}$, so that a bandlimited function is uniquely specified among elements in this space by its values on the sampling lattice. Typically, one also requires the sampling to be stable, in the sense that there is some constant $C>0$ for which

where $\|\phi\|$ is the Hilbert space norm of the bandlimited function $\phi$ (we will assume that our bandlimited functions form a subspace of some Hilbert space). The stability requirement is important in applications to ensure that noise in the signal $\phi$ does not produce arbitrarily large errors after sampling and reconstruction. Noise present in a signal may not always be bandlimited, but typically one can ensure the signal which one samples is bandlimited by first applying an appropriate filter.

There have been important results in identifying properties of sampling lattices which are necessary and/or sufficient for the reconstruction of particular classes of bandlimited function spaces. Often these involve concocting different notions of density of a given sampling lattice (see Section 1.2 of [122] for an overview). One of the most important results is a theorem of Beurling and Malliavin [127, 128]. They employ the following notion of density: for a given lattice $\Delta:=\{x_{n}\}_{n\in\mathbb{Z}}\subset\mathbb{R}$, let $\gamma$ be any interval in $\mathbb{R}$ of length $L$, and $N^{-}(L):=\inf_{\gamma}\#\{\Delta\cap\gamma\}$ (i.e., roughly speaking, the smallest number of samples occuring within an interval of length $L$). Then the (lower) Beurling density of $\Delta$ is defined as:

Beurling and Malliavin proved that for functions in $L^{2}(\mathbb{R})$ with Fourier transforms supported on a single interval, $R$, of length $|R|$, a stable sampling is possible for $\Delta$ if and only if $\Delta$ contains a uniformly discrete subset $\Delta_{0}$ with $D^{-}(\Delta_{0})>|R|/2\pi$. (Note: uniformly discrete simply means that the distance between any two distinct points is greater than some fixed positive constant.) Therefore, functions with a larger support in Fourier space require a higher density of sample points in order to obtain a stable reconstruction. This result also shows that the lower Beurling density, which is a kind of average density, is the only relevant feature of a lattice required to establish the possibiliy of a stable reconstruction (at least in principle). For the case of functions bandlimited to $(-\Omega,\Omega)$, the critical density $\Omega/\pi$ was already identified in the work of Nyquist [4], and hence is usually referred to as the Nyquist density in that context.

The work of Beurling and Malliavin was extended by Landau [129, 130], providing a necessary density condition for the case of functions bandlimited to an arbitrary subset of Fourier space, as well as to the case of higher dimensions. Specifically, Landau showed that for the space of functions in $L^{2}(\mathbb{R}^{N})$ whose Fourier transforms are supported in a subset $R$ of Fourier space, if a stable sampling is possible for a lattice $\Delta\subset\mathbb{R}^{N}$, then

where $|R|$ is the volume of the subset $R$ in Fourier space, and $D^{-}(\Delta)$ is a natural analogue of the lower Beurling density in higher dimensions (where one considers the number of sample points in $N$-dimensional balls of some fixed radius instead of intervals in the one-dimensional case, see [129] for details). We will refer to the lower bound of $|R|/(2\pi)^{N}$ as the Beurling-Landau density required for stable reconstruction in the cases considered by Landau.

Some results have also been obtained for bandlimited functions on Riemannian manifolds with curvature [123, 124] (see also discussion in [125, 126]). In the pseudo-Riemannian case, the nature of the corresponding sampling is qualitatively different. We will elaborate upon this for the Minkowski spacetime case in Section 4. An extension of the analysis of the Minkowski case to Friedmann-Robertson-Walker spacetimes was done in [89]. Little is known about sampling on pseudo-Riemannian manifolds beyond these examples.

The above density conditions indicate whether or not reconstruction from a given lattice may be possible in principle, but one may also wish to determine explicitly how the bandlimited functions can be reconstructed from their sample values. Concretely, this would amount to finding an analogue of (1), of the general form:

for an arbitrary function, $\phi$, in the bandlimited subspace. The function $K_{n}(x)$ will be referred to as a reconstruction kernel for this bandlimited space. In the simplest cases, $K_{n}$ will only depend on $n$ through the sample point $x_{n}$, as in (1) with $K_{n}(x)=\operatorname{sinc}[\Omega(x-x_{n})]$. However, generally it may be necessary to use different kernels at different sample points (as we will see later in Subsection 4.2). There are a variety of techniques used for obtaining reconstruction formulas in particular scenarios (see, e.g., [119, 120, 121, 122]).

