Uniqueness Theorems for Fourier Quasicrystals and Temperate Distributions with Discrete Support
aa r X i v : . [ m a t h . F A ] J un UNIQUENESS THEOREMS FOR FOURIER QUASICRYSTALS ANDTEMPERED DISTRIBUTIONS WITH DISCRETE SUPPORT
S.YU.FAVOROV
Abstract.
It is proved that if some points of the supports of two Fourier qua-sicrystals approach each other while tending to infinity, then these quasicrystalscoincide. A similar statement is obtained for a certain class of discrete tempereddistributions.AMS Mathematics Subject Classification: 52C23, 42B10, 42A75
Keywords: tempered distribution, Fourier transform, Fourier qua-sicrystal, measure with discrete support, almost periodic distribution introduction P.Kurasov and R.Shur [6] noted that if zeros of two holomorphic almost periodic func-tions in a strip get closer at infinity, then the zeros sets of these functions coincide. Thisresult can be interpreted as the coincidence of two almost periodic discrete sets if theyget closer at infinity. It is natural to expect the same effect for other almost periodic ob-jects, in particular, for Fourier quasicrystals or, in general, for distributions with discretesupport.Denote by S ( R d ) the Schwartz space of test functions ϕ ∈ C ∞ ( R d ) with finite norms N m ( ϕ ) = sup R d (max { , | x |} ) m max k k k≤ m | ( D k ϕ )( x )) | , m = 0 , , , . . . ,k = ( k , . . . , k d ) ∈ ( N ∪ { } ) d , k k k = k k k ∞ = max { k , . . . , k d } , D k = ∂ k x . . . ∂ k d x d . Thesenorms generate a topology on S ( R d ), and elements of the space S ∗ ( R d ) of continuouslinear functionals on S ( R d ) are called tempered distributions.The Fourier transform of a tempered distribution f is defined by the equality(1) ˆ f ( ϕ ) = f ( ˆ ϕ ) for all ϕ ∈ S ( R d ) , where ˆ ϕ ( y ) = Z R d ϕ ( x ) exp {− πi h x, y i} dx is the Fourier transform of the function ϕ .We will say that a distribution (or a measure) f on R d is discrete , if for each λ ∈ supp f there is ε = ε ( λ ) > B ( λ, ε ) ∩ supp f = { λ } , and uniformly discrete , if there is ε > B ( λ, ε ) ∩ B ( λ ′ , ε ) = ∅ for all λ, λ ′ ∈ supp f, λ = λ ′ ; a measure µ is atomic if µ = P λ ∈ Λ a λ δ λ with a λ ∈ C and countable Λ, in this case we will write a λ = µ ( λ ).Here B ( x, r ) = { y ∈ R d : | y − x | < r } , and δ λ means the unit mass at the point λ ∈ R d .A complex measure µ ∈ S ∗ ( R d ) is a Fourier quasicrystal if µ and ˆ µ are discrete measures,and measures | µ | and | ˆ µ | belong to S ∗ ( R d ).Note that condition µ ∈ S ∗ ( R d ) do not imply | µ | ∈ S ∗ ( R d ) (see [3], [12]).Such measures are the main object in the theory of the Fourier quasicrystals (see [7],[10]-[14], [4]). The corresponding notion was inspired by experimental discovery of non-periodic atomic structures with diffraction patterns consisting of spots, which was madein the mid ’80s. e will say that a complex measure µ is a sparse Fourier quasicrystal, when µ is discrete, µ ∈ S ∗ ( R d ), ˆ µ is atomic, | ˆ µ | ∈ S ∗ ( R d ), and numbers of elements { supp µ ∩ B ( x, } areuniformly bounded in x ∈ R d .Note that, compared with the classical definition of Fourier quasicrystal, we have weak-ened the conditions on the measure ˆ µ and removed the requirement | µ | ∈ S ∗ ( R d ).Clearly, a Fourier quasicrystal with a uniformly discrete support is a sparse Fourierquasicrystal. Theorem 1.
