A Note on the Phase Retrieval of Holomorphic Functions
aa r X i v : . [ m a t h . C A ] O c t A Note on the Phase Retrieval of HolomorphicFunctions
Rolando Perez III
Abstract.
We prove that if f and g are holomorphic functions on an open connecteddomain, with the same moduli on two intersecting segments, then f = g up to themultiplication of a unimodular constant, provided the segments make an angle that isan irrational multiple of π . We also prove that if f and g are functions in the Nevanlinnaclass, and if | f | = | g | on the unit circle and on a circle inside the unit disc, then f = g up to the multiplication of a unimodular constant. The study of phase retrieval involves the recovery of a function f in somefunction space from given data about the magnitude of f (phaseless informa-tion) and other assumptions on f , where these other assumptions may be interms of some transform of f . Phase retrieval problems are widely studiedbecause of their physical applications in fields of science and engineering. Themost natural question asked on phase retrieval is about the uniqueness of thesolution. However, phase retrieval problems generally have large solution sets,so additional assumptions related to the known data are usually added to re-duce the solution set, or to consequently force the uniqueness of the solution.We refer the reader to the survey articles by Grohs et. al. [7] and Klibanovet. al. [12] for a more general perspective on the phase retrieval problem,together with some examples and further references.Phase retrieval problems have been formulated in both finite and infinite-dimensional cases. In turn, they have been solved using a diverse array oftechniques, which include the use of tools from complex analysis. In somecases, the problem shifts to the complex analytic scenario by holomorphicextensions or by integral transforms. For instance, consider f and g to betwo band-limited L functions, then from the Paley-Wiener theorem, theyare entire functions of finite type. Complex analysis was the key tool used byAkutowicz [1,2] (and a few years later independently by Walther [16] and Hof-stetter [8]) to determine all band-limited functions g such that | g | = | f | . Wenow enumerate some further work which used complex analytic tools. Grohset. al. [6] considered the recovery of a function in a modulation space fromphaseless Gabor measurements, where they considered the short-time Fouriertransform and used the Poisson-Jensen formula in their estimates. Wald-spurger et. al. [15] solved a continuous case of the recovery of an L function R. Perez IIIfrom the modulus of its wavelet transform by using Fourier transforms andholomorphic extensions to the upper half-plane. Moreover, McDonald [13]has extended the work of Akutowicz to cover entire functions of finite genus.The solutions were characterized with the help of Hadamard factorization.In [9], Jaming also used the Hadamard factorization to show that an entirefunction of finite order can be reconstructed from its modulus on two lines,where these lines intersect at an angle which is an irrational multiple of π .Bodmann et. al. [4] then used this result with conformal mappings to showthat a polynomial of degree at most n − n − f ∈ L ( R ) such that its Fouriertransform satisfies an exponential decay condition, find all functions g ∈ L ( R )such that | f | = | g | on R and with its Fourier transform satisfying the sameexponential decay condition as f . Using a Paley-Wiener theorem and a con-formal map, the problem can be translated to the Hardy space on the unitdisc. By the inner-outer factorization, the explicit form of the solution wasobtained. One of our motivation stems from one of the coupled phase retrievalproblems from [10, Lemma 4.5], which states that for f and g in the Hardyspace on the unit disc with | f | = | g | on ( − ,