As an illustration of some of the ideas we have presented in this section, we will now turn to a simple derivation of the Shannon sampling theorem (1) for functions on $\mathbb{R}$ bandlimited to $|k|<\Omega$. The method used in this derivation will develop the reconstruction kernel as well as establish the Nyquist density for this bandlimited space. In Section 4, we will then employ an extension of this method due to Kohlenberg [131] to develop a kind of reconstruction formula for bandlimited functions in Minkowski spacetime.

In this subsection, we will only be considering the subspace of $L^{2}(\mathbb{R})$ consisting of functions bandlimited to the region $|k|<\Omega$ of Fourier space. We will first make some helpful definitions and establish some notation. Let $\{x_{n}\}_{n\in\mathbb{Z}}$ denote some discrete set of points in $\mathbb{R}$. The sampling operator corresponding to this set is defined as:

where $|x_{n})$ denotes the position eigenvector²²2Note that we use round brackets $|\phi)$ instead of angled brackets $\ket{\phi}$ to denote an element of a Hilbert space so they are not confused with quantum states. Of course, this discussion applies equally well to bandlimited wavefunctions, but here we have in mind that the Hilbert spaces physically correspond to a space of field configurations which are integrated over in a path integral. centered at $x=x_{n}$. These position eigenvectors are continuum-normalized, $(x|x^{\prime})=\delta(x-x^{\prime})$, form a resolution of identity, $\int dx|x)(x|=1$, and define the position space representation of Hilbert space elements by $\phi(x)\equiv(x|\phi)$. (Note that these position vectors have not yet been projected down to the bandlimited subspace.) As the name suggests, the sampling operator extracts the values of a function at the discrete set of points $\{x_{n}\}_{n\in\mathbb{Z}}$, by: $S|\phi)=\sum_{n\in\mathbb{Z}}\phi(x_{n})|x_{n})$.³³3Note that the sampling operator is not a well-defined operator on $L^{2}(\mathbb{R})$. However, throughout this section, it will only appear in the combination $HS$, and it should be clear in what follows that with our eventual choice of a filter $H$, the operator $HS$ will be well-defined on the space of bandlimited functions. It will simply be helpful for illustrative purposes to identify $S$ separately. We will also define a filter,

as some operator on $L^{2}(\mathbb{R})$ which is diagonal in Fourier space. Our momentum eigenvectors are normalized as $(k|k^{\prime})=2\pi\delta(k-k^{\prime})$, and thus $\int\frac{dk}{2\pi}|k)(k|=1$. The Fourier-space representation of a Hilbert space element is $\tilde{\phi}(k):=(k|\phi)=\int dx\phi(x)e^{-ikx}$, where $(k|x)=e^{-ikx}$.

Now we claim that we can obtain a sampling formula if we can find a particular discrete set of sample points, $\{x_{n}\}_{n\in\mathbb{Z}}$, and a filter, $H(k)$, such that the operator $HS$ is the identity on the bandlimited subspace (where $S$ is the sampling operator associated with this set of sample points). If this can be achieved, then we immediately get a reconstruction formula of the form (13) for any bandlimited function $\phi$:

We can identify the reconstruction kernel as $K_{n}(x)=(x|H|x_{n})$ in this case. Formulating the problem in this way is not conducive to every sampling problem, but it is particularly useful when attempting to find reconstruction formulas with equidistantly-spaced sampling lattices, i.e., when $x_{n}=\frac{n\pi}{\Theta}$ for an appropriately chosen $\Theta$.

The simplest example is the present case of functions in $L^{2}(\mathbb{R})$ bandlimited to $|k|<\Omega$. Let us consider a sampling lattice $\{x_{n}=\frac{n\pi}{\Theta}\}_{n\in\mathbb{Z}}$ for some $\Theta$ to be determined. This fixes the sampling operator, $S=\sum_{n\in\mathbb{Z}}|x_{n})(x_{n}|$, up to the choice of $\Theta$. Now we need to choose $\Theta$ and find a filter $H(k)$ so that $|\phi)\stackrel{{\scriptstyle!}}{{=}}HS|\phi)$ for all $\phi$ in the bandlimited subspace. Let us then examine this condition in Fourier space:

The step from line (21) to (22) uses the assumption that the sample points are equidistantly spaced, so that:

Therefore, our sampling problem $|\phi)\stackrel{{\scriptstyle!}}{{=}}HS|\phi)$ is to find $\Theta$ and $H(k)$ so that for every bandlimited $\phi$ we have:

The sum on the right-hand side of this equation is over copies of $\tilde{\phi}$ translated in Fourier space by integer multiples of $2\Theta$. This sum can be visualized as in Figure 1 (for $\Theta>\Omega$).