If two sparse Fourier quasicrystals µ = P λ ∈ Λ µ ( λ ) δ λ , ν = P γ ∈ Γ ν ( γ ) δ γ under appropriate numbering Λ = { λ n } ∞ n =1 , Γ = { γ n } ∞ n =1 have the properties (2) λ n − γ n → and µ ( λ n ) − ν ( γ n ) → as n → ∞ , then the measures µ, ν coincide. The conditions of the theorem can be significantly weakened. So, the sparseness ofmeasures and conditions (2) can only be checked on the set E = ∪ k B ( x k , r k ), where { B ( x k , r k ) } ∞ k =1 is an arbitrary sequence of mutually disjoint balls with radii r k → ∞ , andmass of measures can be calculated by groups of points. Theorem 2.
Let µ, ν be discrete measures from S ∗ ( R d ) with supports Λ , Γ respectively.Suppose that there exist disjoint sets Λ n , and disjoint sets Γ n such that (3) Λ ∩ E = ∪ n Λ n , Γ ∩ E = ∪ n Γ n , (we do not require Λ n = ∅ or Γ n = ∅ , but always Λ n ∪ Γ n = ∅ ). Also, for some N < ∞ and all x ∈ E (4) { n : Λ n ∩ B ( x, = ∅} ≤ N, { n : Γ n ∩ B ( x, = ∅} ≤ N. and (5) diam { Λ n ∪ Γ n } → as n → ∞ . If (6) ˆ µ, ˆ ν are atomic measures, and | ˆ µ | , | ˆ ν | ∈ S ∗ ( R d ) , (7) µ (Λ n ) − ν (Γ n ) → as n → ∞ , then the measures µ, ν coincide. Remark 1 . In the case of one-point sets Λ n = { λ n } , Γ n = { γ n } for all n conditions(5) and (7) take, respectively, the form λ n − γ n → n → ∞ , µ ( λ n ) − ν ( γ n ) → n → ∞ . Condition (4) means that quantities { supp µ ∩ E ∩ B ( x, } and { supp ν ∩ E ∩ B ( x, } are uniformly bounded. Remark 2 . Conditions µ, ν ∈ S ∗ ( R d ) and (6) can be replaced by the following: thefunctions Z ϕ ( x − t ) µ ( dx ) and Z ϕ ( x − t ) ν ( dx )are almost periodic for every ϕ ∈ C ∞ with compact support. In particular, instead ofdiscrete measures, we can consider discrete almost periodic multisets ([2], [3]) that arediscrete almost periodic measures with integer positive masses. nder some additional conditions, the uniqueness theorem also holds for distributionswith discrete supports. Note that by [5], Proposition 3.1, for every tempered distribution F with discrete support there is m < ∞ such that F = X λ ∈ Λ X k j k≤ m p λ,j D j δ λ , j ∈ ( N ∪ { } ) d , p λ,j ∈ C . Theorem 3.
Let f = X λ ∈ Λ X k j k≤ m p λ,j D j δ λ , g = X γ ∈ Γ X k j k≤ m q γ,j D j δ γ be tempered distributions with discrete supports Λ , Γ . Suppose that there exist disjointsets Λ n , and disjoint sets Γ n with properties (3), (4), and (5). If (8) ˆ f , ˆ g are atomic measures, and | ˆ f | , | ˆ g | ∈ S ∗ ( R d ) , (9) X λ ∈ Λ n p λ,j − X γ ∈ Γ n q γ,j → ∀ j, as n → ∞ , (10) max j sup n X λ ∈ Λ n | p λ,j | < ∞ , max j sup n X γ ∈ Γ n | q γ,j | < ∞ , then the distributions f, g coincide. Measures and their Fourier transforms can be interchanged in Theorems 2 and 3. Recallthat the support of the Fourier transform of a measure µ is called a spectrum of µ . Theorem 4.
