Prolate Spheroidal Wave Functions Associated with the Quaternionic Fourier Transform
PProlate Spheroidal Wave Functions Associated with theQuaternionic Fourier Transform (cid:73)
Cuiming Zou a , Kit Ian Kou b, ∗ , Joao Morais c a Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macao,China. Email: [email protected] b Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macao,China. Email:[email protected] c J. Morais is with the Departamento de Matem´aticas, Instituto Tecnol´ogico Aut´onomo de M´exico R´ i oHondo Abstract
One of the fundamental problems in communications is finding the energy distributionof signals in time and frequency domains. It should, therefore, be of great interestto find the most energy concentration hypercomplex signal. The present paper findsa new kind of hypercomplex signals whose energy concentration is maximal in bothtime and frequency under quaternionic Fourier transform. The new signals are a gen-eralization of the prolate spheroidal wave functions (also known as Slepian functions)to quaternionic space, which are called quaternionic prolate spheroidal wave functions.The purpose of this paper is to present the definition and properties of the quaternionicprolate spheroidal wave functions and to show that they can reach the extreme case inenergy concentration problem both from the theoretical and experimental description.In particular, these functions are shown as an e ff ective method for bandlimited signalsextrapolation problem. Keywords:
Quaternionic analysis, quaternionic Fourier transform, hypercomplexsignal, energy concentration problem, quaternionic prolate spheroidal wave functions,bandlimited extrapolation ∗ Corresponding author
Preprint submitted to Elsevier November 7, 2018 a r X i v : . [ m a t h . C A ] S e p . Introduction The energy distribution problem [1, 2] is one of the fundamental problems in com-munication engineering. It aims at finding the energy distribution of signals in timeand frequency domains. In particular, researchers take particular note of finding thesignals with maximum energy concentration in both the time and frequency domainssimultaneously. In the early 1960’s, D. Slepian, H. Landau and H. Pollak [3, 4, 5, 6]have found that the prolate spheroidal wave functions (PSWFs) are the most optimalenergy concentration functions in a Euclidean space of finite dimension. The PSWFs,also known as Slepian functions, are bandlimited and exhibit interesting orthogonalityrelations. They are normalized versions of the solutions to the Helmholtz wave equa- tion in prolate spheroidal coordinates. From then on, theory and numerical applicationsof these functions have been developed rapidly. They are often regarded as somewhatmysterious, with no explicit or standard representation in terms of elementary functionsand too di ffi cult to compute numerically.The one-dimensional PSWFs have so far mainly been developed in two di ff erentdirections. One is that fast and highly accurate methods have been deeply developedfor the approximation of the PSWFs and their eigenvalues [7, 8, 9, 10, 11]. From acomputational point of view, the PSWFs provide a natural and e ffi cient tool for com-puting with bandlimited functions defined on an interval [12, 13]. They are preferableto classical polynomial bases (such as Legendre and Chebychev polynomials). On the other hand, analytical properties of the PSWFs have been proposed of the moregeneral context of a variety of function spaces such as hypercomplex and Cli ff ordianspaces and under di ff erent integral transforms [14, 15, 16, 17]. The frequency domainhas indeed been considered not only under the Fourier transform, but also under moregeneral ones such as the fractional Fourier transform [18, 19], linear canonical trans-form (LCT) [20, 21] and so forth. Over the past few years the PSWFs have come toplay a very active part in some problems arising from applications to physical sciencesand engineering, such as wave scattering, signal processing, and antenna theory. Theseapplications have stimulated a surge of new ideas and methods, both theoretical andapplied, and have reawakened an interest in approximation theory, potential theory and ff erential equations.In the present paper, we discuss the energy concentration problem of bandlimitedquaternionic signals under the quaternionic Fourier transform, which is a generaliza-tion of the Fourier transform to quaternionic signals [22, 23, 24, 25]. Quaternions werefirst applied to the Fourier transform within the general context of solving problemsin the nuclear magnetic resonance imaging by R. R. Ernst in the 1980s [26]. In thequaternionic language the Hamiltonian multiplication rules [27] make the underlyingelements of having a non-commutative property leading to the main di ffi culty for theanalysis of some basic properties of those elements in certain spaces. To the best of ourknowledge, there are no previous works considering the energy concentration problem in a non-commutative structure as in the case of the quaternionic (or, more general, theCli ff ord) algebra. This understanding can be the basis for more generalizations. Forthis reason, the purpose of the present paper is to develop the energy extremal proper-ties between the time and frequency domains involving quaternionic signals. We findthat the quaternionic prolate spheroidal wave functions (QPSWFs) preserve the highenergy concentration in both the time and frequency domains under the quaternionicFourier transform. Although we concentrate mainly on Fourier bandlimited signals, ithowever might also be applied for LCT bandlimited cases as is shown in one of ourpreceding papers [14].The body of the present paper will cover the following sequence of topics: In Section 2, we collect some basic concepts of quaternionic analysis and quaternionicFourier transform to be used throughout the paper. Section 3 introduces the QPSWFs.The QPSWFs are ideally suited to study certain questions regarding the relationshipbetween quaternionic signals and their Fourier transforms. We prove that the QPSWFsare orthogonal and complete on both the square integrable space of finite interval andthe two-dimensional Paley-Wiener space of bandlimited quaternionic signals. Section4 describes the time-limited and bandlimited quaternionic signals and their correspond-ing properties using the QPSWFs. In Section 5, we present the energy extremal proper-ties of the QPSWFs in the time and frequency domains. In particular, if a finite energysignal is given, the possible proportions of its energy in a finite time-domain and a finite frequency-domain are found, as well as the signals which do the best job of simultane-3us time and frequency concentration. We find that the QPSWFs can reach the extremecase of the energy relationships. In Section 6, the QPSWFs are used in the bandlimitedextrapolation problem and achieved good results. Section 7 concludes the paper.
2. The quaternionic Fourier transform (QFT)
Quaternion algebra was discovered by W.R. Hamilton in 1843. It is a kind of hy-percomplex numbers related to the rotations in three-dimensional space, which likescomplex numbers used to represent rotations in the two-dimensional plane. Quater-nion algebra has important applications in a variety of field in physics, biomedical,geometry, image processing and so on. In the present section, to distinguish quater-nions from real numbers, we represent real number, real functions using normal lettersand quaternions and quaternionic functions using boldface letters, respectively. Now,we begin by reviewing some basic definitions and properties of quaternion algebra.The new numbers q : = q + i q + j q + k q ( q , q , q , q ∈ R ) , are called quaternions (or more informally, Hamilton numbers) by W.R. Hamilton.The set of all real quaternions is often denoted by H , in honour of its discoverer. Theimaginary units i , j , k obey the following laws of multiplication i = j = k = ijk = − , and the usual component–wise defined addition. In particular, the elements i , j , k are pairwise anticommute.The neutral element of addition, known as additive identity quaternion , is definedby : = + i + j + k
0. A quaternion can be also represented as q : = Sc ( q ) + q , where Sc ( q ) = q denotes the scalar part and q : = i q + j q + k q is the vector part of q . Likein the complex case, the conjugate of q is defined by q : = Sc ( q ) − q . The modulus of q is defined as | q | : = (cid:112) qq = (cid:112) qq = (cid:16) q + q + q + q (cid:17) (2.1)4nd it coincides with its corresponding Euclidean norm as a vector in R .We consider a kind of hypercomplex signals H -valued signals of the form f : R → H such that f ( x , y ) = f ( x , y ) + i f ( x , y ) + j f ( x , y ) + k f ( x , y ) , (2.2) f i ( x , y ) : R → R ( i = , , , . Properties (like integrability, continuity or di ff erentiability) that are ascribed to f have to be fulfilled by all components f i . Since k = ij , we can also rewrite Eq. (2.2) asfollowing f ( x , y ) = f ( x , y ) + i f ( x , y ) + f ( x , y ) j + i f ( x , y ) j , (2.3)which keeps all the imaginary unit i to the left and j to the right of each term [24]. Let L p ( R ; H ) ( p = ,
2) denote the linear spaces of all H -valued functions in R under leftmultiplication by quaternions such that each component is in the usual L p ( R ): L p ( R ; H ) : = (cid:40) f (cid:12)(cid:12)(cid:12)(cid:12) f : R → H , (cid:90) R | f ( x , y ) | p dxdy < ∞ (cid:41) . (2.4)We further introduce the left quaternionic inner product for two functions f , g ∈L ( R ; H ) as follows < f , g > : = (cid:90) R f ( x , y ) g ( x , y ) dxdy . (2.5)Here, we just consider the left inner product throughout the paper. Because the rightinner product will lead the similar results in the following. The reader should note thatthe norm induced by this inner product as follows (cid:107) f (cid:107) L ( R ; H ) : = (cid:16) < f , f > (cid:17) / = (cid:32)(cid:90) R | f ( x , y ) | dxdy (cid:33) / (2.6)coincides with the usual L -norm of f , considered as a vector-valued function. Inparticular, we note the symbol (cid:107) f (cid:107) L = (cid:107) f (cid:107) L ( R ; H ) for simplicity. The square norm (cid:107) f (cid:107) L : = E is also often called the total energy of the H -valued signal f and is normal-ized so that E = L ( R ; H ) furnished with the inner product Eq. (2.5) is an H -valuedHilbert space and the norm in Eq. (2.6) turns L ( R , H ) into a Banach space [28]. We5ill make use of a special notation to express the scalar part of Eq. (2.5): < f , g > : = Sc (cid:16) < f , g > (cid:17) . (2.7)The real-valued inner product Eq. (2.7) appeared for example in [29] in the contextof complex vector spaces, in [30] for spaces of H -valued functions and it was alsoconsidered in [28] for spaces of Cli ff ord-valued functions. Definition 2.1
Two elements f , g ∈ L ( R ; H ) are orthogonal in the L -sense if < f , g > = . To proceed with, we now define an angle between two H -valued functions. Definition 2.2
The angle between two non-zero functions f , g ∈ L ( R ; H ) is definedby arg( f , g ) : = arccos (cid:32) < f , g > (cid:107) f (cid:107) L (cid:107) g (cid:107) L (cid:33) . (2.8)The superimposed argument is well-defined since, obviously, it holds | < f , g > | ≤ | < f , g > | ≤ (cid:107) f (cid:107) L (cid:107) g (cid:107) L . The extremal values of this angle will be discussed in Section 4 in detail.
