Introduction to Hilbert Space Frames

We present an introduction to Hilbert space frames. A frame is a generalization of a basis that is useful, for example, in signal processing. It allows us to expand Hilbert space vectors in terms of a set of other vectors that satisfy a certain “energy equivalence” condition. This condition guarantees that any vector in the Hilbert space can be reconstructed in a numerically stable way from its frame coefficients. We focus on frames in finite dimensional spaces. 1 Preliminaries 1.1 Hilbert Space We recall here some basic definitions and facts about Hilbert space. Readers who are familiar with Hilbert spaces may skip this section. Definition 1. A Hilbert space is a complete, normed vector space H over the complex numbers C, whose norm is induced by an inner product. The inner product is a function 〈 · , · 〉:H× H C that satisfies 1. Linearity in the second argument: ∀a, b∈C and ∀x, y, z ∈H, 〈x, a y+ b z〉= a 〈x, y〉+ b〈x, z〉. 2. Conjugate symmetry: ∀x, y ∈H, 〈x, y〉= 〈y, x〉 where the overbar denotes complex conjugation. 3. Positivity: ∀x 0∈H, 〈x, x〉> 0. The norm of H is induced by its inner product: ∀x ∈ H, ‖x‖2 = 〈x, x〉. The Hilbert space is required to be complete, which means that every sequence that is Cauchy with respect to this norm converges to a point in H. The most common Hilbert spaces, and the only ones we will be concerned with here, are the Euclidean spaces and the square-integrable function spaces. Example 2. The Euclidean space C is a Hilbert space with an inner product defined by (〈x, y〉)= ∑ i=1 n x̄iyi. The norm induced by this inner product is the standard Euclidean distance; for example, in C we have ‖x‖= x1+x2 √ . We can generalize these Euclidean spaces to infinite dimensions. Our vectors are then functions instead of n-tuples of numbers, and we must introduce the additional requirement that the functions be square-integrable to ensure that the inner product is well defined. 1 Example 3. For a measure space M and a measure μ, define L(M, μ) to be the set of measurable functions f : M→ C such that ∫ |f |2d μ < ∞. This is a Hilbert space with the inner product 〈f , g〉= ∫ f̄ g dμ. We will take μ to be the Lebesgue measure when M is R or an interval on R, and counting measure when M=N or M=Z. In the first case, we use the notation L([a, b]), and the Hilbert space consists of square-integrable functions. In the second case, we use a lowercase l and write l(N). Recall that integration with respect to counting measure is just summation, so the inner product on l is 〈x, y〉= ∑ i=1 ∞ xiyī , and l is the space of square-summable sequences. Hilbert spaces are “nicer” than general Banach spaces because of the additional structure induced by the inner product. The inner product allows us to define “angles” between vectors, and in particular, leads to the concept of orthogonality: Definition 4. Two vectors x and y in a Hilbert space H are said to be orthogonal if 〈x, y〉=0. A set of vectors {xi} is said to be orthogonal if 〈xi, xj〉=0 for i j. In the Euclidean spaces, the inner product is the standard dot product, and two vectors are orthogonal if their dot product is zero. 1.2 Linear Operators on Hilbert Space We use the term operator here to refer to a linear map between two Hilbert spaces. We recall some basic terminology regarding operators. Definition 5. For Hilbert spaces H1 and H2, mapping T :H1→H2, is called a linear operator if, for every x, y ∈H1 and for every c1, c2∈C, we have T (c1x+ c2y)= c1Tx+ c2T y. A linear operator is bounded if there exists a constant K > 0 such that ‖T x‖ 6 K ‖x‖ for all nonzero x∈H1. If T is a bounded operator, then we define the operator norm to be the norm induced by the two Hilbert space norms in the following way: ‖T ‖= inf {K: ‖Tx‖6K‖x‖ for allx 0}. Every linear operator has an adjoint, which is the unique operator T ∗ satisfying 〈T x , y〉= 〈x, T ∗y〉 for all x, y ∈H1. An linear operator is an injection if T x= T y x= y (that is, if T maps distinct elements in H1 to distinct elements in H2). A linear operator is a surjection if range(T ) = H2. An operator that is both surjective and injective is called a bijection. In the finite dimensonal case, linear operators are just matrices; the linear operators from C to C are precisely the m × n matrices in Cm×n. Infinite dimensional linear operators are the subject of functional analysis, and are much more difficult to classify in general. We will be working with a special type of linear operator called a “frame operator” whose norm is bounded above and below by two nonzero constants. 2 Section 1