Let µ, ν be atomic measures with discrete spectra ˜Λ , ˜Γ respectively, | µ | , | ν | ∈ S ∗ ( R d ) , and there be disjoint sets Λ n and disjoint sets Γ n such that conditions (3), (4),(5) are met, with Λ replaced by ˜Λ , Γ by ˜Γ .If either ˆ µ, ˆ ν are measures, and ˆ µ (Λ n ) − ˆ ν (Γ n ) → as n → ∞ , or ˆ µ = X λ ∈ ˜Λ X k j k≤ m ˜ p λ,j D j δ λ , ˆ ν = X γ ∈ ˜Γ X k j k≤ m ˜ q γ,j D j δ γ , and X λ ∈ Λ n ˜ p λ,j − X γ ∈ Γ n ˜ q γ,j → ∀ j, as n → ∞ , max j sup n X λ ∈ Λ n | ˜ p λ,j | < ∞ , max j sup n X γ ∈ Γ n | ˜ q γ,j | < ∞ , then the measures µ, ν coincide. We also give an analogue of Theorem 1 for classical Fourier quasicrystals.
Theorem 5.
If two Fourier quasicrystals µ, ν with discrete ”sparse” spectra ˜Λ , ˜Γ (thismeans that sup y ∈ R d { ˜Λ ∩ B ( y, } < ∞ , sup y ∈ R d { ˜Γ ∩ B ( y, } < ∞ ) under appropriate numbering ˜Λ = { λ n } ∞ n =1 , ˜Γ = { γ n } ∞ n =1 have the properties λ n − γ n → and ˆ µ ( λ n ) − ˆ ν ( γ n ) → as n → ∞ , then the measures µ, ν coincide. . auxiliary results Lemma 1.
Let a positive measure µ belong to S ∗ ( R d ) . Then there is N < ∞ such that µ ( B (0 , r )) = O ( R N ) , and for any Borel function H ( x ) such that sup x ∈ R d | H ( x ) | (1+ | x | T ) < ∞ for all T < ∞ we get Z R d | H ( x ) | µ ( dx ) < ∞ . Proof of the Lemma . Assume the converse. Then there is a sequence R n → ∞ such that µ ( B (0 , R n )) > R nn . We may suppose that R n +1 > R n for all n . Take ϕ ( t ) ∈ C ∞ ( R ) , ≤ ϕ ( t ) ≤ ϕ ( t ) = 1 for t ≤ ϕ ( t ) = 0 for t ≥
2. SetΨ( x ) = X n R − nn ϕ ( | x | /R n ) . Clearly, Ψ ∈ C ∞ and(11) Z R d Ψ( x ) µ ( dx ) ≥ X n R − nn µ ( B (0 , R n )) = ∞ . On there other hand, take any
K < ∞ and x such that 2 R p − < | x | ≤ R p with p > K .We have | x | K Ψ( x ) = X n | x | K R − nn ϕ ( | x | /R n ) < K R K − pp X n ≥ p R pp /R nn . Taking into account that R n > n − p R p and p → ∞ as | x | → ∞ , we obtain | x | K Ψ( x ) < K +1 R K − pp → | x | → ∞ . Similarly, one can check that | x | K Ψ ( k ) ( x ) → K and k ∈ ( N ∪ { } ) d , therefore,Ψ ∈ S ( R d ). Since µ ∈ S ∗ ( R d ), we get the contradiction with (11). Hence there exists N such that M ( R ) := µ ( B (0 , R )) ≤ C max(1 , R N ).Furthermore, let | H ( x ) | ≤ C | x | − N − for | x | ≥
1. Passing to polar coordinates andintegrating in parts, we obtain Z R d | H ( x ) | µ ( dx ) ≤ C + C Z | x | > | x | − N − µ ( dx ) = C + C Z ∞ r − N − M ( dr )= C + C (cid:18) lim R →∞ M ( R ) R N +1 − M (1) + ( N + 1) Z ∞ M ( r ) r N +2 dr (cid:19) < ∞ . Lemma is proved.The proofs of our theorems are also based on the properties of almost periodic functionsand distributions. Recall some definitions related to the notion of almost periodicity(a detailed exposition of the theory of almost periodic functions on R see, for example,in [1] and [9], most of the results can easily be generalized to functions on R d ; almostperiodic measures and distributions were introduced in [8] and [15], see also [12], [13], [2],[5]).A set A ⊂ R d is relatively dense, if there is R < ∞ such that every ball of radius R intersects with A .A continuous function f on R d is almost periodic , if for every ε > ε -almostperiods of f { τ ∈ R d : sup t ∈ R d | f ( t + τ ) − f ( t ) | < ε } is a relatively dense in R d .For example, for arbitrary s n ∈ R d the function f ( t ) = X n a n e πi h t,s n i s almost periodic under the condition P n | a n | < ∞ .It was proved in [1] that a finite family { f j } Mj =1 of almost periodic functions on R has acommon relatively dense set of ε -almost periods for every ε . The same result for almostperiodic functions on R d follows immediately from Bochner’s criterion: a function f ( x ) isalmost periodic if and only if for every sequence x n there is a subsequence x n ′ such thatthe functions f ( x + x n ′ ) converges uniformly in x ∈ R . Its proof in [9] practically withoutchanges is transferred to functions on R d and even to mappings from R d to R M .Now, if each f j satisfies Bochner’s criterion, then the mapping F = ( f ( x ) , . . . , f M ( x ))satisfies this criterion too. It remains to notice that every ε -almost period of F is an ε -almost period of every f j .Next, put ϕ t = ϕ ( x − t ) for any function ϕ on R d .A measure µ is almost periodic , if the function F ( t ) = R ψ t ( x ) µ ( dx ) is almost periodicfor any continuous function ψ ( x ) with compact support.A tempered distribution f is almost periodic , if for every ϕ ∈ S ( R d ) the function F ( t ) = f ( ϕ t ) is almost periodic.It can be proved that every almost periodic measure is an almost periodic tempereddistribution, but there are discrete measures that almost periodic tempered distributionsand not almost periodic measures. Nevertheless, if µ is a positive measure, or satisfiesthe condition sup x ∈ R d | µ ( B ( x, | < ∞ , then almost periodicity of µ in the sense ofdistributions implies almost periodicity in the sense of measures ([8], [12], [3]). Lemma 2. If f ∈ S ∗ ( R d ) , ˆ f is an atomic measure, and | ˆ f | ∈ S ∗ ( R d ) , then f is an almostperiodic distribution.In particular, every Fourier quasicrystal is an almost periodic distribution. Proof of the Lemma . Let ˆ f = P n b n δ s n , then | ˆ f | = P n | b n | δ s n . By (1), we have foreach ϕ ∈ S ( R d )(12) f ( ϕ t ) = Z ˇ ϕ ( y ) e πi h t,y i ˆ f ( dy ) = X n b n ˇ ϕ ( s n ) e πi h t,s n i , where ˇ ϕ ( y ) e πi h t,y i is the inverse Fourier transform of the function ϕ ( x − t ). Since ˇ ϕ ( y ) ∈ S ( R d ), we can apply Lemma 1. Therefore, series in (12) absolutely converge, and thefunction f ( ϕ t ) is almost periodic.3. proofs of the theorems Proof of Theorem 2 . Assume the contrary µ ν . Then there is a ∈ R d such that ν ( a ) = µ ( a ). Take ϕ ∈ S ( R d ) such that 0 ≤ ϕ ≤ , supp ϕ ⊂ B (2), and ϕ ( x ) = 1 for | x | <
