Discrete frames on finite dimensional quaternion Hilbert spaces

An introductory theory of frames on finite dimensional quaternion Hilbert spaces is demonstrated along the lines of their complex counterpart.


Introduction
Frames were first introduced by Duffin and Schaeffer in a study of non-harmonic Fourier series [9]. However, among many others, the pioneering works of Daubechies et al. brought the proper attention to frames [7,8]. Wavelets and coherent states of quantum optics are specific classes of continuous frames [3]. The study of frames has exploded in recent years, partly because of their applications in digital signal processing [4,11] and other areas of physical and engineering problems. In particular, they are an integral part of time-frequency analysis. In this note we are primarily interested in frames on finite dimensional quaternion Hilbert spaces. There has been a constant surge in finding finite tight frames, largely as a result of several important applications such as internet coding, wireless communication, quantum detection theory, and many more [5,6,10,4,13]. It is crucial to find a specific class of frame to fit to a specific physical problem, because there is no universal class of frame that fit to all problems. As technology advances, physicists and engineers will face new problems and thereby our search for tools to solve them will continue.
A Separable Hilbert space possesses an orthonormal basis and each vector in the Hilbert space can be uniquely written in terms of this orthonormal basis. Despite orthonormal bases are hard to find, this uniqueness restricted flexibility in applications and pleaded for an alternative. As a result frames entered to replace orthonormal bases. Frames are classes of vectors in Hilbert spaces. In a finite dimensional Hilbert space a typical frame possesses more vectors than the dimension of the space, and thereby each vector in the space can have infinitely many representations with respect to the frame. This redundancy of frames is the key to their success in applications. The role of redundancy varies according to the requirements of the application at hand. In fact, redundancy gives greater design flexibility which allows frames to be constructed to fit a particular problem in a manner not possible by a set of linearly independent vectors [3,4,7,2].
Hilbert spaces can be defined over the fields R, the set of all real numbers, C, the set of all complex numbers, and H, the set of all quaternions only [1]. The fields R and C are associative and commutative and the theory of functional analysis is a well formed theory over real and complex Hilbert spaces. But the quaternions form a non-commutative associative algebra and this feature highly restricted mathematicians to work out a wellformed theory of functional analysis on quaternionic Hilbert spaces. Further, due to the noncommutativity there are two types of Hilbert spaces on quaternions, called right quaternion Hilbert space and left quaternion Hilbert space. In assisting the study of frames the functional analytic properties of the underlying Hilbert space are essential. In the sequel we shall prove the necessary functional analytic properties as needed.
To the best of our knowledge a general theory of frames on quaternionic Hilbert spaces is not formulated yet. In this part of the thesis we shall construct frames on finite dimensional left quaternionic Hilbert spaces following the lines of [4]. Since noncommutativity of quaternions does not play a bigger role in the construction of frames on finite dimensional quaternion Hilbert spaces, most of the results follow their complex counterparts. While the complex numbers are two dimensional the quaternions are four dimensional; the increase in the dimension expected to give greater flexibility in applications. We are also expected to demonstrate this issue in applications as the thesis progress further.

quaternion Algebra
In this section we shall define quaternions and some of their properties as needed here. For details one may consult [1,14,12].
2.1. Quaternions. Let H denote the field of quaternions. Its elements are of the form q = x 0 + x 1 i + x 2 j + x 3 k, where x 0 , x 1 , x 2 and x 3 are real numbers, and i, j, k are imaginary units such that i 2 = j 2 = k 2 = −1, ij = −ji = k, jk = −kj = i and ki = −ik = j. The quaternionic conjugate of q is defined to be q = Quaternions can also be represented by using 2 × 2 complex matrices. It can be written as the linear combination of the matrices where σ 1 , σ 2 and σ 3 are the usual Pauli matrices. In this notations the quaternions can be written as and q = q † (matrix adjoint). Introducing the polar coordinates: x 0 = r cos θ x 1 = r sin θ sin φ cos ψ x 2 = r sin θ sin φ sin ψ x 3 = r sin θ sin φ cos ψ where r ∈ [0, ∞), θ, φ ∈ [0, π], and ψ ∈ [0, 2π), we may write The matrices A(r) and σ(ñ) satisfy the conditions, The quaternion product is not commutative. (

Frames in quaternion Hilbert space
Assume that the space V L H is together with .|. is a separable Hilbert space. Properties of left quaternion Hilbert spaces as needed here can be listed as follows: For f, g ∈ V L H and p, q ∈ H, we have For an enhanced explanation of quaternions and quaternion Hilbert spaces one may consult [1,12] and the many references listed there.
k=1 is a linearly independent set.
As a consequence of this definition, for every f ∈ V L H there exist unique scalar coeffi- We now introduce the frames on finite dimensional left quaternion Hilbert spaces. We shall show that the complex treatment adapt to the quaternions as well. In this chapter left span means left span over the quaternion scalar field, H. We shall also prove the functional analytic properties for quaternions as needed here, and these proofs are the adaptation of the proofs of the complex cases given in [15]. The theory of frames offered here, more or less, follows the lines of [4].
for all f ∈ V L H . The numbers A and B are called frame bounds. They are not unique. The optimal lower frame bound is the supremum over all lower frame bounds, and the optimal upper frame bound is the infimum over all upper frame bounds. Note that the optimal frame bounds are actually the frame bounds. A frame is said to be normalized if f k = 1, for all k ∈ I. In this chapter we shall only consider finite frames {f k } m k=1 , m ∈ N. With this restriction, Schwartz inequality shows that

