Uniqueness of Gabor series

Appl. Comput. Harmon. Anal. 39 (2015) 545–551

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Applied and Computational Harmonic Analysis www.elsevier.com/locate/acha

Letter to the Editor

Uniqueness of Gabor series Yurii Belov 1 Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 24 September 2014 Received in revised form 25 January 2015 Accepted 18 March 2015 Available online 25 March 2015 Communicated by Christopher Heil

We prove that any complete and minimal Gabor system of Gaussians is a Markushevich basis in L2 (R). © 2015 Elsevier Inc. All rights reserved.

MSC: 30D10 30D15 42A63 41A30 Keywords: Gabor analysis Fock space Uniqueness of Fourier expansions

1. Introduction Let Λ ⊂ R2 be a sequence of distinct points. With each such sequence we associate Gabor system GΛ := {e2πiyt e−π(t−x) }(x,y)∈Λ . 2

(1.1)

Function e2πiyt e−π(t−x) can be viewed as the time–frequency shift of the Gaussian e−πt in the phase space. It is well known that system GΛ cannot be a Riesz basis in L2 (R) (see e.g. [9]). On the other hand, there exist a lot of complete and minimal systems GΛ . A canonical example is the lattice without one point, Λ := Z × Z \ {(0, 0)}. However, the generating sets Λ can be very far from any lattice. For example, in [1] it was shown that there exists Λ ⊂ R × {0} ∪ {0} × R such that GΛ is complete and minimal in L2 (R). If GΛ is complete and minimal, then there exists the unique biorthogonal system {g(x,y) }(x,y)∈Λ . So, for any f ∈ L2 (R) we may write the formal Fourier series with respect to the system GΛ 2

1

E-mail address: [email protected]. Author was supported by RNF grant 14-21-00035.

http://dx.doi.org/10.1016/j.acha.2015.03.006 1063-5203/© 2015 Elsevier Inc. All rights reserved.

2

546

Y. Belov / Appl. Comput. Harmon. Anal. 39 (2015) 545–551



f∼

(f, g(x,y) )L2 (R) e2πiyt e−π(t−x) . 2

(1.2)

(x,y)∈Λ

If Λ = Z × Z \ {(0, 0)}, then it is known that there exists a linear summation method for the series (1.2) (e.g. one can use methods from [8]). In [8] this was proved for certain sequences similar to lattices. The main point of the present note is to show that any series (1.2) defines an element f uniquely. Theorem 1.1. Let GΛ be a complete and minimal system in L2 (R). Then the biorthogonal system {g(x,y) }(x,y)∈Λ is complete. So, any function f ∈ L2 (R) is uniquely determined by the coefficients (f, g(x,y) ). This property is by no means automatic for an arbitrary system of vectors. Indeed, if {en }∞ n=1 is an orthonormal basis in a separable Hilbert space, then {e1 + en }∞ is a complete and minimal system but n=2 its biorthogonal {en }∞ is not complete. A complete and minimal system in a Hilbert space with complete n=2 biorthogonal system is called Markushevich basis. Theorem 1.1 is analogous to Young’s theorem [11] for systems of complex exponentials {eiλn t } in L2 of an interval. However, the structure of complete and minimal systems for Gabor systems is more puzzling than for the systems of exponentials on an interval. For example, if Λ generates a complete and minimal system of exponentials in L2 (−π, π), then the upper density of Λ (= lim supr→∞ #(Λ ∩ {|λ| < r})(2r)−1 ) is equal to 1; see Theorem 1 in Lecture 17 of [7]. On the other hand, if GΛ is a complete and minimal Gabor system, then the upper density of Λ ( = lim supr→∞ #(Λ ∩ {x2 + y 2 ≤ r2 })(πr2 )−1 ) can vary from 2/π to 1; see Theorem 1 in [1]. If, in addition, Λ is a regular distributed set, then the upper density have to be from 2/π to 1; see Theorem 2 in [1]. Note that for some systems of special functions (associated to some canonical system of differential equations) in L2 of an interval completeness of the biorthogonal system may fail (even with infinite defect); see [2, Proposition 3.4]. In the next section we transfer our problem to the Fock space of entire functions. The last section is devoted to the proof of our result. Notations. Throughout this paper the notation U (x)  V (x) means that there is a constant C such that U (x) ≤ CV (x) holds for all x in the set in question, U, V ≥ 0. We write U (x) V (x) if both U (x)  V (x) and V (x)  U (x). 2. Reduction to a Fock space problem Let  F := {F is entire and

|F (z)|2 e−π|z| dm(z) < ∞}; 2

C

here dm denotes the planar Lebesgue measure. It is well known that the following Bargmann transform Bf (z) := 21/4 e−iπxy e 2 |z| π