1. Since Λ and Γ are discrete, we can find ρ ≤ / ∪ Γ other than a in the ball B ( a, ρ ). Therefore for all ρ ′ ≤ ρ Z ϕ (cid:18) x − aρ ′ (cid:19) µ ( dx ) = µ ( a ) = ν ( a ) = Z ϕ (cid:18) x − aρ ′ (cid:19) ν ( dx ) . Let N be a number from (4). Set for j = 1 , . . . , N + 1 f j ( t ) = Z ϕ (cid:18) x − t − j ρ (cid:19) µ ( dx ) , g j ( t ) = Z ϕ (cid:18) x − t − j ρ (cid:19) ν ( dx ) , H j ( t ) = f j ( t ) − g j ( t ) . Using (6) and applying Lemma 2, we obtain that all the functions f j ( t ) and g j ( t ) arealmost periodic, and the functions H j ( t ) too. Moreover, for all jH j ( a ) = µ ( a ) − ν ( a ) = 0 . et ε = | µ ( a ) − ν ( a ) | /
2. Denote by T the set of all common ε -almost periods of thefunctions H j ( t ). We get(13) | H j ( a + τ ) | > ε ∀ τ ∈ T , j = 1 , . . . , N + 1 . Since T is relatively dense and r k → ∞ , it follows that B ( x k , r k ) ⊃ B ( a + τ k , ρ ) for every k > k and some τ k ∈ T . By the conditions of the theorem, Λ ∪ Γ is discrete, hence, wehave min { n : (Λ n ∪ Γ n ) ⊂ B ( x k , r k ) } → ∞ as k → ∞ , therefore, by (5),(14) diam(Λ n ∪ Γ n ) → k → ∞ for Λ n ∪ Γ n ⊂ B ( x k , r k ) . In particular, there is k such that for k > k and Λ n ∪ Γ n ⊂ B ( x k , r k ) we getdiam(Λ n ∪ Γ n ) < − N − ρ, and the set Λ n ∪ Γ n can intersect with only one of the spherical shells B ( a + τ k , − j +1 ρ ) \ B ( a + τ k , − j ρ ) , j = 1 , . . . , N + 1 . On the other hand, by (4), { n : (Λ n ∪ Γ n ) ⊂ B ( a + τ k , ρ ) } ≤ N, hence there is m = m ( k ) , ≤ m ≤ N + 1 , such that(Λ n ∪ Γ n ) ∩ [ B ( a + τ k , − m +1 ρ ) \ B ( a + τ k , − m ρ )] = ∅ if Λ n ∪ Γ n ⊂ B ( a + τ k , ρ ) . Consequently, the sets Λ n , Γ n are either both simultaneously subsets of B ( a + τ k , − m ρ )and ϕ (cid:18) λ − a − τ k − m ρ (cid:19) = ϕ (cid:18) γ − a − τ k − m ρ (cid:19) = 1 , ∀ λ ∈ Λ n , ∀ γ ∈ Γ n , or are not subsets of B ( a + τ k , − m +1 ρ ), and ϕ (cid:18) λ − a − τ k − m ρ (cid:19) = ϕ (cid:18) γ − a − τ k − m ρ (cid:19) = 0 ∀ λ ∈ Λ n , ∀ γ ∈ Γ n . Hence, H m ( a + τ k ) = X λ ∈ Λ ∩ B ( a + τ k , − m ρ ) µ ( λ ) − X γ ∈ Γ ∩ B ( a + τ k , − m ρ ) ν ( γ ) = X n :Λ n ⊂ B ( a + τ k , − m ρ ) [ µ (Λ n ) − ν (Γ n )] . By (7) and (14), we obtain µ (Λ n ) − ν (Γ n ) → k → ∞ . Since a number of membersin the last sum is an most N , we get that the sequence H m ( k ) ( a + τ k ) tends to zero as k → ∞ , which contradicts to (13). Proof of Theorem 3 . Assume the contrary f g . Then there is a ∈ supp f suchthat either a ∈ Γ and p a,j = q a,j for some j ∈ ( N ∪ { } ) d , or a Γ and p a,j = 0for some j ∈ ( N ∪ { } ) d . In the latter case put q a,j = 0. Take ϕ ∈ S ( R d ) such that0 ≤ ϕ ≤ , supp ϕ ⊂ B (0 , ϕ ( x ) = 1 for x ∈ B (0 , ∪ Γ other than a in the ball B ( a, ρ ) for some ρ ≤ /