We will call the term $\tilde{\phi}(k-2\Theta m)$ in the sum the $m^{th}$ copy of $\tilde{\phi}$. We see that the different copies of $\tilde{\phi}$ will be non-overlapping if $\Theta\geq\Omega$. In this case, if we choose

then $H(k)$ simply projects out everything except the $0^{th}$ copy, and thus (25) is satisfied for any bandlimited $|\phi)$. This gives the reconstruction kernel

In the case where we choose $\Theta=\Omega$, we have $x_{n}=\frac{n\pi}{\Omega}$, and hence arrive at the Shannon sampling formula (1). Note that when we choose $\Theta=\Omega$, the copies $m=\pm 1$ meet the $m=0$ copy at $k=\pm\Omega$ (respectively). However, if we constrain our bandlimited functions to vanish at the boundary of the interval, we do not have to be concerned with these points.

If $\Theta$ is chosen strictly larger than $\Omega$, note that $H(k)$ can be chosen arbitrarily in the region $\Omega<|k|<-\Omega+2\Theta$, since none of the copies of $\tilde{\phi}$ have support there. Typically this is exploited in practical scenarios so that $H(k)$ can decay to zero more smoothly outside of the bandlimited interval, rather than the sharp drop to zero at $k=\pm\Omega$ in the idealized case. This can help reduce sources of error which arise in practice [119]. Note, however, that it can only be accomplished by choosing a sample density larger than theoretically necessary.

If we consider choosing $\Theta<\Omega$, as illustrated in Figure 2, we will have different copies overlapping each other.

In this situation, it is not possible to choose a $H(k)$ so that (25) is satisfied for every element in the bandlimited subspace. This is because it is possible to find different bandlimited functions for which the sum in the right-hand side of (25) is the same in the region $|k|<\Omega$ (such functions are called aliases). Therefore, the filter $H(k)$ will not be able to distinguish between these aliases since it simply acts by pointwise multiplication (in $k$-space) outside of the sum in (25). We then conclude that we cannot find a reconstruction formula in this way if $\Theta<\Omega$. This demonstrates the Nyquist density criterion (or that of Beurling-Landau more generally), namely that the minimum density of sample points required for a stable reconstruction occurs when $\Theta=\Omega$.

Recall that the Poincaré-invariant notion of bandlimitation on $(N+1)$-dimensional Minkowski spacetime consists of a restriction of the support of fields in Fourier space to the region $|k^{2}|<\Omega^{2}$:

where $k\cdot x\equiv k^{0}x^{0}-\vec{k}\cdot\vec{x}$. The bandlimited region $|k^{2}|=|(k^{0})^{2}-\vec{k}^{2}|<\Omega^{2}$ is illustrated in Figure 3.

Although the bandlimited region of Fourier space shrinks (eventually to zero) as one travels away from the origin along the lines $k^{0}=\pm|\vec{k}|$, the volume of the entire region is infinite (in $1+1$ as well as higher-dimensional Minkowski spacetime). By our discussion in Subsection 3.1, this would indicate that the Beurling-Landau sample density necessary for the stable reconstruction of bandlimited functions in this space is also infinite. Note that Landau’s result also applies to the Minkowski spacetime case because the metric signature does not play a significant role in the definition of square-integrability nor of Fourier space when the metric is flat. One could view the extra minus sign in the Fourier phase (for example, in (28)) as simply establishing a different Fourier convention for the temporal component.

