PROCEEDINGS OF THE INTERNET ANALYSIS SEMINAR ON THE UNCERTAINTY PRINCIPLE IN HARMONIC ANALYSIS

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1 PROCEEDINGS OF THE INTERNET ANALYSIS SEMINAR ON THE UNCERTAINTY PRINCIPLE IN HARMONIC ANALYSIS 1

2 2 Contents Overview of the Workshop 3 1. Completeness and Riesz bases of reproducing kernels in model subspaces 5 Presented by Kelly Bickel 2. Pólya Sequences, Toeplitz Kernels and Gap Theorems 12 Presented by Anne Duffee 3. Unconditional bases of exponentials and of reproducing kernels 17 Presented by Jingguo Lai 4. Fourier Frames 25 Presented by James Murphy 5. Meromorphic Inner Functions, Toeplitz Kernels and the Uncertainty Principle 30 Presented by Rishika Rupam 6. Spectral Gaps for Sets and Measures 35 Presented by Prabath Silva

3 Overview of the Workshop This workshop was part of the Internet Analysis Seminar that is the education component of the National Science Foundation DMS # held by Brett D. Wick. The Internet Analysis Seminar consists of three phases that run over the course of a standard academic year. Each year, a topic in complex analysis, function theory, harmonic analysis, or operator theory is chosen and an internet seminar will be developed with corresponding lectures. The course will introduce advanced graduate students and post-doctoral researchers to various topics in those areas and, in particular, their interaction. This was a workshop that focused on the uncertainty principle in harmonic analysis. The famous Heisenberg Uncertainty Principle from Quantum Mechanics has a striking formulation in the language of Harmonic Analysis. It essentially says that a non-zero measure (distribution) and its Fourier transform cannot be simultaneously small. Throughout the years this broad statement raised a multitude of deep mathematical questions, each corresponding to a particular sense of smallness. The majority of these questions were inspired and are closely connected to many fundamental problems in approximation theory, inverse spectral problems for differential operators and Krein s canonical systems, classical problems in the theory of stationary Gaussian Processes, signal processing, etc. Many of these problems remained open for more than half a century, and some of them were even considered completely intractable. The participants that presented, presented one of the following papers: [1] Anton Baranov, Completeness and Riesz bases of reproducing kernels in model subspaces, Int. Math. Res. Not. (2006), Art. ID 81530, 34. [2] S. V. Hruščëv, N. K. Nikol skiĭ, and B. S. Pavlov, Unconditional bases of exponentials and of reproducing kernels, Complex analysis and spectral theory (Leningrad, 1979/1980), Lecture Notes in Math., vol. 864, Springer, Berlin, 1981, pp [3] N. Makarov and A. Poltoratski, Meromorphic inner functions, Toeplitz kernels and the uncertainty principle, Perspectives in analysis, Math. Phys. Stud., vol. 27, Springer, Berlin, 2005, pp [4] Mishko Mitkovski and Alexei Poltoratski, Pólya sequences, Toeplitz kernels and gap theorems, Adv. Math. 224 (2010), no. 3, [5] Joaquim Ortega-Cerdà and Kristian Seip, Fourier frames, Ann. of Math. (2) 155 (2002), no. 3, [6] Alexei Poltoratski, Spectral gaps for sets and measures, Acta Math. 208 (2012), no. 1, They were then responsible to prepare three one-hour lectures based on the paper and an extended abstract based on the paper. This proceeding is the collection of the extended abstract prepared by each participant. 3

4 4 The following people participated in the workshop: Debendra Banjade Kelly Bickel Anne Duffee Ishwari Kunwar Michael Lacey Jingguo Lai Alexei Poltoratski James Murphy Robert Rahm Rishika Rupam Prabath Silva Brett D. Wick University of Alabama Georgia Institute of Technology University of Alabama Georgia Institute of Technology Georgia Institute of Technology Brown University Texas A&M University University of Maryland Georgia Institute of Technology Texas A&M University University of Indiana Georgia Institute of Technology

5 5 COMPLETENESS AND RIESZ BASES OF REPRODUCING KERNELS IN MODEL SUBSPACES ANTON BARANOV presented by Kelly Bickel Abstract. This paper examines the behavior of systems of reproducing kernels in model subspaces K Φ := H 2 ΦH 2, where Φ is inner and H 2 is the Hardy space. The first main result is a criterion of completeness for a system of reproducing kernels in K Φ, phrased in terms of the argument of Φ. This result is used to show that the completeness property is stable under small perturbations. The author also derives a different completeness result, which relies on densities defined using a de-branges-clark basis of K Φ. The second class of results examine conditions under which a system of reproducing kernels (indexed by real points) forms a Riesz basis in K Φ. 1. Introduction and Main Results 1.1. Introduction. A function Φ on the upper half plane C + is called inner if Φ is bounded, analytic, and lim y 0 Φ(x + iy) = 1 a.e. on R. For each inner function, one can define the model subspace associated to Φ: K Φ := H 2 ΦH 2 = H 2 ΦH 2. Each K Φ is a closed subspace of the Hardy space H 2 and is a reproducing kernel Hilbert space with reproducing kernel given by k w (z) := 1 2πi 1 Φ(w)Φ(z) w z for each z, w C +. Such model subspaces appear in many important areas of both function theory and operator theory and are arguably most well-known for their role in the Sz.- Nagy-Foias model of Hilbert space contractions. Given Φ inner and Λ C + R, it seems reasonable to ask When is the set K(Λ) := {k λn } complete in K Φ? This question was originally motivated by interest in the completeness of systems of complex exponentials {exp(2πiλ n t)} in L 2 ([0, a]) but is also interesting in its own right. The first section of this paper [1] answers the motivating question: when is K(Λ) complete in K Φ? The second section restricts attention to meromorphic Φ and Λ R. It considers the question: When is K(Λ) a Riesz basis of K Φ? In this extended abstract, we first outline the paper s main results about completeness and Reisz bases of K(Λ) in K Φ. The later two sections will address both the proofs of the main results and several applications.

