Exponential frames and syndetic Riesz sequences Marcin Bownik University of Oregon, USA Frame Theory and Exponential Bases June 4–8, 2018 ICERM, Brown University, Providence, RI Marcin Bownik Exponential frames and syndetic Riesz sequences
Abstract Employing the solution to the Kadison-Singer problem, we deduce that every subset S of the torus of positive Lebesgue measure admits a Riesz sequence of exponentials { e i λ x } λ ∈ Λ such that Λ ⊂ Z is a set with gaps between consecutive elements bounded by C |S| . This talk is based on a joint work with Itay Londner (Tel Aviv University). Marcin Bownik Exponential frames and syndetic Riesz sequences
H separable Hilbert space, I a countable set. Definition { ϕ i } i ∈ I ⊂ H is a frame with bounds 0 < A ≤ B < ∞ if A � f � 2 ≤ |� f , ϕ i �| 2 ≤ B � f � 2 � i ∈ I for all vectors f ∈ H . { ϕ i } i ∈ I is a Bessel sequence if A = 0. Definition { ϕ i } i ∈ I ⊂ H a Riesz sequence in H with bounds 0 < A ≤ B < ∞ if 2 � � | a i | 2 ≤ � � � � � | a i | 2 A a i ϕ i ≤ B � � � � � � i ∈ I i ∈ I i ∈ I H for every finite sequence of scalars { a i } i ∈ I . Marcin Bownik Exponential frames and syndetic Riesz sequences
Exponential systems Definition S ⊂ R set of finite positive Lebesgue measure, Λ ⊂ R a countable λ ∈ Λ in L 2 ( S ). � e i λ x � set. Define exponential system E (Λ) = Marcin Bownik Exponential frames and syndetic Riesz sequences
Exponential systems Definition S ⊂ R set of finite positive Lebesgue measure, Λ ⊂ R a countable λ ∈ Λ in L 2 ( S ). � e i λ x � set. Define exponential system E (Λ) = S bounded and Λ separated set inf λ � = µ | λ − µ | > 0 = ⇒ E (Λ) is Bessel. Marcin Bownik Exponential frames and syndetic Riesz sequences
Exponential systems Definition S ⊂ R set of finite positive Lebesgue measure, Λ ⊂ R a countable λ ∈ Λ in L 2 ( S ). � e i λ x � set. Define exponential system E (Λ) = S bounded and Λ separated set inf λ � = µ | λ − µ | > 0 = ⇒ E (Λ) is Bessel. S ⊂ T = R / 2 π Z set of positive Lebesgue measure ⇒ E ( Z ) is a Parseval frame in L 2 ( S ). = Marcin Bownik Exponential frames and syndetic Riesz sequences
Classical results Theorem (Kahane (1957)) Let I ⊂ R be an interval. If the upper density # (Λ ∩ ( a , a + r )) < | I | D + (Λ) := lim r →∞ sup 2 π, r a ∈ R then E (Λ) is a Riesz sequence in L 2 ( I ) . On the other hand if D + (Λ) > | I | 2 π then E (Λ) is not a Riesz sequence in L 2 ( I ) . Marcin Bownik Exponential frames and syndetic Riesz sequences
Classical results Theorem (Kahane (1957)) Let I ⊂ R be an interval. If the upper density # (Λ ∩ ( a , a + r )) < | I | D + (Λ) := lim r →∞ sup 2 π, r a ∈ R then E (Λ) is a Riesz sequence in L 2 ( I ) . On the other hand if D + (Λ) > | I | 2 π then E (Λ) is not a Riesz sequence in L 2 ( I ) . Theorem (Landau (1967)) Let S be a measurable set. If E (Λ) is a Riesz sequence in L 2 ( S ) then D + (Λ) ≤ |S| 2 π . Marcin Bownik Exponential frames and syndetic Riesz sequences
Question Given a set S , does there exist a set Λ of positive density such that the exponential system E (Λ) is a Riesz sequence in L 2 ( S ) ? Marcin Bownik Exponential frames and syndetic Riesz sequences
Question Given a set S , does there exist a set Λ of positive density such that the exponential system E (Λ) is a Riesz sequence in L 2 ( S ) ? This question may be considered under various notions of density. The first result on this subject is Theorem (Bourgain-Tzafriri (1987)) Given S ⊂ T of positive measure, there exists a set Λ ⊂ Z with positive asymptotic density # (Λ ∩ ( − r , r )) dens (Λ) = lim > c |S| 2 r r →∞ and such that E (Λ) is a Riesz sequence in L 2 ( S ) . Here c is an absolute constant, independent of S . Hence, every set S admits a Riesz sequence Λ with positive upper density, proportional to the measure of S . Marcin Bownik Exponential frames and syndetic Riesz sequences
Definition A subset Λ = { . . . < λ 0 < λ 1 < λ 2 < . . . } ⊂ Z is syndetic if the consecutive gaps in Λ are bounded γ (Λ) := sup ( λ n +1 − λ n ) < ∞ . n ∈ Z Marcin Bownik Exponential frames and syndetic Riesz sequences