Therefore, we see that it is not possible to represent this space of functions on a (uniformly) discrete sampling lattice in Minkowski spacetime. Nevertheless, in [85] it was observed that through a temporal-spatial splitting of the allowed Fourier space region, one can develop certain reconstruction properties for these bandlimited functions. Consider an arbitrary bandlimited function, $\phi(x)$, and write the spatial Fourier transform as $\phi_{\vec{k}}(t)$, i.e., defined so that

Now if we examine a fixed spatial mode $\vec{k}$, in order for the function $\phi(x)$ to be bandlimited in the Poincaré-invariant sense, the time-varying amplitude $\phi_{\vec{k}}(t)$ must be bandlimited in time. Specifically, the temporal frequencies, $k^{0}$, of the amplitude $\phi_{\vec{k}}(t)$ of fixed $\vec{k}$, must satisfy:

If we define $r_{\pm}(\vec{k}):=\sqrt{\vec{k}^{2}\pm\Omega^{2}}$, then we can write the set of allowed temporal frequencies for a fixed $\vec{k}$ as:

The intervals for different choices of $\vec{k}$ can be visualized as vertical sections in Figure 3.

For any fixed $\vec{k}$, there is a finite temporal bandwidth (size of the interval $I(\vec{k})$) for the time-dependence of the Fourier coefficient $\phi_{\vec{k}}(t)$. Therefore, one expects to be able to reconstruct $\phi_{\vec{k}}(t)$ from sample values at a discrete set of points in time, provided that the sample density is greater than the Beurling-Landau density corresponding to the mode $\vec{k}$. Note that this does not contradict the requirement of an infinite sample density required for the reconstruction of the full function $\phi(x)$, since we are only considering a fixed mode $\vec{k}$. For a spatial mode with $|\vec{k}|\leq\Omega$, $\phi_{\vec{k}}(t)$ is bandlimited to $k^{0}\in(-r_{+}(\vec{k}),r_{+}(\vec{k}))$, which can be identified with the one-dimensional Shannon case. Therefore, for this mode we can use (1) as the reconstruction formula, where we identify the bandlimit as $r_{+}(\vec{k})$ and reconstruct $\phi_{\vec{k}}(t)$ from samples at the points $\{t_{\vec{k},n}=n\pi/r_{+}(\vec{k})\}_{n\in\mathbb{Z}}$. For spatial modes with $|\vec{k}|>\Omega$, the function $\phi_{\vec{k}}(t)$ is bandlimited to $k^{0}\in(-r_{+}(\vec{k}),-r_{-}(\vec{k}))\cup(r_{-}(\vec{k}),r_{+}(\vec{k}))$. An explicit reconstruction formula is also known for this case of the union of two intervals using samples at the Beurling-Landau density $[r_{+}(\vec{k})-r_{-}(\vec{k})]/\pi$ [131]. We will employ the method of [131] in the next subsection, and simply refer the reader to this reference for the one-dimensional case which we will not need to state here. The point is that in both cases $|\vec{k}|\leq\Omega$ and $|\vec{k}|>\Omega$, there is a known reconstruction formula for $\phi_{\vec{k}}(t)$ from samples in time taken at the minimal density.

How does the sample density change with $\vec{k}$? For the spatial mode $\vec{k}=0$, we have a temporal bandwidth of $2\Omega$. As we increase $|\vec{k}|$, the temporal bandwidth increases to a maximum of $2\sqrt{2}\Omega$ for spatial modes with $|\vec{k}|=\Omega$. The bandwidth then decreases monotonically, and tends to zero as $|\vec{k}|\to\infty$. Therefore, fixed spatial modes with $|\vec{k}|=\Omega$ (i.e., at the Planck scale) require the largest temporal sample density. Modes with much larger spatial frequencies require a temporal sampling which is much less dense, and which decreases to zero density as $|\vec{k}|\to\infty$. Thus, although these bandlimited functions can have arbitrarily small wavelengths, those shorter than the Planck length have a small temporal bandwidth.