 = 21/4

2



f (t)e2πiyt e−π(t−x) dt 2

R

f (t)e−πt e2πtz e− 2 z dt, 2

π

2

z = x + iy,

R

is a unitary map between L2 (R) and the Fock space F; see [5,6] for the details. Moreover, the time–frequency shift of the Gaussian is mapped to the normalized reproducing kernel of F

Y. Belov / Appl. Comput. Harmon. Anal. 39 (2015) 545–551

21/4 B(e2πiut e−π(t−v) )(z) = e−π|w| 2

2

/2 πwz

e

=

kw (z) , kw F

w = u − iv,

547

¯ kw (z) := eπwz .

(2.1)

The existence of such a transformation allows us to apply methods from the theory of entire functions. For that reason the results about time–frequency shifts of the Gaussians are stronger than for the time–frequency shifts of other elements of L2 (R). Lemma 2.1. The system GΛ is complete and minimal in L2 (R) if and only if the system of reproducing  λ (z)  kernels kk is complete and minimal in F. λ∈Λ λ Proof. The system GΛ is complete and minimal if and only if the system GΛ is complete and minimal. Now Lemma 2.1 immediately follows from the unitarity of Bargmann transform. 2 In many spaces of entire functions the system biorthogonal to the system of reproducing kernels can be described via the generating function; see e.g. Theorem 4 in Lecture 18 of [7] (this idea goes back to Paley and Wiener). Lemma 2.2. The system {kλ }λ∈Λ is complete and minimal in F if and only if there exists an entire function (z) F such that F has simple zeros exactly at Λ, Fz−λ belongs to F for some (any) λ ∈ Λ and there is no non-trivial entire function T such that F T ∈ F. Proof. Necessity. The system {kλ }λ∈Λ has a biorthogonal system which we will call {Fλ }λ∈Λ . We know 1 that Fλ1 (z) z−λ z−λ2 ∈ F for any λ1 , λ2 ∈ Λ. This function vanishes at the points λ ∈ Λ \ {λ2 } and so it equals Fλ2 up to a multiplicative constant. Hence, the function cλ Fλ (z)(z − λ) does not depend on λ for suitable coefficients cλ . Denote it by F . It is easy to see that F satisfies the required properties. F (z) Sufficiency. Assume that such F exists. From the inclusion z−λ ∈ F we conclude that the system 0 {kλ }λ∈Λ\{λ0 } is not complete. On the other hand, if the whole system {kλ }λ∈Λ is not complete, then there exists T such that F T ∈ F. 2 The function F from Lemma 2.2 is called a generating function of Λ. So, the following theorem is the reformulation of Theorem 1.1 in terms of the Fock space. Theorem 2.3. If {kλ } is a complete and minimal system of reproducing kernels in F and F is the generating  (z)  function of this system, then the system Fz−λ is also complete. λ∈Λ In the last section we will prove this theorem. 3. Completeness of biorthogonal system 3.1. Preliminary steps Let σ be the Weierstrass σ-function associated to the lattice Z = {z : z = m + in, m, n ∈ Z},  

σ(z) = z

λ∈Z\{0}

 z z2 z 1− e λ + λ2 . λ

2

It is well known that |σ(z)| dist(z, Z)eπ|z| /2 ; see e.g. [10, p. 108]. From this estimate it is easy to see   that system kkw is a complete and minimal system and σ0 (z) := σ(z) z is its generating function. w  w∈Z\{0} σ0 (z) Indeed, we have z−w ⊥ kw2 (z), w1 , w2 ∈ Z \ {0}, w1 = w2 and, hence, the system {kw }w∈Z\{0,w0 } is not 1 complete for any w0 ∈ Z \ {0}. On the other hand, it is known that the system {kw }w∈Z\{0} is complete.