1) and on some segment insidethe disc, g can be obtained uniquely from f . For our first result, we extendthis uniqueness result to holomorphic functions on open connected domains.On the other hand, our next objective is to improve the result of Bocheet. al. [3, Theorem 3], which states that given functions f and g in theHardy space on the disc without singular inner part, if | f | = | g | on the unitcircle and on a smaller circle inside the unit circle, then g can be uniquelydetermined from f . Moreover, they also showed by construction that theBlaschke product associated with g can be uniquely recovered by its moduluson a smaller circle inside the unit circle. For our other result, we extend thisuniqueness result to all functions in the Nevanlinna class, regardless of thepresence of the singular parts.This work is organized as follows. Section 2 includes a quick review ofdefinitions and results related to spaces of holomorphic functions on the disc,and the statements of our results. Section 3 is devoted to the proofs of ourresults. Let D be the unit disc and T := ∂ D be its boundary. We denote by D ( a, r )the disc centered at a ∈ C with radius r > r D = D (0 , r ) and r T = ∂D (0 , r ).We denote by H ∞ ( D ) the space of bounded holomorphic functions on D . The Note on the Phase Retrieval of Holomorphic Functions N = (cid:26) ϕ ∈ Hol( D ) : ϕ = fg , f, g ∈ H ∞ ( D ) (cid:27) . Note that the radial limit given by ϕ ∗ ( ζ ) = lim r → ϕ ( rζ ) exists almost every-where in T and log | ϕ ∗ | ∈ L ( T ). For ϕ ∈ N , ϕ has a factorization [5, Theorem2.9] of the form ϕ = e iγ B ϕ S ν O ϕ S ν (1)where e iγ ∈ T , B ϕ is the Blaschke product formed from the zeros of ϕ , S ν and S ν are singular inner functions, and O ϕ is the outer part of ϕ . Here, theBlaschke product is defined for all z ∈ D as B ϕ ( z ) = z k Y α ∈ Λ α | α | α − z − ¯ αz where Λ is the set of nonzero zeros of ϕ counted with multiplicity, which satisfythe Blaschke condition P α ∈ Λ (1 − | α | ) < ∞ . The singular inner function isgiven by S ν ( z ) = exp (cid:18)Z T z + ζz − ζ d ν ( ζ ) (cid:19) , where ν is a finite positive singular measure on T . Finally, the outer part of ϕ is given by O ϕ ( z ) = exp (cid:18) π Z T ζ + zζ − z log | ϕ ∗ ( ζ ) | | d ζ | (cid:19) . It is easy to see that for f ∈ N , if f ( z ) = 0 then f ( z ) z − z ∈ N . We also recallthe subclass of N called the Smirnov class, defined by N + = (cid:26) ϕ ∈ Hol( D ) : ϕ = fg , f, g ∈ H ∞ ( D ) , g is outer (cid:27) . Furthermore, the Generalized Maximum Principle [14, Section 3.3.1, (g)]states that if ϕ ∈ N + with radial limit ϕ ∗ ∈ L p ( T ), then ϕ belongs to clas-sical Hardy space on the disc H p ( D ), 1 ≤ p ≤ ∞ . Note that every functionin H p ( D ) has a factorization as in (1) with S ν = 1, and this factorization isunique. We begin with a simple observation. Let ω, Ω be open connected sets suchthat ω ⊂ Ω, let f, g ∈ Hol(Ω) and suppose that | f | = | g | on ω . Then, forsome c ∈ T , g = cf on Ω. Indeed, we can assume that | f | = | g | on a closeddisc D , hence f and g have the same zeros with the same multiplicities on D . Consequently, F = f /g is a holomorphic function on D and | F | = 1.Therefore 0 = ∆ | F | = | F ′ | on D and hence F = c for some c ∈ T . R. Perez IIIIn the same spirit when ω is not open, we have shown in the paper [10,Lemma 4.5] that if f, g ∈ H ( D ) and | f ( z ) | = | g ( z ) | , z ∈ ( − , ∪ e iα ( − , α / ∈ π Q , then f and g are equal up to the multiplication of a unimodularconstant. For this result, uniqueness was established by showing that theBlaschke products, singular inner parts, and outer parts of f and g are equal.Our first result consists in showing that this is true for arbitrary holomorphicfunctions in an open connected domain. Theorem 2.1
Let Ω be an open connected domain. Let f, g ∈ Hol(Ω) andsuppose that | f ( z ) | = | g ( z ) | , z ∈ I ∪ I α , (2) where I and I α are segments inside Ω , I α is the α -rotation of I about themidpoint of I , and α / ∈ π Q . Then g ( z ) = cf ( z ) for all z ∈ Ω and for some c ∈ T . Let us now see what is happening if segments are replaced by circles. To doso, recall that if f and g are outer functions in N such that | f | = | g | almosteverywhere on T , then f is equal to g up to the multiplication of a unimodularconstant. Now, Boche et. al. [3, Theorem 3] solved a more general problem:if f, g ∈ H ( D ) have no singular parts (i.e. f = B f O f , g = B g O g ) and | f | = | g | almost everywhere on T and | f | = | g | on ρ T for some 0 < ρ < g is uniquely determined by f . The heart of their proof is the explicitconstruction of the Blaschke product associated to g , as the equality of theouter parts immediately follow. For our next result, with the same equalitiesof the moduli on the aformentioned circles, we improve the result by Boche et.al. by showing that uniqueness holds for all functions in N . We emphasizethat in this result, we may either have the presence or the absence of thesingular inner part. Theorem 2.2
Let f, g ∈ N and let ρ ∈ (0 , . If | f ( ζ ) | = | g ( ζ ) | , a.e. ζ ∈ T and | f ( z ) | = | g ( z ) | , z ∈ ρ T (3) then g ( z ) = cf ( z ) for all z ∈ D and for some c ∈ T . In this section, we present the proofs of our results, and some immediateconsequences of them.
Observe that replacing f ( z ) by f ( z + rze iβ ) with z , r, β appropriately chosen,we may assume that– (1 + ε ) D ⊂ Ω for ε > I = ( − , I α = e iα ( − , Note on the Phase Retrieval of Holomorphic Functions f, g ∈ H ( D ) so that we could apply [10, Lemma 4.5] andobtain g = cf , for some c ∈ T . We will give an alternative simpler proof.Note that, as the zeros of f and g are isolated, by choosing r small enough,we can assume that they have at most one zero in (1 + ε ) D which is at 0. Wecan write f ( z ) = z k e ϕ ( z ) and g ( z ) = z l e ψ ( z ) , z ∈ D (4)where ϕ, ψ ∈ Hol((1 + ε ) D ) and k, l are nonnegative integers. As | f ( x ) | = | g ( x ) | for x ∈ ( − ,