2. Put ψ ( x ) = ϕ ( x/ρ ) x j /j ! . Clearly, ( D j ψ )(0) = 1 and ( D j ψ )(0) = 0 for j = j . Therefore, f ( ψ a ) = p a,j = g ( ψ a ).Using (8) and applying Lemma 2, we obtain that the functions f ( ψ t ) , g ( ψ t ) are almostperiodic, and the function H ( t ) = f ( ψ t ) − g ( ψ t ) too. Moreover, H ( a ) = p a,j − q a,j = 0.Set ε = | H ( a ) | /
2. Denote by T the set of all ε -almost period of the functions H ( t ). Weget(15) | H ( a + τ ) | > ε ∀ τ ∈ T . ince T is relatively dense and r k → ∞ , we see that B ( x k , r k ) ⊃ B ( a + τ k , ρ ) for every k > k and some τ k ∈ T . Set M k = { n : (Λ n ∪ Γ n ) ∩ B ( a + τ k , ρ ) = ∅} . Note that Λ ∪ Γ is discrete, hence,(16) min M k → ∞ as k → ∞ . Furthermore, for sufficiently large k (17) f ( ψ a + τ k ) − g ( ψ a + τ k ) = X n ∈ M k X j " X λ ∈ Λ n p λ,j ( D j ψ )( λ − a − τ k ) − X γ ∈ Γ n q γ,j ( D j ψ )( γ − a − τ k ) . All derivatives of ψ are uniformly continuous, hence it follows from (5) and (10) that forsome fixed points b n ∈ Λ n ∪ Γ n and each j X λ ∈ Λ n p λ,j (cid:2) ( D j ψ )( λ − a − τ k ) − ( D j ψ )( b n − a − τ k ) (cid:3) → n → ∞ , and X γ ∈ Γ n q γ,j (cid:2) ( D j ψ )( γ − a − τ k ) − ( D j ψ )( b n − a − τ k ) (cid:3) → n → ∞ . All derivatives of ψ are uniformly bounded, hence it follows (9) that for each j " X λ ∈ Λ n p λ,j − X γ ∈ Γ n q γ,j ( D j ψ )( b n − a − τ k ) → n → ∞ . Hence, X λ ∈ Λ n p λ,j ( D j ψ )( λ − a − τ k ) − X γ ∈ Γ n q γ,j ( D j ψ )( γ − a − τ k ) → n → ∞ . Note that the number of values of j does not exceed ( m + 1) d and M k ≤ N . Thus weobtain from (16) and (17) | H ( a + τ k ) | = | f ( ψ a + τ k ) − g ( ψ a + τ k ) | → k → ∞ , which contradicts to (15).Theorem 4 follows from Theorems 2 and 3, if only we change the Fourier transform tothe inverse Fourier transform. Theorem 1 follows from Theorem 2, and Theorem 5 followsfrom Theorem 4.The author is grateful to Mikhail Sodin for drawing the attention of the author to thearticle [6], and to Pavel Kurasov for his interest in the research of the author and usefuldiscussion. References [1] H. Bohr, Almost periodic functions, OGIZ (1934), (Russian).[2] Favorov,S., Kolbasina,Ye.: Almost periodic discrete sets.// Journal of Mathematical Physics, Anal-ysis, Geometry. v.6 (2010), No.1, 1-14.[3] Favorov,S., Kolbasina,Ye.: Perturbations of discrete lattices and almost periodic sets. Algebra andDiscrete Mathematica, 2010, v.9, No.2, 48-58.[4] Favorov, S.Yu.: Large Fourier quasicryals and Wiener’s Theorem. Journal of Fourier Analysis andApplications, Vol. 25, Issue 2, (2019), 377-392.[5] Favorov, s.Yu.: Tempered distributions with discrete support and spectrum. Bulletin of the HellenicMathematical Society, v.62, (2018), 66-79.
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Sergii Favorov,Karazin’s Kharkiv National UniversitySvobody sq., 4,61022, Kharkiv, Ukraine
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