6 Completeness Results. Baranov s first completeness result is phrased using the principle argument of Φ, denoted arg Φ, and the Hilbert transform. To state the theorem, let Π denote the Poisson measure on R, i.e. dπ(t) = dt. Then for g L 2 (Π), the Hilbert 1+t 2 transformed is defined as g(x) = 1 ( 1 π lim ɛ 0 x t + t ) g(t)dt. t x t >ɛ The following result characterizes completeness of K(Λ) for Λ C + : Theorem 1. Let Λ = {λ n } C +. Then K(Λ) is not complete in K Φ if and only if there are (1) a nonnegative m L 2 (R) with log m L 1 (Π) (2) a measurable Z-valued function k and a γ R such that arg Φ arg B Λ = 2log m + 2πk + γ a.e. on R, where B Λ is the Blaschke product with zeros {λ n }. The paper separately considers the situation where some of the points defining reproducing kernels are real. To ensure such reproducing kernels exist, Φ is assumed to be analytic in a neighborhood of each such real point. In this situation, Baranov shows: Theorem 2. Let Λ = {λ n } C + and T = {t n } R and let Φ be analytic in a neighborhood of each t n. Then: K(Λ) K(T ) is not complete in K Φ if and only if there exist (1) an inner function J with {J = 1} = T (2) a nonnegative m L 2 (R) with log m L 1 (Π) (3) a measurable Z-valued function k and a γ R such that arg Φ arg B Λ arg J = 2log m + 2πk + γ a.e. on R, where B Λ is the Blaschke product with zeros {λ n }. These criteria can be used to show that the completeness property is stable under both small perturbations of Φ and small perturbations of Λ. The last completeness result relies on a density criterion and follows from Theorem 2. It only applies to meromorphic inner functions Φ, which are always of the form exp(iaz)b Λ (z) where a 0 and the zeros Λ of the Blaschke product B Λ do not accumulate on R. In this case, Φ extends analytically to R and Φ(t) = exp(iψ(t)) for ψ C and increasing on R. Definition 3. To define the density associated to Φ, let Φ be a meromorphic inner function with increasing, smooth argument function ψ. Define the sequence {s n } by ψ(s n ) = 2πn n. As long as 1 Φ L 2 (R), then the set of reproducing kernels {k sn } is an orthogonal basis in K Φ and is called a de Branges-Clark basis. Now let T = {t n } be another sequence in R. Then the upper and lower densities D + and D of T with respect to Φ are defined as follows: D + (T, r) = sup #{m : t m [s n, s n+r )} and D (T, r) = inf #{m : t m [s n, s n+r )} n n

7 and D + (T, r) D (T, r) D + (T ) = lim and D (T ) = lim. r r r r Intuition suggests that if T is much denser than {s n }, then K(T ) is complete in K Φ, and if T is much sparser than {s n }, then K(T ) is not complete. With additional restrictions on Φ and the points {s n }, that is precisely Baranov s result: Theorem 4. Assume Φ is meromorphic, inner and Φ L (R). Further, assume {s n } satisfy ( 1 sup + s ) k n s n s k 1 + s 2 k <. k n Taking D +, D, and T as before: (1) If D (T ) > 1 then K(T ) is complete in K Φ. (2) If D + (T ) < 1 then K(T ) is not complete in K Φ Riesz Bases Results. A set of vectors {h n } is a Riesz basis for a Hilbert space H if H = Span n h n and there are constants A, B > 0 such that for every finite sum h = n c nh n, A c n 2 2 c n h n B c n 2. n n H n To obtain criteria for K(T ) to be a Riesz basis in K Φ, Baranov uses the connections between meromorphic inner functions Φ and Hermite-Biehler functions E, defined as follows: Definition 5. A function E is in the Hermite-Biehler class HB if E is entire and E(z) > E( z) z C +. For each Hermite-Biehler function, one can define the de Branges space H(E), which is the Hilbert space of entire functions F such that F/E and F /E are in H 2, where F (z) := F ( z). The norm on H(E) is given by F E := F/E 2, where 2 is the H 2 norm. If E HB, then Φ := E /E is meromorphic inner and similarly, if Φ is meromorphic inner, then there is an HB function E such that Φ = E /E. Moreover, as shown in [3], the map F F/E maps H(E) unitarily onto K Φ. The following result uses an associated de Branges space H(E) to characterize the Riesz bases of K Φ : Theorem 6. Let Φ = E /E be be meromorphic inner with E HB and let T = {t n } R. Then K is a Riesz Basis for K Φ if and only if there is a meromorphic inner Φ 1 = E 1/E 1 such that (1) H(E) = H(E 1 ) as sets with equivalent norms (2) T = {Φ 1 = 1} and Φ 1 1 L 2 (R) Notice that (2) says K(T ) is a de Branges-Clark basis for Φ 1. However, (1) is more mysterious and sufficient conditions for (1) are discussed later. In the following two sections, we will highlight the main proof techniques for these results and discuss several applications. 7

8 8 2. Main Result 1: Completeness Criteria 2.1. Proofs of Theorems 1 and 2. The proofs of Theorems 1 and 2 rely on the following representations of arguments of inner and outer functions using Hilbert transforms: Remark 7. Argument Representations. Assume m 0, m L 2 (R), and log m L 1 (Π). Then f := exp(log m + ilog m) is an H 2 outer function and every H 2 outer function is of this form with m = f. Thus, arg f = log f + 2πk, for a measurable Z-valued function k. Similarly, if I is inner, then 1 I is outer and since I = (1 I)/(1 Ī), we have arg I = 2log 1 I + 2πk + π, for a measurable Z-valued function k. To illustrate the use of such representations, we summarize the proof of the forward implication of Theorem 1. The converse uses similar ideas and is omitted: Proof: Assume K(Λ) is not complete, so there is an f K Φ that vanishes on Λ. We can assume f = B Λ g, where g is outer. As f K Φ, this implies ΦB Λ g H 2 and there is an inner function I so that ΦB Λ g = Ig. Taking arguments and rearranging gives: arg Φ arg B Λ = 2 arg g + arg I + 2πk + π = 2 log m + 2πk + γ, where m = g 1 I L 2 (R) and log m L 1 (Π). The proof of Theorem 2 is similar but requires the construction of an inner function J such that {J = 1} = T. Remark 8. Constructing the Inner Function. Define a Poisson-finite, positive measure ν supported on T. If δ x denotes the Dirac measure at the point x, just select: ν = ν n ν n δ tn where ν n > 0 and <. 1 + t 2 n n n Then construct a meromorphic Herglotz function G using ν as follows ( 1 G(z) = R t z t ) dν = ( 1 ν 1 + t 2 n t n n z t ) n 1 + t 2 n and the function J = (G i)/(g + i) is meromorphic inner with {J = 1} = T. The only additional technicality in the proof of Theorem 2 is that the {ν n } must be sufficiently small so that, if f K Φ is the function vanishing on Λ T, then f/(1 J) H 2. This construction appears in [5] Application I: Proving Theorem 4. Theorems 1 and 2 are particularly useful because previous results by Baranov, Havin, and Mashreghi [2, 4] show that every mainly increasing function can be written as 2log m + 2πk, where m 0, m L 2 (R), and log m L 1 (Π) and k is a Z-valued measurable function. Definition 9. If f C 1 (R, then f is mainly increasing if there is an increasing sequence {d n } R such that lim n d n = and