The QFT we have used throughout the paper is as follows [22, 24, 25]
Definition 2.3
Let f ∈ L ( R ; H ) . The two-sided QFT of f is defined by F ( f )( u , v ) : = π (cid:90) R e − i ux f ( x , y ) e − j vy dxdy , (2.9) where ( x , y ) , ( u , v ) are points in R . Here, ( x , y ) will denote the space and ( u , v ) the angular frequency variables.Under suitable conditions, the original quaternionic function f can be reconstructedfrom F ( f ) by the inverse transform. 6 efinition 2.4 The inverse (two-sided) QFT of f ∈ L (cid:84) L ( R ; H ) , if applicable, isdefined by F − ( f )( x , y ) : = π (cid:90) R e i ux f ( u , v ) e j vy dudv . (2.10)Since f = f + i f + f j + i f j with f i : R → R , ( i = , , , F ( f ) have asymmetric representation as follows F ( f ) = F ( f ) + i F ( f ) + F ( f ) j + i F ( f ) j , (2.11)where F ( f i ) , ( i = , , ,
3) are H -valued functions. However, we can’t use the modulusin Eq. (2.1) because the F ( f i ) , ( i = , , ,
3) are not real-valued functions. For thisreason, a Q modulus of F ( f ) is introduced as follows [31] |F ( f ) | Q : = |F ( f ) | + |F ( f ) | + |F ( f ) | + |F ( f ) | . (2.12)Let f ( x , y ) and F ( f )( u , v ) ∈ L ( R ; H ). The Parseval’s theorem for the QFT isgiven as follows [31] (cid:107) f (cid:107) L = (cid:107)F ( f ) (cid:107) Q , (2.13)where (cid:107)F ( f ) (cid:107) Q : = (cid:82) R |F ( f )( u , v ) | Q dudv < ∞ . From the Parseval’s identity Eq. (2.13), we can find that the total energy of a H -valued signal in the time-domain is the same as that in the frequency-domain under the Q modulus.
3. The Quaternionic Prolate Spheroidal Wave Functions
The present section introduces the quaternionic prolate spheroidal wave functions(QPSWFs) and discusses some of their general properties.First we introduce the notations we need in this part. Let T : = [ − T , T ] × [ − T , T ] ⊂ R be the time-domain , and W : = [ − W , W ] × [ − W , W ] ⊂ R the frequency-domain sothat W is a scaled version of T . We write W = c T , where x ∈ c T if and only if x / c ∈ T with c a positive constant. For simplicity of presentation, we represent the integralnotations (cid:82) T − T (cid:82) T − T and (cid:82) W − W (cid:82) W − W in the abbreviated notation (cid:82) T and (cid:82) W , respectively. Let7 W : = { f ∈ L ( R ; H ) | F ( f )( u , v ) = , ( u , v ) ∈ R \ W } be the Paley-Wiener spaceof H -valued functions that are bandlimited to W . The space B W will be discussed indetail in Section 4. We are now ready to introduce the quaternionic prolate spheroidal wave functionsin the finite quaternionic Fourier transform setting.
Definition 3.1 (Finite-QFT form)
Given a real c > , the quaternionic prolate spheroidalwave functions (QPSWFs) ψ n : R → H ( n = , , . . . ) , are the solutions of the integralequation µ n i n ψ n ( x , y ) j n : = (cid:90) T e i csx ψ n ( s , t ) e j cty dsdt , (3.14) where µ n are the complex parameters corresponding to the eigenfunction ψ n ( x , y ) .Moreover, the functions { ψ n ( x , y ) } ∞ n = are complete in the class of W -bandlimited func-tions B W . In the notations used above we have concealed the fact that both the ψ n ( x , y )’s andthe µ n ’s depend on the parameter c . When it is necessary to make this dependenceexplicit, we write µ n = µ n ( c ) and ψ n ( x , y ) = ψ n ( x , y ; c ), n = , , · · · , where c >
0. Naturally, considerable simplification occurs when ψ n ( x , y ) is even or odd with n . Due to the symmetry of the domain T , one can easily shows that ψ n ( − x , y ), ψ n ( x , − y )and ψ n ( − x , − y ) are also solutions to the integral Eq. (3.14). Consequently, they aresolutions of Eq. (3.14) as well ψ ee ( x , y ) : = ψ n ( x , y ) + ψ n ( − x , − y ) , ψ e ( x , y ) : = ψ n ( x , y ) + ψ n ( − x , y ) , ψ e ( x , y ) : = ψ n ( x , y ) + ψ n ( x , − y ) , ψ o ( x , y ) : = ψ n ( x , y ) − ψ n ( − x , y ) , ψ o ( x , y ) : = ψ n ( x , y ) − ψ n ( x , − y ) , ψ oo ( x , y ) : = ψ n ( x , y ) − ψ n ( − x , − y ) . { ψ n ( x , y ) } ∞ n = possess a number of special properties that make them most usefulfor the study of bandlimited functions. They are also the eigenfunctions of the integralEq. (3.16) (see Theorem 3.1 below).Before introduce the Theorem 3.1, a lemma that connects sinc-functions and anintegral is listed here without proof, which can be easily checked. Lemma 3.1
Let ( x , y ) , ( u , v ) be points in R . There holds π ) (cid:90) W e i u ( x − s ) e j v ( y − t ) dudv = sin W ( x − s ) π ( x − s ) sin W ( y − t ) π ( y − t ) . (3.15) Theorem 3.1 (Low-pass filtering form)
The solutions of Eq. (3.14) are also the so- lutions of the integral equation λ n ψ n ( x , y ) : = (cid:90) T ψ n ( s , t ) sin W ( x − s ) π ( x − s ) sin W ( y − t ) π ( y − t ) dsdt , (3.16) where λ n are eigenvalues corresponding to the eigenfunctions ψ n ( x , y ) . Proof.