1) we conclude that k = l .It remains to show that the zero-free factors of f and g are equal up to aunimodular constant. First, we note that (2) is equivalent toRe ϕ ( t ) = Re ψ ( t ) and Re ϕ ( te iα ) = Re ψ ( te iα ) , t ∈ ( − , . Since ϕ, ψ ∈ Hol( D ) ∩ C ∞ ( D ), Re ϕ and Re ψ are harmonic and ϕ ( z ) = X n ≥ | z | n b ϕ ( n ) e inθ and ψ ( z ) = X n ≥ | z | n b ψ ( n ) e inθ (5)for z = | z | e iθ ∈ D . It follows thatRe ϕ ( z ) = Re b ϕ (0) + X n ∈ N | z | n b ϕ ( n ) e inθ + b ϕ ( n ) e − inθ ψ ( z ) = Re b ψ (0) + X n ∈ N | z | n b ψ ( n ) e inθ + b ψ ( n ) e − inθ . Thus (2) and (4) together with the sums above imply that Re ϕ ( t ) = Re ψ ( t )for t ∈ ( − , b ϕ ( n ) = Re b ψ ( n ) , n ∈ N , and Re ϕ ( te iα ) = Re ψ ( te iα ) if and only ifRe b ϕ (0) = Re b ψ (0) and Re ( b ϕ ( n ) e inα ) = Re ( b ψ ( n ) e inα )for all n ∈ N . In other words Re (cid:16) b ϕ ( n ) − b ψ ( n ) (cid:17) = D b ϕ ( n ) − b ψ ( n ) , E C = 0 , Re (cid:16)(cid:16) b ϕ ( n ) − b ψ ( n ) (cid:17) e inα (cid:17) = D b ϕ ( n ) − b ψ ( n ) , e − inα E C = 0 , (6)for all n ∈ N . Since α / ∈ π Q , { , e − inα } is a basis for C when n = 0 so by (6),we have b ϕ ( n ) = b ψ ( n ). On the other hand, as Re b ϕ (0) = Re b ψ (0) there exists λ ∈ R such that b ψ (0) = b ϕ (0) + iλ . It follows from (5) that ψ = ϕ + iλ thus g ( z ) = e iλ f ( z ) for all z ∈ D ⊂ Ω. As Ω is connected and f, g ∈ Hol(Ω), thisimplies that g ( z ) = e iλ f ( z ) also holds on Ω. R. Perez III We begin with a simple observation. If z ∈ ρ T is a zero of f , we write f ( z ) = ( z − z ) k ˜ f ( z ) and g ( z ) = ( z − z ) j ˜ g ( z ) with nonnegative integers j, k and ˜ f , ˜ g ∈ N nonvanishing at z . Then | f | = | g | on ρ T reads | ( z − z ) k ˜ f ( z ) | = | ( z − z ) j ˜ g ( z ) | , z ∈ ρ T and this implies that k = j . Therefore, f and g have the same zeros on ρ T with the same multiplicities. We may thus write f = P f and g = P g with P a polynomial which has all the zeros in ρ T and f , g ∈ N nonvanishing on ρ T . Then | f | = | g | on ρ T ∪ T . In other words, we may assume that f and g do not vanish on ρ T .Let { a , . . . , a n } and { b , . . . , b m } be the zeros of f and g on ρ D respectively,counted with multiplicities. For all z ∈ C , write P f ( z ) = n Y i =1 ρ ( z − a i ) ρ − ¯ a i z and P g ( z ) = m Y i =1 ρ ( z − b i ) ρ − ¯ b i z . Notice first that if z ∈ ρ T , | P f ( z ) | = | P g ( z ) | = 1. Further fP f and gP g do notvanish in ρ D . By the Poisson-Jensen formula, for all z ∈ ρ D we havelog (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) P f ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = 12 π Z π − π Re (cid:18) ρe iθ + zρe iθ − z (cid:19) log | f ( ρe iθ ) | d θ = 12 π Z π − π Re (cid:18) ρe iθ + zρe iθ − z (cid:19) log | g ( ρe iθ ) | d θ = log (cid:12)(cid:12)(cid:12)(cid:12) g ( z ) P g ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . But then (cid:12)(cid:12)(cid:12)(cid:12) g ( z ) P g ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) P f ( z ) (cid:12)(cid:12)(cid:12)(cid:12) for all z ∈ ρ D . Thus we get that there is some˜ c ∈ T such that g ( z ) P g ( z ) = ˜ c f ( z ) P f ( z ) for all z ∈ ρ D . Finally, as an identitybetween holomorphic functions on D , this is valid for all z ∈ D . In particular,taking z = re iθ and r −→
1, we get g ( z ) P g ( z ) = ˜ c f ( z ) P f ( z ) for almost every z ∈ T .By (3), we have (cid:12)(cid:12)(cid:12)(cid:12) P f (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) P g (cid:12)(cid:12)(cid:12)(cid:12) on T as well. Hence, by the following lemma,we get 1 P f = cP g for some c ∈ T , which implies g = c ˜ cf . Lemma 3.1