9 (a) f(d n+1 ) f(d n ) 1 n. (b) There is a constant C such that sup f(s) f(t) C and s,t (d n,d n+1 ) sup f (s) f (t) C n. s,t (d n,d n+1 ) To see the usefulness of mainly increasing functions, consider this (very) briefly sketch of the proof of Theorem 4 : Proof: First, prove part (2). Use the density assumption to construct an inner function J with {J = 1} = T such that arg Φ arg J is mainly increasing. Then Theorem 2 implies that K(Λ) K(T ) is not complete in K Φ. To prove part (1), similarly construct an inner function with arg J arg Φ mainly increasing, so it is of the form 2log m+2πk. Simultaneously, assume that K(Λ) K(T ) is not complete and use Theorem 2. This results in a contradiction Application:Perturbations of Λ. Theorem 2 also has implications about the stability of these completeness properties. For example, Baranov proves the following corollary: Corollary 10. Let Λ = {λ n } C + and M = {µ n } C +. Assume K(M) is complete in K Φ. If for some choice of arguments ψ Λ of B Λ and ψ M of B M, (ψ Λ ψ M ) L 1 (Π) and then K(Λ) is also complete in K Φ. (ψ Λ ψ M ) L (R) This result follows quickly from Theorem 2: Proof: Assume K(Λ) is not complete, so arg Φ ψ Λ = 2log m + 2πk + γ. The assumption (ψ Λ ψ M ) = u L (R) implies ψ Λ ψ M = ũ = 2 log m 1, for m 1 = e u/2. Since u L (R), then m 1 L (R) and clearly log m 1 L 1 (Π). So arg Φ ψ M = arg Φ ψ Λ + (ψ Λ ψ M ) = 2 log m 1 m + 2πk + γ, which means K(M) is not complete in K Φ, a contradiction Main Result 2: Riesz Basis Criteria 3.1. Proof of Theorem 6. The proof of Theorem 6 relies on the following condition for a system of reproducing kernels K(T ) to be a Riesz basis in K Φ, which can be found in [6]: Remark 11. The system K(T ) is a Riesz basis for K Φ if and only if for each {c n } satisfying c n 2 <, there is a unique function f K k tn 2 Φ such that f(t n ) = c n 2 n and each f 2 2 c n 2 / k tn 2 2. Such a set T is called a complete interpolating set for K Φ. If Φ = E /E, where E is Hermite-Biehler, an equivalent condition is: T is a a complete interpolating set for H(E), which means that for each {d n } satisfying d n 2 <, there is a unique function F H(E) with F (t E(t n ) 2 k tn 2 n ) = d n 2 n and each F 2 E d n 2 / E(t n ) 2 k tn 2 2.

10 10 The ( ) direction of Theorem 6 s proof is almost immediate: Proof: Assumption (2) coupled with the above remark implies that T is a complete interpolating set for H(E 1 ). Assumption (1) implies not only that H(E 1 ) = H(E) and F E F E1 but also gives E(t n ) 2 k tn 2 2 E 1 (t n ) 2 k 1 t n 2 2 n, where kz 1 denotes the reproducing kernel of K Φ1. Working through the definitions, it follows that T is a complete interpolating set for H(E), so K(T ) is a Riesz basis for K Φ. The ( ) direction of the proof uses the following result of Ortega-Cerda and Seip [7]: Theorem 12. Let E HB and let T = {t n } R. If T is a sampling set for K Φ, then there exist entire functions E 1 and E 2 where E 1 HB and E 2 HB or is constant such that (1) H(E) = H(E 1 ) with norm equivalence. (2) If Φ 1 = E 1/E 1 and Φ 2 = E 2/E 2, then {Φ 1 Φ 2 = 1} = T. (3) (1 Φ 1 Φ 2 ) L 2 (R) Given that, here is a brief proof sketch: Proof: Assume T is a complete interpolating set for K Φ ; this implies T is a sampling set of K Φ, so there are E 1 and E 2 as in Theorem 12. To obtain the desired result, Baranov shows E 2 is constant. This relies on factoring: H(E 1 E 2 ) = E 2 H(E 1 ) E 1H(E 2 ). Facts (2) and (3) about E 1, E 2 imply that T is a complete interpolating set for H(E 1 E 2 ). Assumption (1) implies that T is also a complete interpolating set for E 2 H(E 1 ). Combining the two results, one can show that H(E 1 E 2 ) = E 2 H(E 1 ), and so E 2 must be constant Using Theorem 6. To demonstrate the usability of Theorem 6, in the last section of the paper, Baranov examines when inner functions Φ and Φ 1 can be factored using HB functions E, E 1 so that H(E) = H(E 1 ) with equivalent norms. One sufficient condition for H(E) = H(E 1 ) with equivalent norms is for E(z) E 1 (z) for all z C +. Baranov finds necessary and sufficient conditions for this in the following theorem: Theorem 13. Let Φ, Φ 1 be meromorphic inner with increasing arguments ψ,ψ 1. Assume (ψ ψ 1 ) L 1 (Π).Then there are E 1, E HB such that Φ = E /E, Φ 1 = E 1/E 1 and E(z) E 1 (z) z C + if and only if (ψ ψ 1 ) L (R). Proof: The idea is to construct an entire function S that is real on R such that S = E 1 E w for an outer function w with w, w 1 H and continuous on C +. Then E 2 := SE satisfies Φ = E 2/E 2 and E 1 (z) E 2 (z) on C +. The condition (ψ ψ 1 ) L (R) allows Baranov to define w by w(t) := exp( f(t) + if(t)) on R and extend w to an outer function on C + with the desired properties.

11 References [1] A. D. Baranov, Completeness and Riesz bases of reproducing kernels in model subspaces, Int. Math. Res. Not. Art. ID 81530, [2] A.D. Baranov and V.P. Havin, Admissible majorants for model subspaces and arguments of inner functions, Funktsional. Anal. i Prilozhen. 40 (2006), 4, [3] V.P. Havin and J. Mashreghi, Admissible majorants for model subspaces of H 2. I. Slow winding of the generating inner function., Canad. J. Math. 55 (2003), 6, [4] V.P. Havin and J. Mashreghi, Admissible majorants for model subspaces of H 2. II.Fast winding of the generating inner function., Canad. J. Math. 55 (2003), 6, [5] N. Makarov and A. Poltoratski, Meromorphic inner functions, Toeplitz kernels and the uncertainty principle, Perspectives in analysis, , Math. Phys. Stud., 27, Springer, Berlin, [6] N. K. Nikolski, Treatise on the Shift Operator, Springer, Berlin, [7] J. Ortega-Cerda and K. Seip, Fourier Frames, Ann. of Math. (2) 155 (2002), 3, Georgia Institute of Technology, Atlanta, GA address: kbickel3@math.gatech.edu 11

12 12 PÓLYA SEQUENCES, TOEPLITZ KERNELS AND GAP THEOREMS MISHKO MITKOVSKI AND ALEXEI POLTORATSKI presented by Anne Duffee Abstract. We summarize the paper on Pólya sequences by Poltoraski and Minkovski [5]. A sequence Λ = {λ n } is a Pólya sequence if any entire function of zero exponential type bounded on Λ is constant. We seek to fully characterize these Pólya sets and show that this is equivalent to the Beurling gap problem on Fourier transform. We achieve this via techniques using Toeplitz kernels and de Branges spaces of entire functions Introduction. 1. Introduction and Main Results Presented Definitions. An entire function F is said to have exponential type zero if lim sup z log F (z) z A sequence is separated if it satisfies λ n λ m δ > 0, (n m). A separated real sequence Λ = {λ n } n= is Pólya if any entire function of exponential type zero that is bounded on Λ is constant. It is our goal in this paper to fully characterize Pólya sets Examples. (i) Λ = Z is a Pólya sequence. (ii) λ n := n + n/ log ( n + 2), n Z is a Pólya sequence. (iii) λ n = n 2 is the zero set of the zero type function F (z) := cos 2πz cos 2πz and thus is not a Pólya sequence Background. From Levinson [3], we have that if λ n n p(n), where p(t) satisfies p(t) log t dt < and some smoothness conditions, then Λ = {λ 1+t 2 p(t) n} is a Pólya sequence. For each such p(t) that also satisfies p(t) dt =, there exists Λ = {λ 1+t 2 n } n= that is not Pólya. How do we close the gap between our sufficient condition and the counterexamples? From de Branges [1], we have that Λ is Pólya if p(t) dt <. However, there exists a 1+t 2 Pólya sequence, Λ when p(t) dt =. Thus we have yet to fully characterize the sets of 1+t 2 Pólya sequences. However, de Branges work [1] contains the following necessary condition: if Λ is Pólya, then the complement of Λ, {I n }, is short, i.e.: I n dist 2 (I n, 0) <. n = 0.