Using Eq. (3.15) and having in mind that sin W ( x i − y i ) π ( x i − y i ) ( i = ,
2) is real value,straightforward computations on the right side show that (cid:90) T sin W ( x − s ) π ( x − s ) ψ n ( s , t ) sin W ( y − t ) π ( y − t ) dsdt = π ) (cid:90) T (cid:90) W e i u ( x − s ) ψ n ( s , t ) e j v ( y − t ) dudvdsdt = π ) (cid:90) W e i ux (cid:32)(cid:90) T e − i us ψ n ( s , t ) e − j vt dsdt (cid:33) e j vy dudv . Then we use the definition of QPSWFs to the integral above, µ n (2 π ) (cid:90) W e i ux (cid:18) ( − i ) n ψ n ( uc , vc )( − j ) n (cid:19) e j vy dudv = c µ n (2 π ) ( − i ) n (cid:32)(cid:90) T e i cu x ψ n ( u , u ) e j cu y du du (cid:33) ( − j ) n = c µ n (2 π ) ( − i ) n i n ψ n ( x , y ) j n ( − j ) n = λ n ψ n ( x , y ) . Derived from the above process, we find that the relationships between µ n and λ n are λ n : = c µ n (2 π ) , n = , , . . . . (cid:3) Theorem 3.1 gives a straight forward derivation of QPSWFs from finite-QFT form to alow-pass filtering system. The low-pass filtering form of QPSWFs is important to study9he properties of them. Since the noncommunicative of H -valued signals with the QFT kernel, the low-pass filtering form, which connects the H -valued signals with two real-valued kernel, provides an easy way to study the QPSWFs. Most of the properties ofQPSWFs are deduced from the low-pass filtering form. To state the properties of QPSWFs, we shall need some new notations and basicfacts about integral equations at first. Let the real-valued kernel function k ( x , y , s , t ) : = sin W ( x − s ) π ( x − s ) sin W ( y − t ) π ( y − t ) . (3.17) Definition 3.2 (Quaternion Hermitian operator)
For any f ∈ L ( R , H ) , the quater-nion Hermitian operator is defined by ( K f )( x , y ) : = (cid:90) T k ( x , y , s , t ) f ( s , t ) dsdt . (3.18) K is a Hermitian operator, because for any f , g ∈ L ( R , H ) (cid:90) R f ( x , y )( K g )( x , y ) dxdy = (cid:90) R ( K f )( x , y ) g ( x , y ) dxdy . Here, since this operator acting on the H -valued functions, we call it the quaternion Hermitian operator. In particular, when the function is complex-valued , the role ofthis Hermitian operator is the original Hermitian operator.For any ψ ∈ L ( R , H ) the Hermitian operator Eq. (3.18) is indeed a linear combi-nation of Hermitian operators of four real-valued signals K ψ = K ψ + i K ψ + K ψ j + i K ψ j = λ ψ ,ψ i : R → R ( i = , , , . (3.19)We can now use the properties of Hermitian kernels for real-valued signals to studythe properties of the Hermitian kernel function Eq. (3.17) and of its eigenvalues, thatis, values λ of for which the homogeneous Fredholm integral equation of the form λϕ = K ϕ ( ϕ ∈ R ) has a non-trivial solution.From the general theory of integral equations, we conclude the following propertiesfor the kernel function [32, 33]: k ( x , y , s , t ) is a non-vanishing, continuous, Hermitian kernel, then k ( x , y , s , t )has at least one eigenvalue;(ii) If k ( x , y , s , t ) is a complex-valued, non-vanishing, continuous, Hermitian kernel,then the eigenvalues associated to k ( x , y , s , t ) are real;(iii) Let k ( x , y , s , t ) be a non-zero, continuous, Hermitian kernel. If λ m and λ n are anytwo di ff erent eigenvalues, then their corresponding eigenfunctions ϕ m and ϕ n areorthogonal;(iv) Let k ( x , y , s , t ) be a non-zero, continuous, Hermitian kernel. The set of all of itseigenfunctions { ϕ n } ∞ n = forms a mutually orthogonal system in T : (cid:90) T ϕ n ( x , y ) ϕ m ( x , y ) dxdy = λ n δ mn . (3.20)Here, λϕ = K ϕ , for any ϕ n , ϕ m ∈ R ; (v) Let k ( x , y , s , t ) be a non-zero, continuous kernel defined in T . Let Λ k denote theset of eigenvalues of the kernel k ( x , y , s , t ). Then Λ k is at most countable, and itcannot have a finite limit point. Proposition 3.1
Given a real c > , we can find a countably infinite set of H -valuedsignals { ψ n } ∞ n = and a set of numbers { λ n } ∞ n = with the following properties: The eigenvalues λ n ’s are real and monotonically decreasing in (0 , , λ ≥ λ ≥ λ ≥ · · · , (3.21) and such that lim n →∞ λ n = ; The { ψ n } ∞ n = are orthogonal in T , (cid:90) T ψ n ( x , y ) ψ m ( x , y ) dxdy = λ n δ mn ; (3.22)3. The { ψ n } ∞ n = are orthonormal in R , (cid:90) R ψ n ( x , y ) ψ m ( x , y ) dxdy = δ mn . (3.23) Proof.
Property 1. follows from the general theory of integral equations for real-valued signals [32, 33] and is stated without proof.11e now prove Statement 2. From Eq. (3.20) in Property (iv), we know that thereal-valued eigenfunctions of the integral equation λψ i = K ψ i i = , , , T . For the integral equation K ψ = λ ψ , ( ψ ∈ L ( R , H )), weconclude that (cid:90) T ψ n ( x , y ) ψ m ( x , y ) dxdy = (cid:90) T (cid:0) ψ n , + i ψ n , + ψ n , j + i ψ n , j (cid:1) (cid:0) ψ m , − i ψ m , − ψ m , j + i ψ m , j (cid:1) dxdy = λ n δ mn = λ n δ mn . Here, we use the fact that for di ff erent eigenvalues λ n , λ m , the eigenfunction are or-thogonal, i.e., (cid:82) T ψ n , i ( x , y ) ψ m , j ( x , y ) dxdy = λ n , m = n , i = j , i , j = , , , , , m (cid:44) n , i , j = , , , . For the Statement 3, the orthogonality of the QPSWFs on the region R can bededuced as follows (cid:90) R ψ n ( x , y ) ψ m ( x , y ) dxdy = λ n λ m (cid:90) R (cid:90) T (cid:90) T ψ n ( s , t ) sin W ( x − s ) π ( x − s ) sin W ( y − t ) π ( y − t )sin W ( x − z ) π ( x − z ) sin W ( y − w ) π ( y − w ) ψ m ( z , w ) dsdtdzdwdxdy = λ n λ m (cid:90) T (cid:90) T ψ n ( s , t ) sin W ( s − z ) π ( s − z ) sin W ( t − w ) π ( t − w ) ψ m ( z , w ) dsdtdzdw = λ m (cid:90) T ψ n ( z , w ) v m ( z , w ) dzdw = λ m λ m δ mn = δ mn . Here, we have substituted ψ n ( x , y ) and ψ m ( x , y ) by Eq. (3.16) for the first and the third equality. We use the equation π (cid:82) R e i ( z − s ) x dx = δ ( z − s ) for the second equality. Forthe last equality, we utilize the orthogonality of PSWFs on the finite region. Thus theorthogonality of the ψ n over T implies orthogonality over R and vice-versa. (cid:3) Remark 3.1
Since (cid:107) ψ n (cid:107) L = and (cid:107) ψ n (cid:107) L ( T , H ) = λ n , a small value of λ n implies that ψ n has most of its energy outside the interval T while a value of λ n near implies that ψ n is concentrated largely in T . roposition 3.2 (All-pass filtering form) The QPSWFs satisfy the following integralequation ψ n ( x , y ) = (cid:90) R ψ n ( s , t ) sin W ( x − s ) π ( x − s ) sin W ( y − t ) π ( y − t ) dsdt , (3.24) which is called the all-pass filtering form of QPSWFs. Proof.
This proposition can easy be obtain by the special case of the quaternion con- volution theorem in [34]. Taking the QFT to the both sides of the Eq. (3.16), we have λ n F ( ψ n )( u , v ) = F ( ψ n χ T )( u , v ) χ W ( u , v ) , where χ T ( x , y ) : = , ( x , y ) ∈ T , , otherwise . and χ W ( u , v ) : = , ( u , v ) ∈ W , , otherwise . From thisequation, we know that F ( ψ n )( u , v ) = F ( ψ n )( u , v ) χ W ( u , v ) . Then taking the inverse QFT on both sides, we have ψ n ( x , y ) = (cid:90) R ψ n ( s , t ) sin W ( x − s ) π ( x − s ) sin W ( y − t ) π ( y − t ) dsdt . (3.25) (cid:3) This all-pass filtering form of QPSWFs extent the space domain from T to R .