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The system

 kw 

σ0 (w)

·

σ0 (z)  z−w

is the biorthogonal system. With any function S ∈ F we can associate its  k 

formal Fourier series with respect to the system 

S∼

w∈Z\{0}

w

kw 

w∈Z\{0}



kw , bw kw 

bw :=

kw  σ0 (z) · S(z),  σ0 (w) z − w

 . F

This series is more regular than an arbitrary Fourier series (1.2). For example this series admits a linear summation method. In particular, we know that the sequence {bw } is non-trivial. We need the following estimate of coefficients 2 kw  σ0 (z) 2  S2 · wσ0 (z) . · |bw |2 ≤ S2 · σ  (w) z − w z−w 0 The first estimate is the Cauchy–Schwartz inequality. From Taylor formula σ0 (w + ε) = σ0 (w)ε + O(ε2 ), w ∈ Z \ {0} we get |σ0 (w)| the second estimate. Further,

2

eπ|w| /2 . |w|

2

On the other hand, we have kw 2 = kw (w) = eπ|w| . This gives us

 2 wσ0 (z) 2 |wσ0 (z)|2 e−π|z| = dm(z)  z−w |w − z|2 C



 +

|z−w|<1/2

|z|<2|w|,|z−w|>1/2

 + |z|>2|w|

= I1 + I2 + I3 . π|z|2 /2

From the estimate |σ0 (z)| dist(z, Z \ {0}) e 1+|z| we get that |I1 |  1,  |I2 |  |z|<2|w|,|z−w|>1/2

|w|2 dm(z)  log(1 + |w|), (1 + |z|2 )|w − z|2



|I3 |  |z|>2|w|

|w|2 dm(z)  1. (1 + |z|2 )|z|2

So, |bw |2  log(1 + |w|),

wσ0 (z) 2 z − w  log(1 + |w|),

w ∈ Z \ {0}.

(3.1) (3.2)

Lemma 3.1. If F is the generating function of a complete and minimal system of reproducing kernels {kλ }λ∈Λ in F and Λ ∩ Z = ∅, then for any triple λ1 , λ2 , λ3 ∈ Λ we have 

F (z) ,S (z − λ1 )(z − λ2 )(z − λ3 )

 = F

 w∈Z\{0}

F (w)bw (w − λ1 )(w − λ2 )(w − λ3 )kw 

(3.3)

for any S ∈ F. 2

Proof. It is well known that for any function H ∈ F we have w∈Z\{0} |H(w)| kw 2 < ∞ (see e.g. [4]). So,  F (w)    bw 2 2 (w−λ1 )kw  ∈  . From (3.1) we conclude that (w−λ2 )(w−λ3 ) ∈  . Hence, the series on the right hand side of (3.3) converges and defines a bounded linear functional on F. On the other hand, the left hand side

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549

and the right hand side of (3.3) coincides if S is a finite linear combination of {kw }w∈Z\{0} . Completeness of the system {kw }w∈Z\{0} means that the set of a finite linear combination of {kw }w∈Z\{0} is dense in F. So, bounded linear functionals on the left and the right handside of (3.3) coincide. 2 3.2. Proof of Theorem 1.1 Assume the contrary. Then there exists a function S ∈ F such that S ⊥ loss of generality we can assume that Λ ∩ Z = ∅ and S ⊥ lattice Z + ε for some small ε. From the identity

σ0 (z) z−w ,

F (z) z−λ

for any λ ∈ Λ. Without

w ∈ Z \ {0}. Indeed, we can use shifted

 ck 1 = (z − λ1 )(z − λ2 )(z − λ3 ) z − λk 3

k=1

we get that for any triple λ1 , λ2 , λ3 ∈ Λ 

F (z) ,S (z − λ1 )(z − λ2 )(z − λ3 )

 = 0.

Fix two arbitrary points λ1 , λ2 ∈ Λ. Put L(z) =

 w∈Z\{0}

F (w)bw , (z − w)(w − λ1 )(w − λ2 )kw 

T (z) = L(z) ·

σ0 (z)(z − λ1 )(z − λ2 ) . F (z)

Using Lemma 3.1 we get that the meromorphic function L vanishes at Λ \ {λ1 , λ2 } and, hence, T is an entire function. So,  w∈Z\{0}

F (z)T (z) F (w)bw = . (z − w)(w − λ1 )(w − λ2 )kw  (z − λ1 )(z − λ2 )σ0 (z)

(3.4)

Comparing the residues of both sides of (3.4) we get F (w)T (w) F (w)bw = , (w − λ1 )(w − λ2 )kw  (w − λ1 )(w − λ2 )σ0 (w) Since F (w) = 0 we conclude that T (w) = bw

σ0 (w) kw  ,

w ∈ Z \ {0}.

Assume that T has at least two zeros t1 , t2 . Since S ⊥ So, L(t1 ) = L(t2 ) = 0. Further, L(z) − L(t1 ) F (z)T (z) = (z − λ1 )(z − λ2 )(z − t1 )σ0 (z) z − t1  1 = z − t1

w∈Z\{0}

=

 w∈Z\{0}

w ∈ Z \ {0}.