13 But this condition is not sufficient as we saw in the examples above, λ n = n 2 is the zero set of the zero type function F (z) := cos 2πz cos 2πz, which is non constant on C + but bounded on Λ = {λ n }, and the complement of this Λ is short. What this paper shows, then, is that a separated real sequence Λ is not Pólya iff there exists a long sequence of intervals {I n } such that #(Λ I n ) I n 0, where a sequence of intervals is long if it is not short Beurling-Malliavin densities. We will define the interior and exterior densities in terms of Toeplitz kernels and the exponential function S a (z) = e iaz : The interior BM (Beurling-Malliavin) density is defined as D (Λ) = 1 2π sup{a : N[ ΘS a ] = 0}, where Θ(z) denotes some/any meromorphic inner function with {Θ = 1} = Λ. The exterior BM density is defined as D (Λ) = 1 2π sup{a : N[ S a Θ] = 0}, where Θ(z) denotes some/any meromorphic inner function with {Θ = 1} = Λ. An equivalent definition of the interior BM density follows: let γ : R R be a continuous function such that γ( ) = ±. We say that γ is almost decreasing if the family of intervals BM(γ) is short, where BM(γ) is the collection of the connected components of the open set Background Theorems. { x R : γ(x) max γ(t) t [x,+ ) Theorem 1 ([2]). Let µ be a positive measure on R satisfying dµ(t)/(1 + t 2 ) <. Then there exists a short de Branges space B E contained isometrically in L 2 (µ), with de Branges function E(z) being of Cartwright class and having no real zeros. Moreover, if there exists such a space B E with E(z) of positive exponential type, then there also exists such a space B E that is contained properly in L 2 (µ). Theorem 2 ([4, Section 4.2]). Suppose that Θ(z) is a meromorphic inner function with the derivative of arg Θ(t) bounded on R. Then for any meromorphic inner function J(z), we have }. N + [ ΘJ] 0 ɛ > 0, N[ S ɛ ΘJ] 0. Theorem 3 ([4, Section 4.3]). Suppose γ (t) > const. (i) If γ is not almost decreasing, then for every ɛ > 0, N + [S ɛ e iγ ] = 0. (ii) If γ is almost decreasing, then for every ɛ > 0, N + [ S ɛ e iγ ] 0. As noted in [4], part (i) of Theorem 3 holds without the assumption γ (t) > const.

14 Main Results Presented. Theorem 4. Let Λ = {λ n } n= R be a separated sequence of real numbers. The following are equivalent: (i) Λ = {λ n } is a Pólya sequence. (ii) There exists a non-zero measure µ of finite total variation, supported on Λ, such that the Fourier transform of µ vanishes on an interval of positive length. (iii) The interior Beurling-Malliavin density of Λ, D (Λ), is positive. (iv) There exists a meromorphic inner function Θ(z) with {Θ = 1} = {λ n } such that N[ ΘS 2c ] 0, for some c > 0. Corollary. Let Λ = {λ n } n= be a separated sequence of real numbers. Then Λ is a Pólya sequence if and only if for every long sequence of intervals {I n } the sequence #(Λ In) I n is not a null sequence, i.e., #(Λ In) I n 0. With regard to the gap problem we will prove the following result. Let M denote the set of all complex measures of finite total variation on R. For µ M its Fourier transform ˆµ(x) is ˆµ(x) = e ixt dµ(t). If X is a closed subset of the real line denote by GX) the gap characteristic of X: G(X) := sup{a µ M, µ 0, supp µ X, such that ˆµ = 0 on [0, a]}. Theorem 5. The following are true: (i) For any separated sequence Λ R, G(Λ) 2πD (Λ). (ii) For any closed set X R, G(X) 2πD (X). Corollary. For separated sequences Λ R, G(Λ) = 2πD ( L). Corollary. Let X be a closed subset of the real line. If there exists a long sequence of intervals {I n } such that #(X I n ) I n then any measure µ of finite total variation supported on X, whose Fourier transform vanishes on an interval of positive length, is trivial. The formula for G(X) for a general closed set X is more involved, and thus in the proof, we will consider only a closed set X equal to a separated real sequence Λ. Another immediate consequence of Theorem 2 is the following extension of Beurling s gap theorem: If X R is closed, define 0 T (X) = sup{a meromorphic inner Θ(z) with {Θ = 1} X and N[ ΘS a ] 0}. Theorem 6. For any closed X R, T(X) = G(X).

15 2.1. Technical Lemmas. 2. Main Results Lemma 7. Let Θ(z) be a meromorphic inner function with 1 Θ(t) / L 2 (R) and let σ be the corresponding Clark measure. If N[ ΘS 2a ] 0 for some a > 0, then for any ɛ > 0 there exists h L 2 (σ) such that h(t) lim y ± exy t iy dσ(t) = 0 for every x ( a + ɛ, a ɛ) and the measure hdσ has finite total variation. Proof. The idea of the proof is as follows: If N[ ΘS 2a ] 0 for some a > 0 then there exists some f K Θ such that f/s 2a 2ɛ N[ ΘS 2a 2ɛ ] and thus is in N[ ΘS 2a ]. We can then find our measure hdσ from the Clark representation of f with h L 2 (σ) and of finite total variation, and the limit follows from f/s a ɛ N[ ΘS a ɛ ]. 15 Lemma 8. Let µ be a measure with finite total variation. Then the Fourier transform of µ vanishes on [ a, a] if and only if for every x [ a, a]. lim y ± exy dµ(t) t iy = 0, 2.2. Proofs of Main Results. We will present the proofs of the main theorems in reverse order. Proof of Theorem 6. The inequality T(X) G(X) follows from the lemmas. To prove the opposite inequality, let G(X) = a. Then for any ɛ > 0 there exists a non-zero complex measure of total variation no greater than 1 supported on X whose Fourier transform vanishes on [0, a ɛ]. Consider the set of all such measures. Since this set is closed, convex and contains non-zero elements, by the Krein-Milman theorem it has an extreme point, a nonzero measure ν. We can show that the extremality of ν implies that it is supported on a discrete subset of X. Let Θ(z) be the meromorphic inner function whose Clark measure is ν. Then {Θ = 1} X. It is left to notice that the function f(z) = 1 Θ(z) dν(t) 2πi t z belongs to K Θ and is divisible by S a ɛ (as follows, for instance, from the proof of Lemma 8). Hence f/s a ɛ N[ ΘS a ɛ ] 0. Proof of Theorem 5. By Theorem 6 it is enough to prove that T(Λ) = 2πD (Λ). (i) Suppose that D (Λ) = a/2π. By the definition given of D the function φ(x) = 2πn Λ (x) + (a ɛ)x is almost decreasing for any ɛ > 0. Consider a meromorphic inner function Θ with {Θ = 1} = Λ and bounded derivative on R. Then arg ΘS a ɛ differs from φ by a bounded function. Hence arg ΘS a 2ɛ is almost decreasing. By Theorem 2 and Theorem 3, N[ ΘS a 3ɛ ] 0.