Definition A subset Λ = { . . . < λ 0 < λ 1 < λ 2 < . . . } ⊂ Z is syndetic if the consecutive gaps in Λ are bounded γ (Λ) := sup ( λ n +1 − λ n ) < ∞ . n ∈ Z Theorem (Lawton (2010) and Paulsen (2011)) Given a set S ⊂ T of positive measure, TFAE: (i) There exists r ∈ N and a partition Z = � r j =1 Λ j such that E (Λ j ) is a Riesz sequences in L 2 ( S ) for all j = 1 , . . . , r. (ii) There exists d ∈ N and a syndetic set Λ ⊆ Z with γ (Λ) = d such that E (Λ) is a Riesz sequence in L 2 ( S ) . Marcin Bownik Exponential frames and syndetic Riesz sequences
Definition A subset Λ = { . . . < λ 0 < λ 1 < λ 2 < . . . } ⊂ Z is syndetic if the consecutive gaps in Λ are bounded γ (Λ) := sup ( λ n +1 − λ n ) < ∞ . n ∈ Z Theorem (Lawton (2010) and Paulsen (2011)) Given a set S ⊂ T of positive measure, TFAE: (i) There exists r ∈ N and a partition Z = � r j =1 Λ j such that E (Λ j ) is a Riesz sequences in L 2 ( S ) for all j = 1 , . . . , r. (ii) There exists d ∈ N and a syndetic set Λ ⊆ Z with γ (Λ) = d such that E (Λ) is a Riesz sequence in L 2 ( S ) . Remark (ii) = ⇒ (i) can take r ≤ d by considering translates of Λ. (i) = ⇒ (ii) no upper bound on d in terms of r . Marcin Bownik Exponential frames and syndetic Riesz sequences
Remark Statement (i) is known as the Feichtinger conjecture for exponentials. The Feichtinger conjecture in its general form states that every bounded frame can be decomposed into finitely many Riesz sequences. It has been proved to the Kadison-Singer problem by Casazza-Christensen-Lindner-Vershynin (2005) and Casazza-Tremain (2006). The latter has been solved by Marcus, Spielman and Srivastava (2013). Marcin Bownik Exponential frames and syndetic Riesz sequences
Solution of Kadison-Singer Problem Theorem (Marcus-Spielman-Srivastava (2013)) If ε > 0 and v 1 , . . . , v m are independent random vectors in C d with m � � v i � 2 � � E [ v i v ∗ finite support such that i ] ≤ I d and E ≤ ε for all i =1 i, then �� m � � 1 + √ ε � � � 2 � v i v ∗ � P � ≤ > 0 . � � i � � � i =1 Marcin Bownik Exponential frames and syndetic Riesz sequences
Improvement for support of size 2 Theorem (B.-Casazza-Marcus-Speegle (2016)) 0 , 1 and v 1 , . . . , v m are independent random vectors in C d � � If ε ∈ 2 m � E [ v i v ∗ with support of size 2 such that i ] ≤ I d and i =1 � � v i � 2 � E ≤ ε for all i, then �� m � � � � � v i v ∗ � P � ≤ 1 + 2 ε (1 − ε ) > 0 . � � i � � � i =1 Marcin Bownik Exponential frames and syndetic Riesz sequences
Feichtinger conjecture Theorem (B.-Casazza-Marcus-Speegle (2016)) Let ε > 0 and suppose that { u i } i ∈ I is a Bessel sequence in H with bound 1 that consists of vectors of norms � u i � 2 ≥ ε . Then there exists a universal constant C > 0 , such that I can be partitioned into r ≤ C ε subsets I 1 , . . . , I r such that every subfamily { u i } i ∈ I j , j = 1 , . . . , r is a Riesz sequence in H . Moreover, if ε > 3 / 4 , then r = 2 works. Marcin Bownik Exponential frames and syndetic Riesz sequences
Feichtinger conjecture Theorem (B.-Casazza-Marcus-Speegle (2016)) Let ε > 0 and suppose that { u i } i ∈ I is a Bessel sequence in H with bound 1 that consists of vectors of norms � u i � 2 ≥ ε . Then there exists a universal constant C > 0 , such that I can be partitioned into r ≤ C ε subsets I 1 , . . . , I r such that every subfamily { u i } i ∈ I j , j = 1 , . . . , r is a Riesz sequence in H . Moreover, if ε > 3 / 4 , then r = 2 works. Corollary There exists a universal constant C > 0 such that for any subset S ⊂ T with positive measure, the exponential system E ( Z ) can be |S| Riesz sequences E (Λ j ) in L 2 ( S ) decomposed as a union of r ≤ C for j = 1 , . . . , r. Moreover, if |S| > 3 / 4 , then r = 2 works. Marcin Bownik Exponential frames and syndetic Riesz sequences
Lawton’s Theorem and the solution of Kadison-Singer problem (Feichtinger conjecture) yield syndetic Riesz sequences of exponentials in L 2 ( S ). Marcin Bownik Exponential frames and syndetic Riesz sequences
Lawton’s Theorem and the solution of Kadison-Singer problem (Feichtinger conjecture) yield syndetic Riesz sequences of exponentials in L 2 ( S ). Question (Olevskii) What is the bound on a gap γ (Λ) for syndetic Λ ⊂ Z such that E (Λ) is a Riesz sequence in L 2 ( S ) ? Marcin Bownik Exponential frames and syndetic Riesz sequences
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