Phase-Retrieval in Shift-Invariant Spaces with Gaussian Generator

We study the problem of recovering a function of the form $f(x) = \sum _{k\in \mathbb{Z} } c_k e^{-(x-k)^2}$ from its phaseless samples $|f(\lambda )|$ on some arbitrary countable set $\Lambda \subseteq \mathbb{R} $. For real-valued functions this is possible up to a sign for every separated set with Beurling density $D^-(\Lambda )>2$. This result is sharp. For complex-valued functions we find all possible solutions with the same phaseless samples.

In many measurements only non-negative amplitudes or intensities are recorded rather than the physical quantity itself. The problem then is to recover the underlying object (signal, image) from these phaseless measurements, in other words, to retrieve the phase from the magnitude. This is the problem of phase-retrieval. Particular problems concern the recovery of a function f from its phaseless Fourier measurements |f (ξ)| or the recovery of a function f from its phaseless (or unsigned) samples |f (λ)|. For general information about the phase-retrieval problem we refer to the surveys [13,17].
The question of phase-retrieval is usually ill-posed or even meaningless, unless one imposes additional assumptions on the function to be reconstructed, such as a particular signal model. In this note we consider the problem of phase-retrieval in the particular class of shift-invariant spaces generated by a Gaussian. The prototype of a shift-invariant space is the Paley-Wiener space of bandlimited functions {f ∈ L 2 (R) : suppf ⊆ [−1/2, 1/2]}. More general shift-invariant spaces serve as a substitute for bandlimited functions and have received wide attention in approximation theory and sampling theory [5,10].
Given a generator g ∈ L 2 (R), p ∈ [1, ∞], and a mesh parameter or step size β > 0, let One of the versions of phase-retrieval in shift-invariant spaces is the recovery of a function f (up to a scalar) from its phaseless samples |f (λ)| on some set Λ. The question then is whether the additional information that f belongs to the shift-invariant space V p β (g) suffices to determine f up to a sign. For the prototype of a shift-invariant space, namely the bandlimited functions, this question was solved [20]. Recently Q. Sun and his collaborators have developed a general theory for phase-retrieval in shift-invariant spaces in a series of articles [7][8][9]. A typical result asserts that the samples of |f | on a sufficiently dense union of shifted lattices suffice to recover real-valued functions in some V 2 (g). These papers also cover some of the numerical aspects of phase-retrieval. The problem of phase-retrieval in shift-invariant spaces from Fourier measurements is studied in [18].
We study the problem of phase-retrieval in the shift-invariant space generated by a Gaussian φ γ (x) = e −γx 2 . Precisely, we will assume that f is a linear combination of shifts of a Gaussian and belongs to the shift-invariant space We will allow samples from an arbitrary separated set Λ ⊂ R and measure the density of Λ with the standard notion of the lower Beurling density defined as We will first consider consider real-valued functions and unsigned samples |f (λ)|. For this case we prove the following uniqueness theorem.
Theorem 1 says that for real-valued f, g ∈ V ∞ β (φ γ ) satisfying |f (λ)| = |g(λ)| for all λ ∈ Λ we have either g = f or g = −f , thus the only ambiguity is the (global) sign of f . Our proof yields a reconstruction procedure (see Section 2), but it is well-known that the phase-retrieval problem in infinite dimensions is necessarily ill-posed [4,6,13].
In the second part we will study complex-values functions in V ∞ β (φ γ ). In this case there is no uniqueness, instead we will classify all functions g ∈ V ∞ β (φ γ ) that satisfy |g(λ)| = |f (λ)| on Λ.
Before proceeding, let us discuss some of the fine points of Theorem 1. (i) Remarkably, the uniqueness holds even for bounded coefficients, and not just for square-summable coefficients. We can therefore not rely on the Hilbert space theory for phase-retrieval, but technically we need to rely on the Banach space set-up of Alaifari and Grohs [4].