16 16 (ii) Again we will prove that T(X) 2πD (X). If T(X) = a then for any ɛ > 0 there exists a meromorphic inner Θ(z) such that Γ := {Θ = 1} X and N[ ΘS a ɛ ] 0. By Theorem 2 (and remark after it) this means that arg ΘS a 2ɛ is almost decreasing. Hence 2πn Γ (x) + (a 3ɛ)x is almost decreasing. Since ɛ is arbitrary, D (X) a/2π. Proof of Theorem 4. (ii) (iii) follows from Theorem 5 and (ii) (iv) from Theorem 6. (i) (iii) Assume (iii) is not true, i.e. for every meromorphic inner function Θ(z) with {Θ = 1} = Λ, N[ ΘS 2c ] = 0 for every c > 0. We construct a non-constant zero type entire function which is bounded on Λ, which will mean that Λ is not a Pólya set. Define a measure µ to be the counting measure of Λ. Then by Theorem 1 there exists a short de Branges space B E contained isometrically in L 2 (µ). If E(z) has type zero then all functions in B E have type zero. If the type of E is positive, then by Theorem 1 we can assume that B E is contained properly in L 2 (µ). Suppose that F (z) B E has positive type, and assume that F (iy) grows exponentially in y as y. Since B E L 2 (µ), there exists g L 2 (µ) with ḡ B E. Then for any w C. Thus the function G(z) := 1 Θ(z) 1 2πi t z g(t)dµ(t) can be represented as G(z) = S c (z)h(z) for some nonzero h(z) H 2 (C + ) and c > 0, and belongs to K Θ, where Θ(z) is the inner function corresponding to the measure µ. Hence h N[ ΘS c ] and we have a contradiction. Therefore any F (z) B E has zero type. It is left to notice that F (λ n ) F (λ m ) 2 = F L 2 (µ) <, which means that F (z) is bounded on Λ. (ii) (i) from de Branges proof [1]. m References [1] De Branges, L. Some applications of spaces of entire functions, Canadian J. Math. 15 (1963), [2] De Branges, L. Some Hilbert spaces of entire functions II, Trans. Amer. Math. Soc. 99 (1961), [3] Levinson, N. Gap and density theorems, AMS Colloquium Publications, 26 (1940) [4] Makarov, N., Poltoratski, A. Meromorphic inner functions, Toeplitz kernels, and the uncertainty principle, in Perspectives in Analysis, Springer Verlag, Berlin, 2005, [5] Mitkovski, M., Poltoratski A. Pólya sequences, Toeplitz kernels and gap theorems. Adv. Math. 224 (2010), University of Alabama, Tuscaloosa, AL address: linden.duffee@gmail.com

17 17 UNCONDITIONAL BASES OF EXPONENTIALS AND OF REPRODUCING KERNELS S.V. HRUŠČEV, N.K.NIKOL SKII, B.S.PAVLOV presented by Jingguo Lai Abstract. We give several concise characterizations on when a given family of exponentials and reproducing kernels form an unconditional bases. The method using to analyze the problem is of independent interest. 1. Introduction A well-known fact in Fourier analysis is: the family {e inx } n Z forms a complete orthogonal system in L 2 (0, 2π). So for any f L 2 (0, 2π), we can write f(x) = n Z a n e inx in L 2 (0, 2π) and this sum converges unconditionally. A natural generalization would be: given an interval I R and a family of complex frequencies Λ = {λ n } n Z, can we make good sense of f(x) = n Z a n e iλnx, for every f L 2 (I) in terms of Λ? Our enemy here is: the family E Λ = {e iλx } n Z might NOT be complete or orthogonal. A first progress on this type of problem is Theorem 1. (Paley-Wiener, 1934) The system E Λ = {e iλx } n Z forms a Riesz basis in L 2 (0, 2π) if λ n R and sup n n λ n < π 2. This result has been repeatly revised and generalized. The most equisite formulation is Theorem 2. (Ingham-Kadec 1/4-Theorem) Let δ > 0. Every family E Λ = {e iλx } n Z satisfying λ n R and sup n λ n n = δ forms a Riesz basis in L 2 (0, 2π) iff δ < 1/4. This theorem will be a consequence of our main results. We should mention the definition of Riesz basis and unconditional bases. Definition 1. A family {ψ n } n Z of non-zero vectors in a Hilbert space H is called an unconditional basis in H if (1) the family {ψ n } n Z spans the space H; (2) n a nψ n 2 n a n 2 ψ n 2. If, in addition, ψ n 1, then the family {ψ n } n Z is called a Riesz basis.

18 18 2. Functional Model 2.1. H p spaces and Nevanlinna spaces. Let us work on the upper-half plane C +. The H p spaces and Nevanlinna spaces consists of analytic functions on C + such that H p (C + ) : f p = sup f(x + iy) p dx <, 0 < p <. y>0 R H (C + ) : f = sup f(z) <. z C + N : sup log + f(x + iy) dx <, log + f(z) = max{log f(z), 0}. y>0 R Note that the first two are norms on their corresponding spaces, but not the third one. Theorem 3. (Fatou) H p (C + ) = {f L p (R) : f is supported on [0, )}, p 1, and we can recover by f(x + iy) = f P y (x). So we sometimes identify H p, p 1 as a closed subspace of L p. Theorem 4. (Riesz-Nevanlinna factorization) If f N, f 0, then f(z) = cbf e S 1 /S 2, where c = 1, B is the Blaschke product, S 1, S 2 are singular inner functions and f e is outer function. If f H p, f 0, then f(z) = cbf e S. Definition 2. A Blaschke produce B with zeros Λ = {λ n } n Z is an infinite product B(z) = n Z z λ n ɛ n = b n (z), z λ n n Z where signs ɛ n, ɛ n = 1 make each factor b n (z) nonnegative at z = i. A well-known Blaschke condition: Blaschke product to converge. n Z Definition 3. A singular inner function S is of the form { S(z) = exp 1 } tz + 1 πi t z dµ(t), Iλ n λ n+i < is necessary and sufficient for the 2 where µ is a non-negative finite measure on R = R { } and is singular with respect to the Lebesgue measure on R. Example. If taking µ = πaδ, then S(z) = e iaz is a singular inner function. Definition 4. An outer function f e of a function f is defined by { } 1 tz + 1 log f(t) f e (z) = exp πi t z t dt. Lemma 5. (1) B = S = 1 a.e. on R. (2) f is an outer function iff log f(z) = 1 I(z) log f(t) dt. π R z t 2 R There is a parallel theory on the unit disc D, which we skip. Their relation is the isometry of L p (T) onto L p (R) U p f(x) = 1 ( ) 1 x i f, x R. π 1 p (x + i) 2 p x + i R