4. The Time-limited and Bandlimited spaces
In the present section, we consider two kinds of H -valued functions and their cor-responding spaces. Definition 4.1
We say that a H -valued function f ( x , y ) with finite energy is time-limitedif it vanishes for all ( x , y ) ∈ R \ T = : T c . Definition 4.2
We say that a H -valued function f ( x , y ) with finite energy is bandlimitedif F ( f )( u , v ) ≡ for all ( u , v ) ∈ R \ W = : W c . For any function f ∈ L ( R , H ), we can define two operators such that f becomeseither a time-limited and bandlimited function. We define the time-limited operator D T as follows D T f ( x , y ) : = f ( x , y ) χ T ( x , y ) , (4.26)13here χ T ( x , y ) : = , ( x , y ) ∈ T , , otherwise . The operator D T is linear D T ( α f + β g )( x , y ) = α D T f ( x , y ) + β D T g ( x , y ) , for any α, β ∈ R . Hence there holds < D T f , D T c f > = , for all f ∈ L ( R , H ) . The bandlimited operator B W is B W f ( x , y ) : = π (cid:82) W e i ux F ( f )( u , v ) e j vy dudv . Notedthat B W can be decomposed as follows B W f ( x , y ) = F − (cid:16) D W F ( f )( u , v ) (cid:17) = F − (cid:16) D W ( F ( f ) + i F ( f ) + F ( f ) j + i F ( f ) j ) (cid:17) = B W f + i B W f + B W f j + i B W f j . Here, f i : R → R , ( i = , , , B W is also linear. Similarly, there holds < B W f , B W c f > = , for all f ∈ L ( R , H ) . (4.27) Remark 4.1
For real-valued functions f i ( x , y ) , i = , , , , the inverse QFT F − ( D W F ( f i )( u , v )) = B W f i . In most cases, F − ( D W F ( f i )) , i = , , , are H -valued functions, but here weconclude that it is real-valued function. SinceB W f i ( x , y ) = π (cid:90) W e i ux F ( f i )( u , v ) e j vy dudv = π ) (cid:90) W (cid:90) R e i ux e − i us f i ( s , t ) e − j vt e j vy dsdtdudv = (cid:90) R sin W ( x − s ) π ( x − s ) f i ( s , t ) sin W ( y − t ) π ( y − t ) dsdt , form the third equality, we can find that B W f i ( x , y ) is just a filter transform for thesereal-valued functions f i ( x , y ) , i = , , , . That means B W f i ( x , y ) are also real-valuedfunctions. Now, we discuss the spaces induced by these two kinds of operators. The spaceinduced by D T is D T : = { f ∈ L ( R , H ) | f ( x , y ) ≡ , ( x , y ) ∈ T c } . (4.28)It is easy to see that D T is a linear subspace of L ( R , H ).14 emma 4.1 D T is complete. Proof.
We must show that if a set of H -valued functions { f n ( x , y ) } ∞ n = is a Cauchysequence in D T , then it converges in D T . Due to the completeness of L ( R , H ), thereis a function f ∗ ∈ L ( R , H ) such that (cid:107) f n − f ∗ (cid:107) L → as such n → ∞ . We nowdefine the new function D T f ∗ ( x , y ) = f ∗ ( x , y ) χ T ( x , y ) , (4.29)where χ T is the characteristic function on T . It can be verified that (cid:107) f n − D T f ∗ (cid:107) L → as n approaches infinity and, moreover, D T f ∗ ∈ D T . Thus, D T is complete. (cid:3) Similarly, the space induced by B W form a linear subspace B W of L ( R , H ), whichis also complete.Now we consider the extremal values of the angle between time-limited and ban-dlimited functions. The question then arises as to what are the extremal values of arg( f , g ) between g ∈ D T and f ∈ B W under the QFT? The following lemma andtheorem find this extremal value. Since the proof is analogous to the one-dimensionalcase in [4]. We restate the similar results for H -valued functions without proof. Lemma 4.2 If f ∈ B W is fixed, then arg( f , g ) between f and any g ∈ D T satisfies inf g ∈D T arg( f , g ) > . This infimum equals arccos (cid:107) D T f (cid:107) L (cid:107) f (cid:107) L and is assumed by g = kD T f for any positive constant k. We proceed now to find the least arg( f , g ) of arbitrary f ∈ B W and g ∈ D T . Weshow that the spaces B W and D T form indeed a least angle. Theorem 4.1
Let f ∈ B W and g ∈ D T . There exists a least angle between B W and D T that satisfies inf f ∈B W , g ∈D T arg( f , g ) = arccos (cid:112) λ (4.30) if and only if f = ψ and g = D T ψ , where λ is the largest eigenvalue of (3.16), and ψ is the corresponding eigenfunction. The proof is analogous to the one-dimensional case [4], which is omitted here.15e have thus found that the two subspaces B W and D T of L ( R , H ), which haveno functions except in common, actually have a minimum angle between them, sothat, in fact, a time-limited function and a bandlimited function cannot even be veryclose together.
5. The Energy Extremal Theorem
In the present section, we develop the relationship between the energy concentra-tion of a H -valued signal in the time and frequency spaces. In particular, if a finite energy H -valued signal is given, the possible proportions of its energy in a finite time-domain and a finite frequency-domain are found, as well as the signals which are themost optimal energy concentration signals.Throughout this section let g ( x , y ) be a H -valued function, T and W be two speci-fied square regions such that W = c T with c a positive constant. We form the energyratios in the time-domain and frequency-domain, respectively, as follows ξ : = (cid:107) D T g (cid:107) L (cid:107) g (cid:107) L and η Q : = (cid:107) D W F ( g ) (cid:107) Q (cid:107)F ( g ) (cid:107) Q . (5.31)Assume for simplicity that E : = (cid:107) g (cid:107) L = g ( x , y ) ranges overall functions in L ( R , H ), the pair ( ξ, η Q ) takes various values of the unit square inthe ( x , y )-plane. In the following, we will discuss the set of possible pairs ( ξ, η Q ). Inparticular, we will determine the functions that reach the extremal values of ( ξ, η Q ). Lemma 5.1
A non-zero H -valued function g ( x , y ) ∈ L ( R , H ) cannot be both ban-dlimited and time-limited. Proof.
From the uncertainty principle of [35, 31], it follows that | T || W | ≥ (1 − ε T − ε W ) , where g ( x , y ) has an essential support on T and W . In our case ε T = ε W = | T || W | ≥ g ( x , y ) is bandlimited, then g ( x , y ) cannot be identically zero in the region T . (cid:3) Remark 5.1
If the H -valued function is bandlimited, i.e. η Q = , then ξ (cid:44) by Lemma5.1. Analogously, if the function is time-limited, i.e. ξ = , then η Q (cid:44) .
16e now discuss the possible values of ( ξ, η Q ) if the underlying signal is bandlim-ited. Theorem 5.1
For any non-zero bandlimited function g ( x , y ) ∈ B W , there holds ξ ≤√ λ . Proof.
If a H -valued function is bandlimited, then η Q =
1. Since { ψ n ( x , y ) } ∞ n = is acomplete basis in B W we can expanse g ( x , y ) as follows g ( x , y ) = ∞ (cid:88) n = a n ψ n ( x , y ) , (5.32)where a n ∈ H . Clearly, ξ = (cid:82) T | g ( x , y ) | dxdy = (cid:80) ∞ n = λ n | a n | = ≤ λ (cid:80) ∞ n = | a n | , because λ ≥ λ n for all n . Hence, the extremal condition is g ∗ ( x , y ) = ψ ( x , y ). Itfollows that ξ = | g ∗ | = λ . For any other linear combination of ψ n , ξ is less than √ λ . (cid:3) As we know, ξ take values in the interval [0 , η Q when ξ is fixed. We start by considering the case ξ = Theorem 5.2
If the time energy ratio ξ equals to on T , the frequency energy ratio η Q is larger or equals to and less than on W for a H -valued signal. Proof.