Let F and G be meromorphic on C without poles on T ∪ ρ T for < ρ = 1 . Suppose | F ( ζ ) | = | G ( ζ ) | and | F ( ρζ ) | = | G ( ρζ ) | , ζ ∈ T . (7) Then F and G have the same zeros and poles in C \ { } , with the samemultiplicities. In particular, if F and G are rational functions that satisfy (7) , then G = cF outside the poles with c ∈ T . Note on the Phase Retrieval of Holomorphic Functions Proof
Let F be meromorphic on C and z ∈ C . Write F ( z ) = ( z − z ) k ˜ F ( z )and for some k ∈ Z with ˜ F ( z ) = 0. Define the multiplicity k := m F ( z ),where k > z is a zero of F , and k < z is a pole of F . In particular, formeromorphic functions F , F on C , we have m F F = m F + m F , m F /F = m F − m F and if F = F in the neighborhood of a pole z , then m F = m F .First, note that if ρ >
1, we replace F and G by f ( z/ρ ) and g ( z/ρ ), andreplace ρ by 1 /ρ <
1. We thus assume that ρ <
1. Observe that (7) isequivalent to F ( z ) F (cid:18) z (cid:19) = G ( z ) G (cid:18) z (cid:19) , z ∈ T and F ( z ) F (cid:18) ρ ¯ z (cid:19) = G ( z ) G (cid:18) ρ ¯ z (cid:19) , z ∈ ρ T . As an identity between meromorphic functions in C , these equations are alsovalid for z ∈ C not a pole of any of the functions involved. As poles areisolated, we have for z = 0, m F ( z ) + m F (cid:18) z (cid:19) = m G ( z ) + m G (cid:18) z (cid:19) (8)and m F ( z ) + m F (cid:18) ρ ¯ z (cid:19) = m G ( z ) + m G (cid:18) ρ ¯ z (cid:19) . (9)Now, (8) gives m F ( z ) − m G ( z ) = m G (cid:18) z (cid:19) − m F (cid:18) z (cid:19) = m F ( ρ z ) − m G ( ρ z )with (9) applied to 1 / ¯ z , for z ∈ C \{ } . If z = 0 is such that m F ( z ) = m G ( z )then 0 = m F ( z ) − m G ( z ) = m F ( ρ z ) − m G ( ρ z ) = · · · = m F ( ρ k z ) − m G ( ρ k z ) = m F/G ( ρ k z )for all k ∈ N . But then F/G is meromorphic and either has ρ k z as a zero( m F/G ( ρ k z ) >
0) or as a pole ( m F/G ( ρ k z ) <
0) for every k . Letting k −→ ∞ we have ρ k z −→
0. As z = 0, this contradicts the fact that zerosand poles of F/G are isolated. Hence, F and G have the same nonzero zerosand poles with the same multiplicities.Furthermore, if F and G are rational functions, then they have same zerosand poles in C \ { } , thus there exists c ∈ T and m ∈ Z such that G = cz m F .But then (7) implies that— on one hand | F ( ρe it ) | = | c | ρ m | F ( ρe it ) | for all t thus | c | ρ m = 1.— on the other hand | F ( e it ) | = | c || F ( e it ) | thus | c | = 1.As ρ < m = 0 and then G = cF . R. Perez III Corollary 3.2
Let
F, G be two meromorphic functions in C with no poleat such that | F ( ζ ) | = | G ( ζ ) | and | F ( ρζ ) | = | G ( ρζ ) | , ζ ∈ T , < ρ = 1 . Then there exists c ∈ T such that G = cF . Proof
As seen in the previous proof, we can assume that ρ < F and G have the same nonzeropoles in C . Note that there is at most a finite number of such poles in aneighborhood of D . We can factor them out and write F = ˜ F /P , G = ˜ G/P with ˜
F , ˜ G having no pole in a neighborhood of D and P a polynomial. Butthen ˜ F , ˜ G ∈ N and | ˜ F | = | ˜ G | on T ∪ ρ T . The previous theorem shows thatthere exists c ∈ T such that ˜ G = c ˜ F thus G = cF on D and thus on C . Acknowledgements
The author is supported by the CHED-PhilFrance scholarship from CampusFrance and the Commission of Higher Education (CHED), Philippines. Theauthor also wants to thank his supervisors in the Universit´e de Bordeaux,Philippe Jaming and Karim Kellay, for their very helpful comments and sug-gestions.
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