Now that we know there is a reconstruction formula for each fixed spatial mode $\phi_{\vec{k}}(t)$, can we combine them into a reconstruction formula for $\phi(x)$? Schematically, let us write the reconstruction of mode $\phi_{\vec{k}}(t)$ from an appropriate lattice $\{t_{\vec{k},n}\}_{n\in\mathbb{Z}}$ as

where $K_{\vec{k},n}(t)$ is an appropriate reconstruction kernel (which is generally different for each $|\vec{k}|$). Then we can write the full function as

In order to turn this into a sampling of $\phi(x)$ at a discrete set of times, we would like to do an inverse spatial Fourier transform of $\phi_{\vec{k}}(t_{\vec{k},n})$. However, the mode-dependence of the sample times $t_{\vec{k},n}$ is an obstruction to identifying $\phi_{\vec{k}}(t_{\vec{k},n})$ as the spatial Fourier transform of $\phi(x)$ at some time. To overcome this, we notice that because there is a maximum temporal bandwidth of $2\sqrt{2}\Omega$ (for modes with $|\vec{k}|=\Omega$), we could try to eliminate the mode dependence of the sample times by sampling all of the modes at the density corresponding to this bandwidth. The expense is that, for most of the modes, this sampling is inefficient as they are being sampled at a density larger than required. However, the gain is that if we can find a reconstruction of all of the $\phi_{\vec{k}}(t)$’s from some single temporal lattice $\{t_{n}\}_{n\in\mathbb{Z}}$, then this would allow us to write:

The function in the square brackets is a kind of reconstruction kernel, depending on $n$, $t$, and $\vec{x}-\vec{x}^{\prime}$. This allows for a reconstruction of the bandlimited function $\phi(x)$ from samples consisting of the functions $\phi(t_{n},\vec{x})$ on a discrete set of constant-time hypersurfaces. Although this is not a sampling and reconstruction of the traditional kind as in (1) and (13), we expect to have a kind of reconstruction from samples in time, by sampling all of the spatial modes at the same density of $\sqrt{2}\Omega/\pi$. Note that despite being able to sample in time, the spacetime sample density remains infinite because the samples require knowledge of the function on the entire constant-time hypersurfaces, and these functions can have arbitrarily short wavelengths.

What about an analogous sampling in space (rather than in time)? It turns out there is an analogue of (33), but not of (35), for a sampling in space in dimensions $3+1$ and higher. Suppose for an arbitrary bandlimited function, $\phi(x)$, we write the temporal Fourier transform as $\phi_{k^{0}}(\vec{x})$, so that

Now we examine a fixed temporal mode $k^{0}$. This mode has spatial dependence $\phi_{k^{0}}(\vec{x})$, and, similar to above, in order for $\phi(x)$ to be bandlimited, the spatial frequencies of $\phi_{k^{0}}(\vec{x})$ must satisfy:

The intervals for different choices of $k^{0}$ can be visualized as horizontal sections in Figure 3.

In order to determine the spatial sample density required to reconstruct a fixed temporal mode, $\phi_{k^{0}}(\vec{x})$, the Beurling-Landau condition tells us to find the volume of its support in $\vec{k}$-space. Unlike the case of fixing a spatial mode, the volume of the $\vec{k}$-space region $|\vec{k}|\in I(k^{0})$ depends on the spatial dimension $N$, since we are only restricting the magnitude of $\vec{k}$ to lie in $I(k^{0})$. The volume for each fixed $k^{0}$ is:

where $V_{N}=\pi^{N/2}/\Gamma(N/2+1)$ is the volume of the unit ball in $N$ dimensions. We see that for every fixed $k^{0}$, this volume is finite, and thus we expect to be able to reconstruct the temporal mode $\phi_{k^{0}}(\vec{x})$ from a discrete spatial lattice with density proportional to $\text{Vol}(I(k^{0}))$. Since this can be done for each of the temporal modes, we get an analogue of (33) for sampling in space.

In order to get a reconstruction formula analogous to (35), we need to be able to sample all of the modes from the same lattice in space. For this to be possible, we need to find a spatial sample density which is large enough to reconstruct all of the temporal modes. In $1+1$ dimensions, as we saw for the spatial modes, $\text{Vol}(I(k^{0}))$ has an upper bound of $2\sqrt{2}\Omega$ which occurs at $|k^{0}|=\Omega$. In $2+1$ dimensions, $\text{Vol}(I(k^{0}))$ has a maximum of $2\pi\Omega^{2}$, which occurs for all $|k^{0}|\geq\Omega$. However, in $3+1$ dimensions and higher, $\text{Vol}(I(k^{0}))$ diverges as $|k^{0}|\to\infty$, thus the required spatial sample density increases without bound for modes of large temporal frequency. Therefore, we see that although we should be able to reconstruct each temporal mode from a discrete spatial lattice, there is no uniformly discrete spatial lattice of sufficiently high density to reconstruct all of the temporal modes in $3+1$ dimensions and higher, and hence no analogue of (35) in these cases.