See Appendix A. (cid:3)
From the property of symmetry of the QFT we conclude that all the properties forbandlimited functions proved so far have corresponding time-limited counterparts. Forexample, if ξ =
0, then we can get 0 ≤ η Q <
1. If η Q =
0, then it follows that 0 ≤ ξ < ξ =
1. For anynon-zero time-limited function g ( x , y ) ∈ D T , i.e. for which ξ =
1, we conclude that η Q ≤ √ λ . If η Q = √ λ , then g ∗ ( x , y ) = D T ψ ( x , y ) √ λ .We now prove that for arbitrary H -valued signals for which 0 < ξ < √ λ , η Q is not limited. Lemma 5.2
For < ξ < √ λ , η Q can take any value in the interval [0 , for non-zero H -valued signals. roof. For 0 < ξ < √ λ , the sequence { λ n } ∞ n = is monotone decreasing in the interval(0 , λ n → n approaches infinity. Hence there exists an eigenvalue suchthat λ n < ξ . We consider the signal g ∗ ( x , y ) = (cid:112) ξ − λ n ψ ( x , y ) + (cid:112) λ − ξ ψ n ( x , y ) √ λ − λ n , (5.33)where ψ n ( x , y ) is the eigenfunction corresponding to the eigenvalue λ n . We can findthat g ∗ ( x , y ) ∈ B W since ψ ( x , y ) , ψ n ( x , y ) ∈ B W . Straightforward computations showthat (cid:107) g ∗ (cid:107) L = λ − λ n (cid:104) ( ξ − λ n ) + ( λ − ξ ) (cid:105) = , (cid:107) D T g ∗ (cid:107) L = λ − λ n (cid:104) ( ξ − λ n ) λ + ( λ − ξ ) λ n (cid:105) = ξ , which implies that g ∗ ( x , y ) ∈ A . We conclude that η Q = (cid:107)F ( B W g ∗ ) (cid:107) Q = (cid:88) i = (cid:90) W |F ( g ∗ i ) | dudv = (cid:88) i = (cid:90) R | B W g ∗ i | dxdy = λ − λ n ) (cid:104) ( ξ − λ n ) + ( λ − ξ ) (cid:105) = . Here, the last equality used the following result that for i = , , , (cid:90) R | B W g ∗ i | dxdy = λ − λ n ) (cid:90) R (cid:16) (cid:113) ξ − λ n ψ , ( x , y ) + (cid:113) λ − ξ ψ n , ( x , y ) (cid:17)(cid:16) (cid:113) ξ − λ n ψ , ( x , y ) + (cid:113) λ − ξ ψ n , ( x , y ) (cid:17) dxdy = λ − λ n ) (cid:16) (cid:90) R ( ξ − λ n ) ψ , ( x , y ) ψ n , ( x , y ) dxdy − (cid:90) R (cid:113) ( ξ − λ n )( λ − ξ ) ψ , ( x , y ) ψ n , ( x , y ) dxdy − (cid:90) R (cid:113) ( ξ − λ n )( λ − ξ ) ψ n , ( x , y ) ψ , ( x , y ) dxdy + (cid:90) R ( λ − ξ ) ψ n , ( x , y ) ψ n , ( x , y ) dxdy ) (cid:17) = ( ξ − λ n ) + ( λ − ξ )4( λ − λ n ) . Thus, if 0 < ξ < √ λ , then there exists a signal such that η Q =
1. The verification of0 ≤ η Q < (cid:3)
18e conclude this section by studying the range of possible values of η for which √ λ ≤ ξ < Theorem 5.3
The maximum of η Q is assumed by arccos ξ + arccos η Q = arccos (cid:112) λ , (5.34) as such √ λ ≤ ξ < , where λ is the largest eigenvalue of Eq. (3.16). Proof.
See Appendix B. (cid:3)
Here, the result is the same as the real-valued case. But the functions we considered are H -valued. That means, we generalized the classical results to covered a bigger space. Remark 5.2
From Theorem 5.3 it follows that the set of all possible pairs ( ξ, η Q ) is theregion enclosed by Figure 1 bounded by the curve arccos ξ + arccos η Q ≥ arccos √ λ .Although this curve depends on the parameter c, for simplicity of presentation we takec = . Figure 1: The relationship between ξ and η Q for c = emark 5.3 By means of the classical one-dimensional PSWFs we now construct aspecial case of a QPSWFS as follows ψ ( x , y ) = ϕ ( x ) ϕ ( y )(1 + i + j + k ) , (5.35) where ϕ is the first real-valued PSWFs of zero order in one-dimension. Table 1: The values of ( ξ, η Q ) and λ for the first six QPSWFs. Functions ψ ψ ψ ψ ψ ψ λ ξ η Q We will compare six pairs of ( ξ, η Q ) of most energy concentrate function under thecondition c = . For example, we compare the signals ψ ( x , y ) and ψ ( x , y ) . Thesignal ψ ( x , y ) is the most energy concentrate function in the frequency domain, while ψ ( x , y ) is not concentrate in the centre of the frequency domain. In Table 1 we list thefirst six pairs of ( ξ, η Q ) and λ for the first QPSWFs. The energy of ψ ( x , y ) , ψ ( x , y ) and ψ ( x , y ) in the frequency domain is , while the others are less than . This ismainly because these signals ψ i ( i = , , , . . . ) have a single peak in the frequencydomain, and the corresponding energy is mainly concentrate in this peak. For ψ i ,i = , , , . . . , there are four peaks. In this case the energy is dispersed in a largerbandwidth. As shown in Figure 2, the function ψ ( x , y ) has four peaks in the frequencydomain.The results in Table 1 also support the obtained energy concentration proportionsof a signal as such arccos ξ + arccos η ≥ arccos √ λ . In fact, when arccos η = arccos 1 = , it follows that the corresponding smallest value is arccos ξ = arccos 0 . = . > arccos √ λ = . . igure 2: The modulus of QPSWFs ψ ( x , y ) and ψ ( x , y ) in time and frequency domains.
6. QPSWFs in Bandlimited Extrapolation Problem
In the present paper, the QPSWFs have found to be the most energy concentrationsignals and for any W -bandlimited H -valued signal f can be expanded into a series f ( x , y ) = ∞ (cid:88) j = a j ψ n ( x , y ) . (6.36)However, the W -bandlimited f is finite segment in space domain empirically (assumethe domain known is D : = [ − d , d ] × [ − d , d ]), i.e., g ( x , y ) = f ( x , y ) χ D ( x , y ) , (6.37)where χ D ( x , y ) : = , ( x , y ) ∈ D , , otherwise . How to get the W -bandlimited f in the outside ofthe segment areas is known as an extrapolation problem. There is an iteration methodto get the f in Figure 3. igure 3: The flow of the iteration. In this iteration method, the unknown information for f outsider the D is filled stepby step. The n -th step is F ( f n )( u , v ) = F ( g n − )( u , v ) χ W ( u , v ) ↔ f n ( x , y ) = g n − ( x , y ) ∗ (cid:32) sin W x π x sin Wy π y (cid:33) , where g n − ( x , y ) = g ( x , y ) + f n − ( x , y )(1 − χ D ( x , y )) = g ( x , y ) , ( x , y ) ∈ D , f n − ( x , y ) , otherwise . Wealso show the first step for the scalar part of a H -valued signal f ( x , y ) in Figure 4.Clearly, there is some new information joined in g ( x , y ) in the first step of the iterationprocess. Specifically, g ( x , y ) becomes g ( x , y ) in Figure 4.Now we will show that the iteration method is e ff ective, i.e., f n → f , n → ∞ . Inorder to get this result, we first show the following lemma. Lemma 6.1
The function of the n-th iteration is given by f n ( x , y ) = f ( x , y ) − ∞ (cid:88) j = a j (1 − λ j ) n ψ j ( x , y ) , (6.38) where λ j are the eigenvalues of QPSWFs ψ j ( x , y ) . Proof.
