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Duffin and Schaeffer invented frames while working on non-harmonic Fourier series . Later, despite the fact that many others investigated the frame theory, the mechanism brought by Daubechies et al. gave a strong place to frames in harmonic analysis [2,3]. Coherent states of quantum optics and wavelets are subclasses of continuous frames . In the modern era, frames became standard tool in many areas of engineering and physical problems partly because of their success in digital signal processing, particularly one can name the time-frequency analysis [5,6]. 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 [5,7,8,9,10]. There is no fixed class of frames which is appropriate for all physical problems. Each problem is different and a solution to it demands a specific frame. The current nature of technology advancement constantly faces new problems, therefore, the research to find tools to solve them will continue.
Any vector in a separable Hilbert space can be written in terms of an orthonormal basis of the Hilbert space and the so-written expression is unique. Besides, finding an orthonormal basis for a separable Hilbert space is hard, the uniqueness of the expression of vectors, in terms of an orthonormal basis, is undesirable in applications. In fact, this uniqueness limited flexibility in applications. In order to overcome this strain practitioners looked for a substitute. At this juncture, frames appeared as a replacement to orthonormal bases. In a finite dimensional Hilbert space, usually, a frame contains more vectors than any orthonormal basis and this surplus allowed vectors to have enormously many expressions. This flexibility of frames is the key to their success in applications. How flexible a frame should be varies according to the nature of each problem. Flexibility of frames provides design freedom and which permits one to build frames adequate to a particular problem in a way that is not possible by an orthonormal basis [2,4,5,11].
We can only define Hilbert spaces over the fields and H, which are the set of all real numbers, the set of all complex numbers, and the set of all quaternions respectively . The fields and are commutative and associative. Functional analytic properties of real and complex Hilbert spaces well-studied. However, the field of quaternions is non-commutative but associative. Due to this non-commutativity of quaternions, a systematic study of functional analytic properties of quaternionic Hilbert spaces is not concluded yet. Further, as a result of this non-commutativity we have two types of Hilbert spaces, namely left quaternionic Hilbert space and right quaternionic Hilbert space. The functional analytic properties of the underlying Hilbert space are crucial to the study of frames. In this regard, we shall examine some functional analytic properties of finite dimensional quaternion Hilbert spaces as required for the development of the frame theory.
As far as we know a general theory of discrete frames on quaternionic Hilbert spaces is not available in the literature. In this respect we shall construct discrete frames on finite dimensional left quaternionic Hilbert spaces following the lines of . In the construction of frames on finite dimensional quaternion Hilbert spaces the effect of non-commutativity of quaternions is adaptable, and therefore most of the results can be manipulated from their complex counterparts. Finally, as a scope of the construction, four dimensional quaternions may provide more feasibility in applications than its two dimensional complex counterpart.
2. Quaternion Algebra
In this section we shall define quaternions and some of their properties as needed here. For details one may consult [12,13,14].
Let H denote the field of quaternions. Its elements are of the form where and are real numbers, and are imaginary units such that , , and . The quaternionic conjugate of is defined to be .
2.2. Properties of Quaternions
The quaternion product allows the following properties. For we have
For each there exists r such that
If then whenever
The quaternion product is not commutative.
3. Frames in Quaternion Hilbert Space
 Let is a vector space under left multiplication by quaternionic scalars, where H stands for the quaternion algebra. For and the inner product
satisfies the following properties:
unless a real norm
Assume that the space together with is a separable Hilbert space. Properties of left quaternion Hilbert spaces as needed here can be listed as follows: For and we have
For an enhanced explanation of quaternions and quaternion Hilbert spaces one may consult [12,13] and the many references listed there.
3.1. Some Basic Facts in Left Quaternion Hilbert Space
(Basis) Let be a finite dimensional left quaternion Hilbert space with an inner product The linearity is assumed in the second entry of the inner product. For a sequence in is called a basis if the sequence satisfies the following two conditions
is a linearly independent set.
As a consequence of this definition, for every there exist unique scalar coefficients such that If is an orthonormal basis, that is then Thereby
We now introduce frames on finite dimensional left quaternion Hilbert spaces. We shall show that the complex treatment adapt to the quaternions as well. In this note 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 . The theory of frames offered here, more or less, follows the lines of .