For the case of temporal sampling, we have so far only argued that we expect there to be a reconstruction formula of the form (35), but we have not determined explicitly how all of the spatial modes can be sampled from the same temporal lattice. In Subsection 4.2, we will develop an explicit reconstruction formula for (35), demonstrating that it is possible to reconstruct the full function $\phi(x)$ from a discrete set of constant-time hypersurfaces. We will see that some care is required in choosing a sampling lattice from which all of the modes can be reconstructed simultaneously. We will focus on the case of temporal sampling since a spatial sampling of $\phi(x)$ of the form (35) is not possible in $3+1$ dimensions.

In this subsection, we will produce a temporal lattice and a reconstruction formula which will show that an arbitrary Minkowski-bandlimited function, $\phi(x)$, can be reconstructed from samples $\phi(t_{n},\vec{x})$, hence explicitly demonstrating (35).

First, let us argue that it is not possible to employ a lattice of the form $\{t_{n}=n\pi/\Theta\}_{n\in\mathbb{Z}}$ (for any $\Theta\in\mathbb{R}^{+}$) using a similar construction to Subsection 3.2. Suppose we were to choose a lattice of this form. The corresponding sampling operator for this lattice is:

We will also define a filter:

Recall from Subsection 3.2 that the aim is to find a $\Theta$ and a function $H(k)$ so that $HS$ is the identity on the bandlimited subspace, i.e., $|\phi)=HS|\phi)$ for all bandlimited $|\phi)$. In Fourier space, this can be written

Note our Fourier convention is $(x|k)=e^{-ik\cdot x}=e^{-ik^{0}t+i\vec{k}\cdot\vec{x}}$, and we have again used

since we have assumed $t_{n}=n\pi/\Theta$. Then, similarly to the one-dimensional case, we have the condition

Figure 4 provides an illustration of the sum on the right-hand side of this equation.

In Subsection 3.2, we argued that in order for the filter to be able to recover the original bandlimited function from the result of the sum over copies, the support of the copies $m\neq 0$ cannot overlap the support of the original ($0^{th}$) copy. In the one-dimensional case this could be achieved by requiring $\Theta\geq\Omega$. However, in the present case, it should be clear from Figure 4 (and can also be shown analytically) that each of the copies will have some overlap with the original, regardless of how they are translated. Therefore, we cannot obtain a reconstruction formula using the above procedure in the form presented.

Fortunately, an extension of the above method was devised by Kohlenberg [131] which applies when the number of copies which overlap in any given region is bounded. In [131], the aim was to develop a reconstruction formula for one-dimensional functions bandlimited to a region in Fourier space of the form $0<\Omega_{1}<|k|<\Omega_{2}$ (called bandpass functions), for which overlap between copies generically occurs when sampling at the minimal density $(\Omega_{2}-\Omega_{1})/\pi$. The idea of [131] is to consider multiple sampling lattices, $\{x_{n}^{(i)}\}_{n\in\mathbb{Z}}$ (where the different lattices are indexed by $i$), with corresponding sampling operators, $S_{i}$, as well as multiple filters, $H_{i}$. A $p^{th}$-order sampling is then said to be possible if one can design a set of lattices and filters so that $\sum_{i=1}^{p}H_{i}S_{i}$ is the identity on the bandlimited function space.

We will show that a temporal sampling and reconstruction of a Minkowski-bandlimited function, $\phi(x)$, can be achieved using a second-order sampling. Recall that the maximum temporal bandwidth of the spatial modes of $\phi(x)$ was found to be $2\sqrt{2}\Omega$, and thus the Beurling density we expect to require for the temporal sampling lattice is $\sqrt{2}\Omega/\pi$. Instead of a single equidistantly-spaced lattice with this density, we will consider two equidistantly-spaced lattices each with half the density, $\Omega/\pi\sqrt{2}$ (so that the union has a Beurling density of $\sqrt{2}\Omega/\pi$), and which are shifted from each other by an amount $\tau$ (which is unspecified at this stage):

where we fix $\Theta=\Omega/\sqrt{2}$. Apart from the abstract expectation, it will become clear during the following analysis that this is the minimal density which can be chosen in order to obtain a reconstruction formula using this method. We have sampling operators corresponding to the two lattices,