As we have known, for any W -bandlimited H -valued signal f can be expandedinto a series f ( x , y ) = (cid:80) ∞ j = a j ψ n ( x , y ) . Without loss of generality, we suppose f ( x , y ) = igure 4: The first step of the scalar part of an input g ( x , y ). m ( x , y ). We shall show that f n ( x , y ) = (1 − (1 − λ m ) n ) ψ m ( x , y ) = : C n ψ m ( x , y ) . For n = f ( x , y ) = (1 − (1 − λ m ) ) ψ m ( x , y ) is true. Suppose that is true for n = k , thenwe must show that is true for k +
1. Since g k ( x , y ) = g ( x , y ) + f k ( x , y )(1 − χ D ( x , y )) = ψ m ( x , y ) χ D ( x , y ) + C k ψ m ( x , y )(1 − χ D ( x , y )) , then f k + ( x , y ) = g k ( x , y ) ∗ (cid:32) sin W x π x sin Wy π y (cid:33) = ( C k ψ m ( x , y ) + (1 − C k ) ψ m ( x , y ) χ D ( x , y )) ∗ (cid:32) sin W x π x sin Wy π y (cid:33) . From the low-pass filtering form and all-pass filtering form of QPSWFs, we have f k + ( x , y ) = C k ψ m ( x , y ) + (1 − C k ) λ m ψ m ( x , y ) = C k + ψ m ( x , y ) . Here, we get an iteration equation about C k and C k + , i.e., C k + = C k + (1 − C k ) λ m . As we know C = λ m , then directly computation shows that C k + = − (1 − λ m ) k + .That means f n ( x , y ) = C n ψ m ( x , y ) = (1 − (1 − λ m ) n ) ψ m ( x , y ) for f ( x , y ) = ψ m ( x , y ).Applying this results to f ( x , y ) = (cid:80) ∞ j = a j ψ n ( x , y ), we conclude that f n ( x , y ) = ∞ (cid:88) j = (1 − (1 − λ j ) n ) a j ψ j ( x , y ) = f ( x , y ) − ∞ (cid:88) j = a j (1 − λ j ) n ψ j ( x , y ) . (cid:3) Since we have know the relationship between f n and f from Lemma 6.1, to show f n → f , with n → ∞ can also be rewritten as f − f n →
0, with n → ∞ . Here wedenote the error e n ( x , y ) between f n and f as e n ( x , y ) : = f ( x , y ) − f n ( x , y ) = ∞ (cid:88) j = a j (1 − λ j ) n ψ j ( x , y ) . (6.39)24s the ψ j ( x , y ) are orthogonal in R , then the energy E n of error e n is E n : = (cid:90) R e n ( x , y ) e ∗ n ( x , y ) dxdy = (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:88) j = a j (1 − λ j ) n ψ j ( x , y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dxdy = ∞ (cid:88) j = a j (1 − λ j ) n . Since E = (cid:80) ∞ j = a j < ∞ , then for any ε >
0, there exists N , such that (cid:80) j > N a j < ε . Onthe other hand, λ i are monotonically decreasing as j →
0. Then 1 − λ j ≤ − λ N , for k ≤ N . Then we have E n = N (cid:88) j = + ∞ (cid:88) j = N + a j (1 − λ j ) n ≤ (1 − λ N ) n N (cid:88) j = a j + ∞ (cid:88) j = N + a j ≤ (1 − λ N ) n E + ε. That means E n → n → ∞ , because 0 < − λ N < e n ( x , y ) can also be obtained as follows e n ( x , y ) = π (cid:90) W e i ux ( F ( f )( u , v ) − F ( f n )( u , v )) e j vy dudv , where F ( f ) and F ( f n ) are the QFT of f and f n , respectively. Then we have | e n ( x , y ) | = π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) W e i ux ( F ( f )( u , v ) − F ( f n )( u , v )) e j vy dudv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π (cid:90) W (cid:12)(cid:12)(cid:12) e i ux (cid:12)(cid:12)(cid:12) dudv (cid:90) W (cid:12)(cid:12)(cid:12) ( F ( f )( u , v ) − F ( f n )( u , v )) e j vy (cid:12)(cid:12)(cid:12) dudv = W π E n . That means 0 ≤ | e n ( x , y ) | ≤ (cid:113) WE n π →
0, with n → ∞ . Then we have shown that f − f n →
0, with n → ∞ .
7. Conclusion
In the present paper, we develop the definitions of QPSWFs. The various proper-ties of QPSWFs such as orthogonality, double orthogonality are established. Using thenew QPSWFs, we established the energy distribution for hypercomplex signal (specialquaternionic signal) in the QFT domain. In particular, if a finite energy quaternionic25ignal is given, the possible proportions of its energy in a finite time-domain and afinite frequency-domain are found, as well as the signals which do the best job of si-multaneous time and frequency concentration. It is shown that the QPSWFs will revealthe characteristics of the extreme energy distribution of the original quaternionic signaland its QFT in di ff erent regions. We also apply the QPSWFs to the bandlimited signalsextrapolation problem. Further investigations on this topic under di ff erent linear inte- gral transformation are now under investigation and will be reported in a forthcomingpaper. Acknowledgment
The first and second authors acknowledge financial support from the National Natu-ral Science Funds No. 11401606 and University of Macau No. MYRG2015-00058-L2-FST and the Macao Science and Technology Development Fund FDCT / / / A3.The third author acknowledges financial support from the Asociaci´on Mexicana deCultura, A. C.
Appendix A
This part proof the Theorem 5.2.
Proof.
As we know if ξ =
0, then η Q (cid:44) η Q cannot getthe value of 1, in the following we find the underlying signal for which η Q is infinitelyclose to 1.Let A : = (cid:110) g ∈ L ( R , H ) : (cid:107) g (cid:107) L = , (cid:107) D T g (cid:107) L = ξ (cid:111) be a given function class.Here, we just consider the condition of ξ =
0. Construct a signal g ∗ ( x , y ) definedas follows g ∗ ( x , y ) = ψ n ( x , y ) − D T ψ n ( x , y ) √ − λ n , (7.40)where λ n is the ( n + ψ n its corresponding eigenfunc-26ion. Direct computations show that (cid:107) g ∗ (cid:107) L = − λ n (cid:90) R (cid:16) ψ n ( x , y ) − D T ψ n ( x , y ) (cid:17)(cid:16) ψ n ( x , y ) − D T ψ n ( x , y ) (cid:17) dxdy = − λ n (cid:16) (cid:90) R ψ n ( x , y ) ψ n ( x , y ) dxdy − (cid:90) R ψ n ( x , y ) D T ψ n ( x , y ) dxdy − (cid:90) R D T ψ n ( x , y ) ψ n ( x , y ) dxdy + (cid:90) R D T ψ n ( x , y ) D T ψ n ( x , y ) dxdy (cid:17) = − λ n + λ n − λ n = , and ξ = (cid:107) D T g ∗ (cid:107) L =
0. Hence g ∗ ∈ A .Now we compute (cid:107) B W g ∗ (cid:107) L . Since ψ n ( x , y ) ∈ B W , it follows that B W g ∗ ( x , y ) = ψ n ( x , y ) − B W D T ψ n ( x , y ) √ − λ n . (7.41)Here, the QFT of B W g ∗ ( x , y ) is given as follows F ( B W g ∗ ) = D W (cid:16) F ( g ∗ ) + i F ( g ∗ ) + F ( g ∗ ) j + i F ( g ∗ ) j (cid:17) , (7.42)and we want to compute (cid:107)F ( B W g ∗ ) (cid:107) Q = (cid:90) W |F ( B W g ∗ ) | Q dudv = (cid:88) i = (cid:90) W (cid:16) |F ( B W g ∗ i ) | (cid:17) dudv , (7.43)where for i = , , , , B W g ∗ i ( x , y ) = ψ n , i ( x , y ) − B W D T ψ n , i ( x , y )2( √ − λ n ) , (7.44)and the QFT of B W g ∗ i ( x , y ) satisfies that F (cid:16) B W g ∗ i (cid:17) = D W (cid:16) F ( g ∗ i ) (cid:17) . Hence, in order tocompute (cid:107)F ( B W g ∗ ) (cid:107) Q in Eq. (7.43), we need to compute each term of (cid:82) R | B W g ∗ i | dxdy ,for i = , , , B W g ∗ i ( x , y ) and their QFT F ( B W g ∗ i ), Parseval’s theo-rem holds [31], then for all i = , , , (cid:90) R | B W g ∗ i | dxdy = (cid:90) W |F ( B W g ∗ i ) | dudv (7.45)Having in mind that for i = , , , B W D T ψ n , i ( x , y ) = π ) i n (cid:90) W (cid:90) T e − i ux e i us ψ n , i ( s , t ) e j vt e − j vy dsdtdudv j n = i n (cid:16) (cid:90) T sin W ( x − s ) π ( x − s ) ψ n , i ( s , t ) sin W ( y − t ) π ( y − t ) dsdt (cid:17) j n . (cid:90) R | B W g ∗ i | dxdy (7.46) = − λ n ) (cid:90) R (cid:16) ψ n , i ( x , y ) − B W D T ψ n , i ( x , y ) (cid:17)(cid:16) ψ n , i ( x , y ) − B W D T ψ n , i ( x , y ) (cid:17) dxdy = − λ n ) (cid:16) (cid:90) R