(iii) A similar statement for bandlimited functions was proved by Thakur [20] and revisited in [4]. Both [20] and Theorem 1 support the intuition that the recovery of phase from magnitude requires twice as many samples as the recovery from the samples f (λ). This is well known for finite-dimensional frames, but in infinite dimensions it is more subtle to formulate and prove. To our knowledge Theorem 1 is one of only two models for which phase-retrieval is possible with a sharp density condition. The investigations in [7][8][9] require a much higher sampling rate or deal with conditions under which phase-retrieval is not even possible.
In Section 1 we collect several statements about the shift-invariant space V ∞ β (φ γ ) and then prove Theorem 1. In Section 2 we study phase-retrieval for complexvalued functions. There we use a factorization of period entire functions whose proof is postponed to Section 3.

Phase-Retrieval for Real-Valued Functions
We set up the proof of Theorem 1. To avoid unnecessary parameters, we set β = 1, without loss of generality. This is possible because , if and only if phase-retrieval on βΛ is possible for V ∞ β (φ γ ). We note that D − (βΛ) = β −1 D − (Λ). Thus it suffices to prove Theorem 1 for β = 1.
Our first use of complex variable methods is the following lemma for Fourier series.
(ii) Conversely, if D(z) is a periodic entire function D(z + k) = D(z) for all z ∈ C and k ∈ Z with growth |D(ξ + iy)| = O(e π 2 y 2 /γ ), then the Fourier series of D(ξ) has coefficients of Gaussian decay d k = c k e −γk 2 for some c ∈ ℓ ∞ (Z).
Proof. For completeness we give the elementary proof.
(ii) If D is entire and periodic, then with uniform convergence on compact sets and exponentially decaying coefficients. See, e.g., [19,Thm. 3.10.3]. Consequently the Fourier coefficients of ξ → D(ξ + iy) are d k e −2πky and satisfy for all k and y, where the last inequality follows from the assumption. Setting y = − γk π yields the desired estimate The analysis of phase-retrieval in V ∞ 1 (φ γ ) involves several steps.
We start with a simple algebraic observation.
we find that This calculation shows that the function |f | 2 belongs to a different shift-invariant space generated by φ 2γ with step size 1/2. Set From these definitions we see that Step 2. A sharp sampling theorem. Our main tool is the following sampling theorem for shift-invariant spaces with Gaussian generator from [12,Thm. 4.4].
Thus the coefficientsr of |f | 2 are uniquely and stably determined by the phaseless samples of f on Λ.
Note that in (11) we have used the norm equivalence sup x∈R |f (x)| 2 ≍ sup n∈Z | r n |.
Step 3. A functional equation. The sampling inequality (11) allows us to recover the coefficientsr from the phaseless samples |f (λ)| 2 . Finally we have to recover the coefficients c and d from the coefficientsr of |f | 2 , or equivalently from the r n = r n e −γn 2 /2 . Letd(ξ) = k∈Z d k e 2πikξ be the Fourier series of d andr be the Fourier series of r. Then equation (7) turns into (12)r(ξ) =d(ξ)d(−ξ) .
Consequently, we have to find the entire function D that satisfies the identity In other words, to every solution of the functional equation (13) corresponds a function g such that |g(λ)| = |f (λ)| for λ ∈ Λ.
Assuming that f is real-valued, we can now prove Theorem 1 quickly.
Proof of Theorem 1. Since f is real-valued by assumption, its coefficients are also real-valued, c =c. This entails that D(−z) = k∈Z d k e −2πik(−z) = k∈Z d k e 2πikz = D(z) and (13) becomes the identity The uniqueness of f up to a sign is now immediate: assume that two entire (nonzero) functions D 1 and D 2 satisfy D 2 Since the ring of entire functions does not have any zero divisors, we conclude that either D 1 = D 2 or D 1 = −D 2 on C. Using formulas (4) -(6) we find that the coefficients c of f , and thus f , are uniquely determined by the phaseless samples |f (λ)|, λ ∈ Λ, up to a sign. Theorem 1 is therefore proved.