σ0 (z) z−w

we get bw = 0 and, hence, t1 , t2 ∈ / Z \ {0}.

 1 F (w)bw 1 − (w − λ1 )(w − λ2 )kw  z − w t1 − w

1 F (w)bw · . (w − λ1 )(w − λ2 )kw  (z − w)(w − t1 )

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Using (3.4) we get,  F (z)T (z) λ1 − t 1 F (z)T (z) = 1− (z − t1 )(z − λ2 )σ0 (z) (z − λ1 )(z − λ2 )σ0 (z) z − t1 =

 w∈Z\{0}

=

 w∈Z\{0}

 λ1 − t 1 F (w)bw 1− (z − w)(w − λ1 )(w − λ2 )kw  w − t1 F (w)bw . (z − w)(w − t1 )(w − λ2 )kw 

Repeating this procedure for the point t2 we get F (z)T (z) = (z − t1 )(z − t2 )σ0 (z)

 w∈Z\{0}

F (w)bw . (z − w)(w − t1 )(w − t2 )kw 

So, F (z)

T (z) = (z − t1 )(z − t2 )

 w∈Z\{0}

F (w)bw |w|1/2 σ0 (z) · . z−w (w − t1 )(w − t2 )|w|1/2 kw 

(3.5)

|w|1/2 σ0 (z) ∈ 2 and estimates |bw |2  log(1 + |w|), z−w  1 (see (3.1), (3.2)) we get that the right hand side of (3.5) belongs to F. This contradicts the completeness of sequence {kλ }λ∈Λ . Hence T has at most one zero. So, T (z) = eP (z) (a1 z − a0 ), where P is a polynomial of degree at most 2.

σ (w) 1/2 This contradicts the estimate |T (w)| = bw 0  log (1+|w|) , w ∈ Z \ {0} (see (3.1)). 2 From the inclusion



 F (w) (w−λ1 )kw 

kw 

|w|

3.3. Concluding remarks 1. The author wonders if the following statement (stronger than Theorem 1.1) is true: Question 1. Any complete and minimal Gabor system is a strong Markushevich basis. Which means that any vector f ∈ L2 (R) belongs to the closed linear span of members of its Fourier series (1.2) (see [3] and references therein). For systems of complex exponentials {eiλn t } in L2 of an interval this is not true; see [3, Theorem 2]. (z) 2. Using our methods one can prove the completeness of the system { Fz−λ }λ∈Λ under weaker assumptions n than in Theorem 2.3 (e.g. if F ∈ F and z F ∈ / F, n ∈ N). Nevertheless we prefer to formulate the result as it is to avoid inessential technicalities.

Acknowledgments A part of the present work was done when author was visiting Norwegian University of Science and Technology, whose hospitality is greatly appreciated. References [1] G. Ascenzi, Yu. Lyubarskii, K. Seip, Phase space distribution of Gabor expansions, Appl. Comput. Harmon. Anal. 26 (2009) 277–282.

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[2] A. Baranov, Yu. Belov, Systems of reproducing kernels and their biorthogonal: completeness or incompleteness?, Int. Math. Res. Not. 22 (2011) 5076–5108. [3] A. Baranov, Y. Belov, A. Borichev, Hereditary completeness for systems of exponentials and reproducing kernels, Adv. Math. 235 (2013) 525–554. [4] J. Buckley, X. Massaneda, J. Ortega-Cerda, Traces of functions in Fock spaces on lattices of critical density, Bull. Lond. Math. Soc. 44 (2012) 222–240. [5] G. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud., vol. 122, Princeton University Press, Princeton, NJ, 1989. [6] K. Gröchenig, Foundations of Time–Frequency Analysis, Birkhäuser, Boston, MA, 2001. [7] B. Levin, Lectures on Entire Functions, Transl. Math. Monogr., vol. 150, Amer. Math. Soc., Providence, RI, 1996. [8] Yu. Lyubarskii, K. Seip, Convergence and summability of Gabor expansions at the Nyquist density, J. Fourier Anal. Appl. 5 (1999) 127–157. [9] K. Seip, Density theorems for sampling and interpolation in the Bargmann–Fock space. I, J. Reine Angew. Math. 429 (1992) 91–106. [10] K. Seip, R. Wallsten, Density theorems for sampling and interpolation in the Bargmann–Fock space. II, J. Reine Angew. Math. 429 (1992) 107–113. [11] R. Young, On complete biorthogonal system, Proc. Amer. Math. Soc. 83 (3) (1981) 537–540.