ψ n , i ( x , y ) ψ n , i ( x , y ) dxdy − (cid:90) R ψ n , i ( x , y ) B W D T ψ n , i ( x , y ) dxdy − (cid:90) R B W D T ψ n , i ( x , y ) ψ n , i ( x , y ) dxdy + (cid:90) R B W D T ψ n , i ( x , y ) B W D T ψ n , i ( x , y ) dxdy (cid:17) = − λ n + λ n − λ n ) , n = k , k = , , · · · + λ n − λ n ) , n = k − , k = , , · · · . Here, the second equality for Eq. (7.46) have four terms integral. The first one is theorthogonality of QPSWFs. Computations of the second integral show that (cid:90) R ψ n , i ( x , y ) B W D T ψ n , i ( x , y ) dxdy = (cid:90) R (cid:90) T ψ n , i ( x , y ) sin W ( s − x ) π ( s − x ) ( − j ) n ψ n , i ( s , t )( − i ) n sin W ( t − y ) π ( t − y ) dsdtdxdy = λ n (cid:90) R ψ n , i ( x , y )( − j ) n ψ n , i ( x , y )( − i ) n dxdy = λ n ( − j ) n ( − i ) n . Similarly, we can immediately obtain the third integral that (cid:90) R B W D T ψ n , i ( x , y ) ψ n , i ( x , y ) dxdy = λ n i n j n . (7.47)Then for the two cross-terms for the second equality in Eq. (7.46) have (cid:90) R ψ n , i ( x , y ) B W D T ψ n , i ( x , y ) dxdy + (cid:90) R B W D T ψ n , i ( x , y ) ψ n , i ( x , y ) dxdy = λ n ( − j ) n ( − i ) n + λ n i n j n = λ n , n = k , k = , , · · · , n = k − , k = , , · · · The last integral for the second equality in Eq. (7.46) used the Lemma 3.1 to28ompute that (cid:90) R B W D T ψ n , i ( x , y ) B W D T ψ n , i ( x , y ) dxdy = (cid:90) R (cid:16) i n (cid:90) T sin W ( x − s ) π ( x − s ) ψ n , i ( s , t ) sin W ( y − t ) π ( y − t ) dsdt j n (cid:17)(cid:16) ( − j ) n (cid:90) T sin W ( x − z ) π ( x − z ) ψ n , i ( z , w ) sin W ( y − w ) π ( y − w ) dzdw ( − i ) n (cid:17) dxdy = i n (cid:16) (cid:90) T (cid:90) T ψ n , i ( s , t ) ψ n , i ( z , w ) sin W ( s − z ) π ( s − z ) sin W ( t − w ) π ( t − w ) dsdtdzdw (cid:17) ( − i ) n = λ n i n (cid:16) (cid:90) T ψ n , i ( z , w ) ψ n , i ( z , w ) dzdw (cid:17) ( − i ) n = λ n . Using the above results of Eq. (7.46), there holds η Q = (cid:107)F ( B W g ∗ ) (cid:107) Q = (cid:88) i = (cid:90) W |F ( g ∗ i ) | dudv = (cid:88) i = (cid:90) R | B W g ∗ i | dxdy = − λ n + λ n − λ n , n = k , k = , , · · · + λ n − λ n , n = k − , k = , , · · · . Therefore g ∗ ∈ A and η Q = (cid:107) B W g ∗ (cid:107) Q = √ − λ n , ∀ n = k , k = , , · · · . As { λ n } ∞ n = is monotone decreasing in the interval (0 , λ n can be arbitrarily close to 0. Thus η Q can be arbitrarily close to 1. On the other hand, it is clear that when both energy ratios ξ and η Q are equal to 0,the underlying quaternionic signal must be the identically zero signal. Nevertheless,we are able to finding a signal that is not identically zero as such η Q is infinitely closeto 0. For any r ∈ R and the g ∗ ( x , y ) in the Eq. (7.40), we note the QFT of g ∗ ( x , y ) as F ( g ∗ ), where F ( g ∗ ) = F ( g ∗ ) + i F ( g ∗ ) + F ( g ∗ ) j + i F ( g ∗ ) j . We want to find a newfunction f ( x , y ) such that the QFT of f ( x , y ) satisfy that F ( f )( u , v ) = F ( g ∗ )( u − r , v ) . (7.48)If there exist that f ( x , y ), then η Q = (cid:107)F ( B W f ) (cid:107) Q = (cid:90) W (cid:88) i = |F ( f i )( u , v ) | dudv = (cid:90) W (cid:88) i = |F ( g i )( u − r , v ) | dudv = (cid:90) W − r − W − r (cid:90) W − W (cid:88) i = |F ( g i )( u , v ) | dudv . η Q is continuous over r for fixed W . For F ( g ∗ )( u , v ) ∈ L ( R ; H ), η Q approacheszero as r approaches infinity. Thus, η Q can be arbitrarily close to 0.Now we need to present the formula of that f ( x , y ) and check whether that f ( x , y ) belong to A . Since f ( x , y ) satisfy that F ( f )( u , v ) = F ( g ∗ )( u − r , v ), it follows that f ( x , y ) = π (cid:90) R e i ux F ( f )( u , v ) e j vy dudv = π (cid:90) R e i ux F ( g ∗ )( u − r , v ) e j vy dudv = π (cid:90) R e i ( u + r ) x F ( g ∗ )( u , v ) e j vy dudv = e i rx π (cid:90) R e i ux F ( g ∗ )( u , v ) e j vy dudv = e i rx g ∗ ( x , y ) . It is easy check that (cid:107) f (cid:107) L = (cid:90) R | e i rx g ∗ ( x , y ) | dxdy = (cid:107) g ∗ (cid:107) L = , and ξ = (cid:90) T | e i rx g ∗ ( x , y ) | dxdy = (cid:90) T | g ∗ ( x , y ) | dxdy = . Thus, f ( x , y ) ∈ A and η Q can be arbitrarily close to 0 when r → ∞ . That is, 0 ≤ η Q <
1. This completes the proof. (cid:3)
Appendix B
This part proof the Theorem 5.3.
Proof.
Let us recall the function class A = (cid:110) g ∈ L ( R , H ) : (cid:107) g (cid:107) L = , (cid:107) D T g (cid:107) L = ξ (cid:111) . Let g ( x , y ) ∈ A and √ λ ≤ ξ <
1. Now, take its projections in D T and B W , respec-tively. We can decompose g as follows g = λ D T g + µ B W g + g ∗ , (7.49)where λ , µ ∈ R , < g ∗ , D T g > L = and < g ∗ , B W g > L = . The g ( x , y ) ∈ A ξ can alsodecomposed by g = g + i g + g j + i g j . (7.50)30hen we need to consider i = , , , g i = λ D T g i + µ B W g i + g ∗ i , (7.51)Now, we compute the scalar inner product Eq. (2.7) of the decomposition Eq. (7.49),respectively, with g i , D T g i , B W g i and g ∗ i . Direct computations show that (cid:107) g i (cid:107) L = λ (cid:107) D T g i (cid:107) L + µ (cid:107) B W g i (cid:107) L + < g i , g ∗ i >, (cid:107) D T g i (cid:107) L = λ (cid:107) D T g i (cid:107) L + µ < B W g i , D T g i > , (cid:107) B W g i (cid:107) L = λ < D T g i , B W g i > + µ (cid:107) B W g i (cid:107) L ,< g i , g ∗ i > = < g ∗ i , g ∗ i > . For every H -valued function f , the equation (cid:107) f (cid:107) L = (cid:107)F ( f ) (cid:107) Q holds. Then we can getthe following results by integrating the equations above i = , , , (cid:107)F ( g ) (cid:107) Q = λ (cid:107)F ( D T g ) (cid:107) Q + µ (cid:107)F ( B W g ) (cid:107) Q + (cid:107)F ( g ∗ ) (cid:107) Q , (cid:107)F ( D T g ) (cid:107) Q = λ (cid:107)F ( D T g ) (cid:107) Q + µ (cid:88) i = < B W g i , D T g i > , (cid:107)F ( B W g ) (cid:107) Q = λ (cid:88) i = < D T g i , B W g i > + µ (cid:107)F ( B W g ) (cid:107) Q . Substituting the facts ξ = (cid:107)F ( D T g ) (cid:107) Q , η Q = (cid:107)F ( B W g ) (cid:107) Q and (cid:107)F ( g ∗ ) (cid:107) Q = (cid:107) g ∗ (cid:107) L intothe equations above, it follows that1 = λξ + µη Q + (cid:107) g ∗ (cid:107) L ,ξ = λξ + µ (cid:88) i = < B W g i , D T g i > ,η Q = λ (cid:88) i = < D T g i , B W g i > + µη Q , By eliminating (cid:107) g ∗ (cid:107) L , λ and µ from the above equations, we obtain that η Q − (cid:88) i = < B W g i , D T g i > − (cid:88) i = < D T g i , B W g i > (7.52) = − ξ + (cid:16) − (cid:107) g ∗ (cid:107) L (cid:17) · − ( (cid:80) i = < B W g i , D T g i > )( (cid:80) i = < D T g i , B W g i > ) ξ η Q , ξη Q (cid:44)