19 Our main results are stated on the upper-half plane C +, but sometimes it is easier to work on D. In this note, we mainly focus on the space H 2 (C + ) or H 2 (D), which are reproducing kernel Hilbert spaces (RKHS). Some results concerning H p for p 2 can be found in Chapter 2, Section 6 of the original paper Two Theorems of Paley-Wiener. Theorem 6. If f L 2 (0, ), then F (z) = f(t)e itz dt satisfies F (z) H 2 (C 0 + ). Conversely, if F H 2 (C + ), then there exists f L 2 (0, ) such that F (z) = f(t)e itz dt. 0 Theorem 7. Let A and C be two positive constants. If f L 2 ( A, A), then F (z) = A A f(t)eitz dt is an entire function of exponential type A, i.e. F (z) Ce A z, and R F (x) 2 dx <. Conversely, if F is an entire function of exponential type A and R F (x) 2 dx <., then there exists f L 2 ( A, A) such that F (z) = A A f(t)eitz dt Functional Model. Now we are in a position to carefully formulate our problem. Let Λ = {λ n } n Z C + and sup n Iλ n < be a fixed subset and a > 0. Let B be the Blaschke product for the sequence Λ if it satisfies the Blaschke condition and identically zero otherwise. We want to find condtions on Λ such that the family E Λ = {e iλnx } n Z forms a Riesz basis in L 2 (0, a). We first note that e iλnx 1, so in this case, Riesz basis means exactly unconditional basis. Note also {e iλnx } is an unconditional basis iff {e iλnx } is an unconditional basis. Let Λ = { λ n } n Z. The inverse Fourier transform gives F (e iλx χ [0, ) )(z) = i. Note that z λ {k(z, λ) = i } are exactly the reproducing kernels of z λ H2 (C + ) and since B is the Blaschke product for Λ, we would have span{k(z, λ n )} n Z = K B = H 2 (C + ) BH 2 (C + ). Hence, F maps span{e iλnx χ [0, ) } n Z onto the subspace K B = H 2 (C + ) BH 2 (C + ). This says {e iλnx } is an uncondition basis of L 2 (0, ) iff {k(z, λ n )} is an unconditional basis. Let θ a (z) = e iaz, we can deduce F L 2 (0, a) = F L 2 (R + ) F L 2 (a, ) = H 2 θ a H 2 = K θ a. So {e iλnx } is an uncondition basis of L 2 (0, a) iff {P Kθ ak(z, λ n )} is an unconditional basis. At this stage, it is natural to consider the following general problem: Let θ be any inner function and let B be a Blaschke product with the sequence Λ. Let P θ = P Kθ be the orthogonal projection onto K θ. Then the function k θ (z, λ) = P θ k(z, λ) is the reproducing kernel for K θ, because for f K θ, f(z), k θ (z, λ) = f(z), P θ k(z, λ) = f(z), k(z, λ) = f(λ). So k θ (z, λ) = i 1 θ(λ)θ(z). In the case of D, k(z, λ) = 1 and k z λ 1 λz θ(z, λ) = 1 θ(λ)θ(z). 1 λz General problem of unconditional bases for reproducing kernel: What is to be assumed about the pair (θ, Λ) for the family {k θ (z, λ)} λ Λ to be an unconditional basis in K θ? 3. Towards a solution (I) 3.1. Carleson s Interpolation Theorem and Carleson Embedding Theorem. Theorem 8. (Carlenson s Interpolation Theorem) Let Λ = {λ n } n Z C + be a sequence, the following are equivalent: 19

20 20 (1) The sequence is an interpolating sequence: every interpolation problem f(λ n )(Iλ n ) 1 2 = an, n Z with {a n } l 2 has solution f H 2 (C + ). (2) Λ satisfies the well-known Carleson condition: inf λ n λ k n λ n λ = δ > 0. (C) k n k (3) Λ is a rare set, i.e. there exists ε > 0, such that D(λ n, εiλ n ) D(λ k, εiλ k ) =, n k, (R) and the measure µ = n Z Iλ nδ λn is a Carleson measure. On the unit disc D, the conditions are slightly different: Carleson condition: inf n k n λn λ k 1 λ k λ = δ > 0. (C) n Rarity condition: D(λ n, ε(1 λ n )) D(λ k, ε(1 λ k )) =, n k. (R) Carleson measure: µ = n Z (1 λ n )δ λn. λ It is not hard to check that condition (R) is equivalent to inf n λ k n k 1 λ k λ δ > 0. To n understand the Carleson measure condition, we need the Carleson Embedding Theorem. Theorem 9. (Carleson Embedding Theorem on the unit disc D) Let µ be a positive measure on D. The following are equivalent: (1) µ is a Carleson measure. (2) sup f 2 dµ <. f H 2 (D), f 1 D (3) sup λ supp(µ) D k(z, λ) 2 dµ(z) < for all normalized reproducing kernel k(z, λ) = (1 λ 2 ) λz Using statement (3) of Carleson Embedding Theorem, µ = n Z (1 λ n )δ λn is a Carleson (1 λ measure on D iff sup n n k 2 )(1 λ k 2 ) <. 1 λ k λ n A first reduction of the problem. We first mention a nice characterization due to N.K.Nikol skii and B.S.Pavlov, which will be appealed to. Theorem 10. (Nikol skii-pavlov) Let Λ = {λ n } n Z C +. The following are equivalent: (1) The family {k(z, λ n )} n Z forms an unconditional basis in its own span in H 2 (C + ). (2) The family {e iλnx } n Z forms an unconditional basis in its L 2 (R + )-span. (3) Λ (C). And it comes our main results: Theorem 11. Let θ be an inner function and Λ = {λ n } n Z C +. If the family {k θ (z, λ n )} n Z is an unconditional basis in K θ, then Λ (C). Theorem 12. Let θ be an inner function and Λ = {λ n } n Z C +. Moreover, assume that sup n θ(λ n ) < 1. The following are equivalent:

21 (1) {k θ (z, λ)} n Z forms an unconditional basis in K θ. (2) Λ (C) and P θ K B maps isomorphically the space K B onto K θ, where B is the Blaschke product for the sequence Λ. We need to understand the meaning of sup n θ(λ n ) < 1. Since (z λ) 1 2 H2 = k(z, λ), k(z, λ) = k(λ, λ) = i λ λ = 1 2Iλ, P θ (z λ) 1 2 H = k θ(λ, λ) = i 1 θ(λ) 2 = 1 θ(λ) 2, 2 λ λ 2Iλ thus sup n θ(λ n ) < 1 is equivalent to k(z, λ n ) H 2 k θ (z, λ n ) H 2. There is yet a theorem without the condition sup n θ(λ n ) < 1 formulated in Chapter 2, Section 4 of the original paper. The above theorem, in particular, implies Theorem 13. Let θ be an inner function, Λ = {λ n } n Z C + and a > 0. The following are equivalent: (1) {e iλnx } n Z is an unconditional basis in L 2 (0, a). (2) Λ (C) and the restriction f fχ (0,a) maps isomorphically the space span L 2 (R){e iλnx } n Z onto L 2 (0, a). 4. Towards a solution (II) We need to undersand when P θ K B maps isomorphically the space K B onto K θ? To do this, we have to do more preparations A lemma from functional analysis. Lemma 14. Let M and N be two closed subspaces of a Hilbert space H. P M N is an isomorphism onto its image (of left invertible) iff P N M < 1. P M N is an isomorphism onto M (or invertible) iff both P N M < 1 and P M N < 1. Proof. For the first assertion, we just need to remember: An operator T : X Y defined on two Banach spaces satisfies ker(t ) = {0} and Range(T ) is closed iff there exists c > 0 such that T x c x for all x X. Thus, P M N is an isomorphism onto its image iff P M x c x, x N. Note that P M N + P M N = Id N, so we have P M N < 1. Finally, because (P M N) = P N M, we conclude P N M < 1. For the second assertion, since (P M N) = P N M, so P M N is an isomorphism onto M iff both P M N and P N M are left invertible. The first assertion then implies the second Hankel operators, Toeplitz operators and Nehari Theorem. Definition 5. Let ϕ L (R). The Toeplitz operator with symbol ϕ is the operator T ϕ on H 2 (C + ) defined by T ϕ f = P + (ϕf) for f H 2 (C + ). The Hankel operator with the same symbol ϕ is the defined by H ϕ f = P (ϕf) for f H 2 (C + ). Hence, we have the simple relation ϕf = H ϕ f + T ϕ f for f H 2 (C + ). Theorem 15. (Nehari) If ϕ L (R), then H ϕ = dist(ϕ, H ). A simple corollary of Nehari Theorem is 21

22 22 Corollary 16. Let ϕ L (R) and ϕ = 1 a.e., then T ϕ is an isomorphism onto its image iff H ϕ = dist(ϕ, H ) < 1, T ϕ is an isomorphism onto H 2 (C + ) iff H ϕ = dist(ϕ, H ) < 1, H ϕ = dist(ϕ, H ) < 1. Proof. Note that the relation ϕf = H ϕ f + T ϕ f gives f 2 = H ϕ f 2 + T ϕ f 2 for f H 2 (C + ). Hence, H ϕ < 1 iff T ϕ f c f, f H 2 (C + ) for some c > 0, which is equivalent to T ϕ being an isomorphism onto its image. Finally, using Nehari Theorem, the first assertion is proved. The second assertion follows When is P θ B invertible? First Lemma 14 tells us P θ B maps isomorphically the space K B onto K θ iff P B Kθ < 1 and P θ KB < 1. To better understant these, we need the following lemma. Lemma 17. Let ϕ be an inner function, then P ϕ = ϕp ϕ. Proof. Write P = ϕp ϕ, then P is idemponent and normal. Idemponence: P P = ϕp ϕϕp ϕ = ϕp ϕ = P. Self-ajoint operator: P f, g = ϕp ϕf, g = f, ϕp ϕg = f, P g. These two imply P is a projection. Moreover, P f = 0 iff f θh 2. So P = P θ. Having this lemma, we can conclude: P B K θ = P B θh 2 = BP B θh 2 = BH Bθ θ θh 2. So P B K θ = H Bθ. Similarly, P θ K B = H Bθ. Hence, by Corollary 16, P θ B maps isomorphically the space K B onto K θ iff H Bθ = dist(bθ, H ) < 1 and H Bθ = dist(bθ, H ) < 1 iff both T Bθ and T Bθ are invertible. 5. Towards a solution (III) Up to now, we have obtained a few non-trivial characterizations. To go further, we need to consider a more general situation. Again, we begin with some preparations Hilbert transform, Helson-Szegö condition and (A 2 )-condition. For v L (R), we define its Hilbert transform ṽ by ṽ(x) = 1 π p.v. here, we need the term The Schwartz formula ( 1 x t + t ) v(t)dt, 1 + t 2 R t to remove the sigularity at infinity. 1+t 2 V (z) = 1 ( 1 πi R t z t ) v(t)dt 1 + t 2 recovers the function V by its real part v only, provided V H and IV (i) = 0. The function ṽ(x) is the boundary value of IV (z). Definition 6. A non-negative function w is called a function satisfying Helson-Szegö condition (w (HS)) if there are functions u, v in L (R) such that v L < π and w = exp(u + ṽ). 2 Definition 7. A unimodular function ϕ on R is called a Helson-Szegö function if there are a unimodular constant λ and an outer function h satisfying ϕ = λ h h and h 2 (HS).

23 Another equivalent form of the Helson-Szegö condition has been obtained by B.Muckenhoupt, R.A. Hunt and R.L.Wheeden. Theorem 18. (Muckenhoupt-Hunt-Wheeden) The (HS)-condition is equivalent to the Mukenhoupt (A 2 )-condition 1 wdx 1 w 1 dx <, where I is the family of all intervals on R. I I sup I I I We now state our main theorem. I Theorem 19. Let ϕ be a unimodular function. The following are equivalent. (1) The Toeplitz operator T ϕ is invertible. (2) dist L (ϕ, H ) < 1 and dist L (ϕ, H ) < 1. (3) There exists an outer function f, f H, satisfying ϕ f L < 1. (4) There exists a branch of the argument α of ϕ, ϕ(x) = e iα(x), such that dist L (α, L + C) = inf{ α ṽ c L : v L (R), c C} < π 2. (5) ϕ is a Helson-Szegö function. The equivalence of (1) and (2) is exactly Corollary 16 from the last Chapter. To obatin a list of invertibility tests for P θ B, it suffices to put ϕ = Bθ in the condition of the theorem. 6. Towards a solution (IV) 6.1. A particular interesting characterization. We want to further refine assertion (4) of Theorem 19. Before doing that, let us mention a simple but important remark. The following isomorphisms in L 2 (0, a) preserves the exponentials: f(x) e iαx f(x), f(x) f(a x), f(x) f(x). Any of these isomorphisms preserves the property of being an unconditional basis. So we always can move a frequency set Λ = {λ n } n Z from C + to C δ = {z C : Iz > δ}, δ > 0. Now, let Λ C δ, δ > 0 and let B be a Blaschke product with zero set Λ. For x R, we have d d argb(x) = I( dx dx log B(x)) = I( d dx log n Z ( x λn x λ n thus it is easy to see that the function α Λ defined by x Iλ n α Λ (x) = 2 dt ax, x R, t λ n 2 0 n Z ) ) = 2 n Z Iλ n x λ n 2, is a continuous branch of argument, up to an additive constant, of the unimodular function Bθ a on R. Now, we refine assertion (4) of Theorem 19 using this specific function α Λ. Theorem 20. Let Λ C δ, δ > 0. Then the family {e iλnx } n Z forms an unconditional basis in L 2 (0, a) iff Λ (C) and dist L (α Λ, L + C) < π 2. The sufficency of this theorem is a consequence of Theorem 13 and Theorem 19. As a corollary, we can easily prove Ingham-Kadec 1/4-Theorem stated in the introduction. 23

