A Brief Introduction to Hilbert Space Frame Theory and Its Applications

This is a short introduction to Hilbert space frame theory and its applications for those outside the area who want an introduction to the subject. We will increase this over time. There are incomplete sections at this time. If anyone wants to add a section or fill in an incomplete section on ”their applications” contact Pete Casazza. 1. Basic Definitions For a more complete treatment of frame theory we recommend the book of Christensen [18], and the tutorials of Casazza [5, 6] and the Memoir of Han and Larson [23]. For a complete treatment of frame theory in time-frequency analysis we recommend the excellent book of Gröchenig [22]. For an excellent introduction to frame theory and filter banks plus applications we recommend Kovačević and Chebira [29]. Hilbert space frames were introduced by Duffin and Schaeffer in 1952 [20] to address some deep questions in non-harmonic Fourier series. The idea was to weaken Parseval’s identity. Parseval’s Identity 1.1. If {fi}i∈I is an orthonormal sequence in a Hilbert space H, then for all f ∈ H we have ∑ i∈I |〈f, fi〉| = ‖f‖. We do not need to have an orthonormal sequence to have equality in Parseval’s identity. For example, if {ei}i∈I and {gi}i∈I are orthonormal bases for a Hilbert space H then { 1 √ 2 ei, 1 √ 2 gi}i∈I satisfies Parseval’s identity. In 1952, Duffin and Schaeffer [20] were working on some deep problems in non-harmonic Fourier series and weakened Parseval’s identity to produce what they called a frame. Definition 1.2. A family of vectors {fi}i∈I is a frame for a Hilbert space H if there are constants 0 < A ≤ B <∞ so that for all f ∈ H (1.1) A‖f‖ ≤ ∑ i∈I |〈f, fi〉| ≤ B‖f‖. The authors were supported by NSF DMS 0704216. 1 2 P.G. CASAZZA AND J.C. TREMAIN A,B are called the lower (respectively, upper) frame bounds for the frame. If A = B this is an A-tight frame and if A = B = 1 this is a Parseval frame. If ‖fi‖ = ‖fj‖ for all i, j ∈ I, this is an equal norm frame and if ‖fi‖ = 1 for all i ∈ I this is a unit norm frame. If we have just the right hand inequality in inequality (9.1) we call {fi}i∈I a B-Bessel sequence. The frame is bounded if infi∈I‖fi‖ > 0. We note that there are no restrictions put on the frame vectors. For example, if {ei}i=1 is an orthonormal basis for a Hilbert space H, then {e1, 0, e2, 0, e3, 0, · · ·} is a Parseval frame for H. Also, {e1, e2 √ 2 , e2 √ 2 , e3 √ 3 , e3 √ 3 , e3 √ 3 , · · ·}, is a Parseval frame for H. Definition 1.3. A family of vectors {fi}i∈I in a Hilbert space H is a Riesz basic sequence if there are constants A,B > 0 so that for all families of scalars {ai}i∈I we have A ∑