Alternatively, one could prove Theorem 1 by verifying the following criterium for phase-retrieval [4,6] We will produce a counter-example to the complement property. Set S = {λ 2j : j ∈ Z} and S c = {λ 2j+1 : j ∈ Z}. Then S and S c are disjoint and D(S) = D(S c ) = 1 2 D(Λ) < 1. By the necessary density condition for sampling in shiftinvariant spaces, e.g. [5], S and S c cannot be sampling sets for V 2 1 (φ γ ), but they are interpolating sets by [12,Thm. 1.3]. These facts imply that f → f (λ) λ∈S is onto ℓ 2 (S) with non-trivial kernel. Consequently, there exist non-zero functions f, g ∈ V 2 1 (φ γ ) ⊆ V ∞ 1 (φ γ ) such that f (λ) = 0 for λ ∈ S and g(λ ′ ) = 0 for λ ′ ∈ S c . By taking the real part of f and g, we also may assume that f and g are real-valued.
By construction, f + g and f − g are linearly independent, and thus sign-retrieval is not unique.

Complex-Valued Functions
Theorem 1 states that the unsigned samples {|f (λ)|} determine a unique realvalued function ±f ∈ V ∞ β (φ γ ). To treat complex-valued functions in V ∞ β (φ γ ), we use a factorization of the entire periodic functions D(z) and R(z) that is intermediate between the Hadamard factorization and a Blaschke product. Lemma 6. Let D be an entire function of order 2 satisfying the periodicity D(z +l) = D(z) for all z ∈ C and l ∈ Z. Let m be the order of the zero at z = 0, W + be the zeros of D in {x + iy = 0 : −1/2 < x ≤ 1/2, y ≥ 0} and W − be the zeros of D in {x + iy : −1/2 < x ≤ 1/2, y < 0}. Then D possesses the factorization for some r ∈ Z and C ∈ C. The product converges uniformly on compact sets.
We postpone the proof of this lemma to Section 3 and first discuss its application to the phase-retrieval problem.
The factorization (15) serves to find all solutions D to the equation D(z)D(−z) = R(z) in (13). The spirit of this argument is similar to the analysis in [1,14,16,20].
To avoid spelling out the product in (15), we use the notation Here W is understood as a sequence {w j : j ∈ N} of zeros, where elements may be repeated according to the (finite) multiplicity of the zero. Since D is of order 2, we know that j∈N |w j | −3 < ∞. See also (21) below. With this understanding we obtain the following convenient formulas for Π(W ).  This implies that the order the zero at 0 is even and that the factor e 2πiz occurs with an even power. Furthermore, since Jw = w for every w = u+iw with u = 0 or u = ±1/2, the zeros on the lines iR and ±1/2+iR also occur with even multiplicity in R. Now assume that R = Π(Z, 2m, 2r) is given so that its zero set is symmetric Z = JZ and that the zeros on the lines iR and 1/2 + iR have even multiplicity.
Let S + = {x + iy : 0 < x < 1/2} and S 0 = iR ∪ (1/2 + iR). Let Z 0 be the zeros of R in Z ∩ S 0 counted with half their multiplicity. In our notation this means that Z 0∪ Z 0 = Z ∩ S 0 . Next choose V ⊆ Z ∩ S + arbitrary and set Then Conversely, every factorization DD * = R arises in this way, because we can always write the zero set of D as W = (W ∩ S + ) ∪ (W ∩ S 0 ) ∪ (W ∩ JS + ) and we may choose V = W ∩ S + to recover W from Z = W ∪ JW .
To summarize, we state the following lemma.
Lemma 7. Let R = R * = Π(Z, 2m, 2r) be a periodic entire function of order 2 with all zeros on iR ∪ 1/2 + iR of even multiplicity. Then every solution of DD * = R is given by some D V , as defined in (19) and (20).
The analysis of the factorization DD * = R is the key tool to find all possible solutions to the phase-retrieval problem for complex-valued functions in V ∞ β (φ γ ). In contrast to the uniqueness among real-valued solutions, there are many substantially different solutions. In principle, these can be found by the following procedure.
(iii) Choose V ⊆ Z ∩ S + and define D V by (20).