From (3.6) it is clear that the upper frame condition is always satisfied with
In order for the lower condition in (3.5) to be satisfied, it is necessary that leftspan{f k } m k=1 = V L H . Let us see this in the following.
Then a mapping ϕ : W −→ R is continuous if and only if for any sequence {x n } in W which converges to x 0 as n → ∞ then ϕ(x n ) converges to ϕ(x 0 ) as n → ∞.
Proof. Suppose that ϕ is continuous on W then ϕ is continuous at x 0 ∈ W . Thereby, for given ǫ > 0 there exists δ > 0 such that Since x n converges to x 0 as n −→ ∞, there exists N ∈ N such that From (3.7) and (3.8), there exists N ∈ N such that Thereby for given ǫ > 0, there exists N ∈ N such that Conversely suppose that, for a sequence {x n } in W which converges to x 0 as n → ∞ then ϕ(x n ) converges to ϕ(x 0 ) as n → ∞. Assume that ϕ is not continuous, then for given δ > 0, there exists ǫ > 0 such that This is a contradiction with our supposition. Hence ϕ is continuous.

Proof. From the Schwartz inequality the upper frame condition is satisfied with
Let W := lef tspan {f k } m k=1 and consider the mapping Now we want to prove that ϕ is continuous. Let {g n } be a sequence in W such that g n −→ g as n −→ ∞. Now, −→ 0 as n −→ ∞ ⌈∵ g n −→ g as n −→ ∞⌉ Thereby ϕ(g n ) converges to ϕ(g) as n −→ ∞. From the lemma (3.5), ϕ is continuous.
Since the closed unit ball in W is compact, from the lemma (3.6), we can find g ∈ W with g = 1 such that It is clear that A > 0 as not all f k are zero. Now given f ∈ W, f = 0, we have Thereby From (3.12) and (3.13) , Hence {f k } m k=1 is a frame for lef tspan {f k } m k=1 .
Proof. Suppose that {f k } m k=1 is a frame for V L H . Then there exist A, B > 0 such that which is a contradiction. Thereby V L H ⊆ lef tspan{f k } m k=1 . Clearly lef tspan{f k } m k=1 ⊆ V L H . Thereby the conclusion follows.
From the above corollary it is clear that a frame is an over complete family of vectors in a finite dimensional Hilbert space.
For a detail explanation we refer the reader to [1].
T is usually called the pre-frame operator, or the synthesis operator. The adjoint operator is called the analysis operator. By composing T with its adjoint we obtain the frame operator Note that in terms of the frame operator, for Proposition 3.9. Assume that {f k } m k=1 is a tight frame for V L H with frame bound A. Then S = AI (here I is the identity operator on V L H ),and Proof. The frame operator S is given by H . The identity operator on V L H is given by f |f k f k as A is real. Proof. For arbitrary f 1 , f 2 ∈ V L H and scalars α, β ∈ H, we have the following computation for every g ∈ V L H , Since this holds for all g ∈ V L H , we conclude that Hence either S * f = 0 or S * f ≤ S f . In either case This proves that S * is bounded, and furthermore, that S * ≤ S .
Proof. We have since g|f = f |g = 0, as f and g are orthogonal.  T * y|x = 0, for all y ∈ V L H .
T * , then x|T * y = 0, for all y ∈ V L H . Thereby T * y|x = 0, for all y ∈ V L H . By the definition (3.12), y|T x = 0, for all y ∈ V L H . Thereby T x|y = 0, for all y ∈ V L H . It follows that T x = 0. Hence x ∈ N T . Therefore, that S −1 (f ) = k and S −1 (g) = h. Thereby f = S(k) and g = S(h). Let α, β ∈ H,then Thereby for all f, g ∈ V L H and α, β ∈ H, Thereby for every f ∈ V L H , Similarly we have Thereby for every f ∈ V L H , From (3.33 ) and (3.34 ), for every f ∈ V L H , From the part (1), Therefore Theorem (3.22 ) is one of the most important results about frames, and We have already seen that, for f ∈ V L H , the frame coefficients { f |S −1 f k } m k=1 have minimal ℓ 2 norm among all sequences {c k } m k=1 for which f = m k=1 c k f k . In the next theorem, let us see that the existence of coefficients minimizing the ℓ 1 norm. Then {f k } 3 k=1 is a frame for V L H . Now the frame operator S is given by f |f k f k .
We obtain that Similarly we can get, Therefore the cannonical dual frame is