24 Entire functions of exponential type. An important tool used to prove Theorem 20 is the properties of entire functions of exponential type. Let F (z) be an entire function of exponential type A, i.e. for every ε > 0, there exists C ε > 0 such that F (z) C ε e (A+ε) z. Definition 8. The 2π-periodic indicator function of F (z) is defined by log F (re iϕ ) h F (ϕ) = lim, ϕ R. r r Definition 9. Let K C be a convex compact set. The supporting function k(ϕ) of the set K is k(ϕ) = sup{r(ζe iϕ ) : ζ K}. A classical result on entire functions of exponential type asserts that: h F (ϕ) is a supporting function of a convex compact set G F, i.e. h F (ϕ) = sup{r(ζe iϕ ) : ζ G F }. The convex compact set G F is called the indicator diagram of F. Write F (z) = n=0 F (z). c n n! z n. The function f(z) = n=0 c n z n+1 is called the Borel transform of The smallest convex compact set containing all singularities of f(z), denoted by G F, is called the conjugate diagram of F (z). We denote by k F (ϕ) the supporting function of this set G F. Theorem 21. (Pólya) For every entire function of exponential type F (z), the relation h F (ϕ) = k F ( ϕ) holds and hence G F = {ζ : ζ G F }. We denote by M a = {all entire functions F of exponential type with G F = [0, ia] and F 2 R (HS) or equivalently (A 2 )}. The most important intermediate step in proving Theorem 20 is: Theorem 22. Let Λ C δ, δ > 0 be a Blaschke set, let B denote the corresponding Blaschke product and let θ a = exp(iaz), a > 0. The following are equivalent: (1) There exists a function of the class M a with simple zeros whose zere set is Λ. (2) The restriction Bθ a R is a Helson-Szegö function, i.e. there exists a unimodular constant c and an outer function h such that h 2 R (HS) and Bh = chθ a a.e. on R. Theorem 2, Theorem 11, Theorem 12, Theorem 19, Theorem 20 and Theorem 22 will be carefully explained during the workshop. References [1] S. V. Hruščev, N. K. Nikol skii, B. S. Pavlov Unconditional bases of exponentials and of reproducing kernels, in Complex Analysis and Spectral Theorey (V. P. Havin and N. K. Nikol skii, Eds.), Lecture Notes in Mathematics, Vol. 864, Springer-Verlag, Berlin, [2] J. B. Garnett Bounded analytic functions, Academic Press, New York, [3] N. K. Nikol skii Treatsie on the shift operator Grundlehren der Mathematischen Wissenschaften [Fundamental Principals of Mathematical Sciences], 273. [4] J. B. Conway A course in functional analysis, GTM 96, Second edition, New York, Spronger-Verlag, [5] B. Y. Levin Lectures on Entire Functions, Trans. of Math. Monographs, v Amer. Math. Soc., RI 151 Thayer Street, Providence, RI 02912, USA address: jglai@math.brown.edu

25 25 FOURIER FRAMES JOACHIM ORTEGA-CERDÀ AND KRISTIAN SEIP Presented by James Murphy Abstract. We give an extended overview of Fourier frames. Our focus shall be on the 2002 paper of Ortega-Cerdà and Seip entitled On Fourier Frames. We give background in the form of major definitions and early results in the theory, and briefly review de Branges theory of Hilbert spaces of entire functions. We then move to the two major results of On Fourier Frames : characterizations of sampling sequences for the Paley-Wiener space. Contents 1. Introduction and background on de Branges spaces 1 2. Main Result 1: Complex Analysis and Sampling 4 3. Main Result 2: Approximation of Subharmonic Functions and Sampling 4 References 5 1. Introduction and background on de Branges spaces Recalling that {e inx } n Z is an orthonormal basis for L 2 ( π, π), we generalize to the case when a family of complex exponentials spans L 2 ( π, π), though perhaps with redundancy. Definition 1.1. A family of complex exponentials {e iλ kx } k Z, with Λ = {λ k } k Z R, is a Fourier frame if there exist 0 < A B < such that f L 2 ( π, π): (1.2) π A f(x) 2 dx π k Z π π f(x)e iλkx dx 2 B π π f(x) 2 dx. The statement (1.2) is a particular instance of the frame condition, which has been generalized to a variety of settings. Of particular relevance to the notion of Fourier frame is the applicability of the Paley-Wiener theorem. There are many formulations of this classical result, but we use the following version [7] to motivate an alternate characterization of Fourier frames. Theorem 1.3 (Paley-Wiener). Let σ > 0 be constant. Then a function F (x) is of the form: F (x) = σ σ f(ξ)e iξx dξ for some f L 2 ( σ, σ) if and only if F (x) L 2 (R) and F can be extended to an entire function of exponential-type at most σ, meaning F extends to an entire function F such that C > 0 with the property that F (z) Ce σ z everywhere.

26 26 We call the space of entire functions of exponential type at most π, whose restriction to R C is square-integrable, the Paley-Wiener space, denoted PW. Definition 1.4. A sequence Λ = {λ k } k Z is sampling for PW if there exist 0 < A B < such that f P W : A f(x) 2 dx f(λ k ) 2 B f(x) 2 dx. R k Z R The Paley-Wiener theorem can be used together with the Plancherel theorem to show that Λ = {λ k } k Z is sampling for PW if and only if {e iλ kx } k Z is a Fourier frame. Hence, we will study sampling sequences in order to understand Fourier frames. Indeed, the two main results of On Fourier Frames are characterizations of sampling sequences for PW; we must keep in mind that such characterizations also describe Fourier frames, our initial objects of study. We now introduce several properties of sequences {λ k } k Z R, which will be used to prove the two main results characterizing sampling sequences. Definition 1.5. Λ = {λ k } k Z is a complete interpolating sequence if the interpolation problem f(λ k ) = a k, k Z has a unique solution f P W for all l 2 (C) data {a k } k Z. We note that an alternate characterization of Λ being a complete interpolating sequence is that Λ is sampling, but Λ \ {λ j } is not, for any j Z. In this sense, complete interpolating sequences can be considered minimal sampling sequences. Definition 1.6. Consider Λ = {λ k } k Z where λ k λ k+1, k Z. Such a sequence is separated if q := inf (λ k+1 λ k ) > 0; q is the separation constant. For a separated sequence, k Z define the associated distribution function n Λ as: n Λ (0) = 0, a < b, n Λ (b) n Λ (a) = Λ (a, b). Theorem 1.7 (Landau s Inequality). If Λ is a separated sampling sequence for PW, then there exist constants A, B such that for all a < b: n Λ (b) n Λ (a) b a A log + (b a) B. Landau s inequality provides a necessary condition for sampling. This sophisticated result can be considered alongside a more elementary inequality, which gives a sufficient condition: n Λ (b) n Λ (a) (1 + ɛ)(b a) C, C, ɛ > 0 independent of a < b = Λ is sampling. Indeed, the example of Λ = {k + log + k } k Z optimizes Landau s inequality. Of paramount importance to the study of Fourier frames is the notion of Beurling density. Definition 1.8. For a separated sequence Λ = {λ k } k Z with associated distribution n Λ, the lower Beurling uniform density is D (Λ) := lim R min x R (n Λ (x + R) n Λ (x)). R