(Frames) A countable family of elements in is a frame for if there exist constants such that
The real numbers A and B are called frame bounds. The numbers A and B are not unique. The supremum over all lower frame bounds is the optimal lower frame bound, and the infimum over all upper frame bounds is the optimal upper frame bound . In fact, optimal frame bounds are the frame bounds. A frame is said to be normalized if In this note we shall only consider finite frames With this restriction, Schwartz inequality shows that
From (3) it is clear that the upper frame condition is always satisfied with . In order for the lower condition in (2) to be satisfied, it is necessary that . Let us see this in the following.
Let be a sequence in and Then a mapping is continuous if and only if for any sequence in W which converges to as then converges to as
Similar to the complex case. ☐
Let be a continuous mapping and M is a compact subset of then assumes a maximum and a minimum at some points of
Similar to the complex case. ☐
Let be a sequence in Then is a frame for left span
From the Schwartz inequality the upper frame condition is satisfied with Thereby
Let and consider the mapping
Now we want to prove that is continuous. Let be a sequence in W such that as Now,
Thereby converges to as From the Lemma (1), is continuous. Since the closed unit ball in W is compact, from the Lemma (2), we can find with such that
It is clear that as not all are zero. Now given we have so Hence
Uniqueness: Assume that there is another family in such that
Hence there exists a unique family in such that
Since for fixed
Since is a basis for is linearly independent. Therefore in (29), Otherwise (), becomes linearly dependent. Hence,
We can give a perceptive clarification of why frames are important in signal transmission. Let us say we want to transmit a signal f that belonging to a left quaternion Hilbert space from a transmitter T to a receiver R. Suppose that both T and R have the knowledge of frame for . Let T transmits the frame coefficients Using the received numbers, the receiver R can reconstruct the signal f using the frame decomposition. If R receives a perturbed (noisy) signal, of the correct frame coefficients, using the received coefficients, R will reconstruct the transmitted signal as
this differs from the correct signal f by the noise Minimizing this noise for various signals with different types of noises has been a hot topic in signal processing. Since the frame is an over complete set, it is possible that some part the noise contribution may sum to zero. At the same time, if is an orthonormal basis this scenario is never a possibility. In that case
so each noise contribution will make the reconstruction worse.
If , defines a norm in . In fact is a complete metric space with respect to this norm.
We have already seen that, for , the frame coefficients have minimal norm among all sequences for which . In the next theorem, let us see that the existence of coefficients minimizing the norm.
Let be a frame for a finite-dimensional left quaternion Hilbert space Given there exist coefficients such that and
Fix . It is clear that we can choose a set of coefficients such that Let Since we want to minimize the norm of the coefficients, we can now restrict our search for a minimizer to sequences belonging to the compact set
Define a function
We can prove is continuous by similar proof of Proposition (2). From (32) and Lemma (2),
Hence for given there exist coefficients such that and
Let , then in view of Proposition (2), the set of vectors is a frame for W. If then using the frame decomposition of the frame one can obtain useful expression for the orthogonal projection onto the subspace W.
Let be a frame for a subspace W of the left quaternion Hilbert space . Then the orthogonal projection of onto W is given by
Define an operator P from to W by
First let us prove that P is onto. For, let and be a frame operator in Since be a frame for the subspace we have
But thereby Since is arbitrary, for given , there exists such that Thereby P is onto. Now we want to prove that P is an orthogonal projection. Hence our claims are
The mapping is given by
Since be a frame for a subspace W of the left quaternion Hilbert space from (7) the frame operator S is given by
From (42) and (44), P is an orthogonal projection. ☐
are called frame coefficients. The frame is called the canonical dual of
The research of K. Thirulogasanthar was partly supported by The Fonds de recherche du Quebec Nature et technologies (FRQNT) and part of this work was done while he was visiting University of Jaffna. He would like to thank their hospitality.
M. Khokulan: The essence of this article is part of the M. Phil thesis of Khokulan. Initial proofs were done by him. K. Thirulogasanthar: Supervised the thesis, provided ideas to the proofs, corrected all the proofs and wrote the article. S. Srisatkunarajah: Supervised the thesis, proof read the article and made corrections.
Conflicts of Interest
The authors declare no conflicts of interest.
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