0. For (cid:80) i = < B W g i , D T g i > and (cid:80) i = < D T g i , B W g i > in the aboveequation, we conclude that < B W g , D T g > = (cid:88) i = < B W g i , D T g i > . (7.53)Since g = g + i g + g j + i g j , D T g = D T g + i D T g + D T g j + i D T g j , B W g = B W g + i B W g + B W g j + i B W g j , and < B W g , D T g > = Sc (cid:32)(cid:90) R B W g ( x , y ) D T g ( x , y ) dxdy (cid:33) = Sc (cid:16) (cid:90) R ( B W g + i B W g + B W g j + i B W g j )( D T g + i D T g + D T g j + i D T g j ) dxdy (cid:17) = (cid:90) R (cid:16) B W g D T g + B W g D T g + B W g D T g + B W g D T g (cid:17) dxdy = (cid:88) i = < B W g i , D T g i > . Then the Eq. (7.52) becomes η Q − < B W g , D T g > = − ξ + (cid:16) − (cid:107) g ∗ (cid:107) L (cid:17)(cid:16) − ( < B W g , D T g > ) ξ η Q (cid:17) . (7.54)Let arg( D T g , B W g ) = arccos < D T g , B W g > (cid:107) D T g (cid:107) L (cid:107) B W g (cid:107) L . By Theorem 4.1, it follows thatarg( D T g , B W g ) ≥ arccos (cid:112) λ . We conclude that η Q − ξη Q cos arg( D T g , B W g ) = − ξ + (cid:16) − (cid:107) g ∗ (cid:107) L (cid:17)(cid:16) − cos arg( D T g , B W g ) (cid:17) ≤ − ξ + sin arg( D T g , B W g ) . Simplifying the above inequality, we obtain (cid:16) η Q − ξ cos arg( D T g , B W g ) (cid:17) ≤ (1 − ξ ) sin arg( D T g , B W g ) , from which it follows that η Q ≤ cos (cid:16) arg( D T g , B W g ) − arccos ξ (cid:17) , and arg( D T g , B W g ) ≥ √ λ . We conclude thatarccos η Q ≥ arccos (cid:112) λ − arccos ξ. (7.55)The equality is attained by setting g ∗ ( x , y ) = p ψ ( x , y ) + qD T ψ ( x , y ) , (7.56)where p = (cid:113) − ξ − λ , and q = ξ √ λ − (cid:113) − ξ − λ . The proof is completed. (cid:3) ReferencesReferences [1] A. Rihaczek, Signal energy distribution in time and frequency, IEEE Transac-tions on Information Theory 14 (3) (1968) 369–374. doi:10.1109/TIT.1968.1054157 .[2] N. Tugbay, E. Panayirci, Energy optimization of band-limited nyquist signals inthe time domain, IEEE Transactions on Communications 35 (4) (1987) 427–434. doi:10.1109/TCOM.1987.1096794 .[3] D. Slepian, H. O. Pollak, Prolate spheroidal wave functions, fourier analysis, and uncertainty–I, Bell System Technical Journal 40 (1) (1961) 43–64. doi:10.1002/j.1538-7305.1961.tb03976.x .[4] H. J. Landau, H. O. Pollak, Prolate spheroidal wave functions, fourier analysisand uncertainty–II, Bell System Technical Journal 40 (1) (1961) 65–84. doi:10.1002/j.1538-7305.1961.tb03977.x .[5] H. J. Landau, H. O. Pollak, Prolate spheroidal wave functions, fourier analysisand uncertainty–III: The dimension of space of essentially time-and bandlimitedsignals, Bell System Technical Journal 41 (4) (1962) 1295–1336. doi:10.1002/j.1538-7305.1962.tb03279.x .[6] D. Slepian, Prolate spheroidal wave functions, fourier analysis and uncertainty–
IV: Extensions to many dimensions; generalized prolate spheroidal functions,33ell System Technical Journal 43 (6) (1964) 3009–3057. doi:10.1002/j.1538-7305.1964.tb01037.x .[7] H. J. Landau, H. Widom, Eigenvalue distribution of time and frequency limiting,Mathematical Analysis and Applications 77 (2) (1980) 469–481. doi:10.1016/0022-247X(80)90241-3 .[8] I. C. Moorea, M. Cada, Prolate spheroidal wave functions, an introduction to theslepian series and its properties, Applied and Computational Harmonic Analysis16 (3) (2004) 208–230. doi:10.1016/j.acha.2004.03.004 .[9] I. C. Moorea, M. Cada, New e ffi cient methods of computing the prolate spheroidal wave functions and their corresponding eigenvalues, Applied andComputational Harmonic Analysis 24 (3) (2008) 269–289. doi:10.1016/j.acha.2007.06.004 .[10] S. Slavyanov, Asymptotic forms for prolate spheroidal functions, USSR Com-putational Mathematics and Mathematical Physics 7 (5) (1967) 50–62. doi:10.1016/0041-5553(67)90093-6 .[11] G. Walter, T. Soleski, A new friendly method of computing prolate spheroidalwave functions and wavelets, Applied and Computational Harmonic Analysis19 (3) (2005) 432–443. doi:10.1016/j.acha.2005.04.001 .[12] Detailed analysis of prolate quadratures and interpolation formulas, Numerical Analysis.[13] Uncertainty principles, prolate spheroidal wave functions, and applications, Ap-plied and Numerical Harmonic Analysis.[14] K. K. J. Morais, Y. Zhang, Generalized prolate spheroidal wave functions foro ff set linear canonical transform in cli ff ord analysis, Mathematical Methods inthe Applied Sciences 36 (9) (2013) 1028–1041. doi:10.1002/mma.2657 .[15] Constructing prolate spheroidal quaternion wave signals on the sphere, Mathe-matical Methods in the Applied Sciences.3416] A. I. Zayed, A generalization of the prolate spheroidal wave functions, Proceed-ings of the American mathematical society 135 (7) (2007) 2193–2203. doi: http://dx.doi.org/10.1090/S0002-9939-07-08739-4 .[17] G. Walter, X. Shen, Wavelets based on prolate spheroidal wave functions,Fourier Analysis and Applications 10 (1) (2004) 1–26. doi:10.1007/s00041-004-8001-7 .[18] Generalized and fractional prolate spheroidal wave functions, Proceedings of the10th International Conference on Sampling Theory and Applications.[19] S. Pei, J. Ding, Generalized prolate spheroidal wave functions for optical finitefractional fourier and linear canonical transforms, Optical Society of America A22 (3) (2005) 460–474. doi:10.1364/JOSAA.22.000460 .[20] J. M. H. Zhao, Q. Ran, L. Tan, Generalized prolate spheroidal wave functions as- sociated with linear canonical transform, IEEE Transactions on Signal Processing58 (6) (2010) 3032–3041. doi:10.1109/TSP.2010.2044609 .[21] D. S. H. Zhao, R. Wang, D. Wu, Maximally concentrated sequences in both timeand linear canonical transform domains, Signal, Image and Video Processing 8 (5)(2014) 819–829. doi:10.1007/s11760-012-0309-1 .[22] A. H. M. Bahria, E. M. Hitzer, R. Ashinob, An uncertainty principle for quater-nion fourier transform, Computers and Mathematics with Applications 56 (9)(2008) 2398–2410. doi:10.1016/j.camwa.2008.05.032 .[23] Quaternion-fourier transfotms for analysis of two-dimensional linear time-invariant partial di ff erential systems, Proceeding of the 32nd Conference on De- cision and Control, San Antonio, Texas.[24] E. M. Hitzer, Quaternion fourier transform on quaternion fields and general-izations, Advances in Applied Cli ff ord Algebras 17 (3) (2007) 497–517. doi:10.1007/s00006-007-0037-8 . 3525] E. M. Hitzer, B. Mawardi, Cli ff ord fourier transform on multivector fields anduncertainty principles for dimensions n = mod
4) and n = mod
4) advances inapplied cli ff ord algebras, Advances in Applied Cli ff ord Algebras 18 (3-4) (2008)715–736. doi:10.1007/s00006-008-0098-3 .[26] Principles of nuclear magnetic resonance in one and two dimensions, Oxford Uni-versity Press. [27] A. Sudbery, Quaternionic analysis, Mathematical Proceedings of the CambridgePhilosophical Society 85 (2) (1979) 199–225. doi:http://dx.doi.org/10.1017/S0305004100055638 .[28] Cli ff ord analysis, London: Pitman Research Notes in Mathematics.[29] Interpolation and approximation, Blaisdell Publishing Company Press, NewYork.[30] Quaternionic analysis and elliptic boundary value problems, Blaisdell PublishingCompany Press, New York.[31] K. K. L. Chen, M. Liu, Pitt’s inequatlity and the uncertainty principle associatedwith the quaternion fourier transform, Mathematical Analysis and Applications
423 (1) (2015) 681–700. doi:10.1016/j.jmaa.2014.10.003 .[32] Integral equations, Clarendon Press / Oxford University Press.[33] The classical theory of integral equations a concise treatment, New York: Birkh.[34] Convolution therorems for quaternion fourier transform: properties and applica-tions, Abstract and Applied Analysis 2013.[35] D. L. Donoho, P. B. Stark, Uncertainty principles and signal recovery, SIAM Jour-nal on Applied Mathematics 49 (3) (1989) 906–931. doi:10.1137/0149053doi:10.1137/0149053