(iv) Determine the Fourier coefficients of D V (ξ): and set c k = d k e γk 2 and for some α ∈ C, |α| = 1. Since |R(ξ+iy)| = O(e 2π 2 y 2 /γ ) by Lemma 2, |D V (ξ+iy)| = O(e π 2 y 2 /γ ) by Lemma 7. By Lemma 2 the Fourier coefficients of D V have Gaussian decay and consequently c k = d k e γk 2 is bounded. It follows that f ∈ V ∞ 1 (φ γ ). By construction |f V (λ)| = |f (λ)| for all λ ∈ Λ. Furthermore, every function h ∈ V ∞ 1 (φ γ ) satisfying |h(λ)| = |f (λ)|, λ ∈ Λ, must be of the form f V . Remarks. 1. If f is real-valued, then D * = D, and its zero set is symmetric, W = JW , consequently the zero set Z = W∪JW contains all zeros of D with double multiplicity. Since every zero has even multiplicity, we can set Z ∩ S + = V∪V , where V are the zeros of R in S + with half the multiplicity. Equation (19) then yields W = V ∪JV ∪Z 0 = JW . The corresponding function D V satisfies D V = D * V , and f V is the real-valued solution of the phase-retrieval problem. We note that other choices of V yield complex-valued functions f V such that |f V (λ)| = |f (λ)|.
For real-valued f the steps (i) -(iv) constitute a reconstruction procedure of f from its unsigned samples |f (λ)|, λ ∈ Λ. Whereas Theorem 1 only asserts the uniqueness up to a sign, the factorization of Lemma 6 also implies a (rather theoretical) reconstruction.
Of course, for the boundedness of these coefficients we still need the analysis that led to Lemma 7.
3. Stability. The procedure in steps (i) -(iv) is only of theoretical interest, because it is well-known that phase-retrieval in infinite dimensions is inherently unstable [3,4,6]. This is completely obvious in the reconstruction procedure in the shift-invariant space V ∞ 1 (φ γ ): numerically, the transition from the relevant coefficient sequence c to d, given by d k = c k e −γk 2 amounts to the truncation of c to a finite sequence. Once the sequence d has been obtained (steps (iii) and (iv) above), the transition c k = d k e γk 2 leads to an amplification of all accumulated errors. Yet, despite the inherent instability, several steps in the above reconstruction of f are stable. The relevant estimate is (11), for the reconstruction ofr from the phaseless samples |f (λ)|. For coefficient sequences c with small support the reconstruction promises to be reasonably effective, which is consistent with the arguments in [3].

Proof of the Factorization Lemma
As we do not know a precise citation for the statement, we include a proof of Lemma 6. We need to show that every periodic entire function D of order 2 (more generally, of finite order) can be factored into a product with factors of the form , where w is a zero of D in the vertical strip S = {x+iy ∈ C : −1/2 < x ≤ 1/2}. The choice of the signs depends on the sign of Im w. As we will see in part (vi) of the proof, the sign is determined by the asymptotic behavior of cot πw as Im w → ±∞.
Proof. (i) Since D is periodic, its zero set is periodic and, by definition of W ± the zero set is w∈W + ∪W − (w + Z). Since D is entire of order 2, the convergence exponent of its zeros is > 2 [19]. This implies that  be the main part of the right-hand side of (15). We first check the convergence of the product. For this we need to verify that is finite for every R > 0. This expression is simply whereas for w ∈ W − we have G(z, w)(1 − e 2πiw ) e 2πiz − e 2πiw = e πz(cot πw−i) e π 2 z 2 /(2 sin 2 πw) .
The left-hand side is an entire function with period 1, whereas the right-hand side is the exponential of a quadratic polynomial. It is now easy to see that e P is periodic for a polynomial P , if and only if P (z) = 2πirz for some r ∈ Z. We conclude that D(z) =D(z)e 2πirz , and this is precisely (15).
Since this sum is over the terms with Imw > 0 and w∈W |w| −3 < ∞, it is finite. For the terms in W − our choice of the signs yields the same conclusion. Finally w∈W | 1 2 sin 2 πw | is finite by the same reason.