2478 J. Opt. Soc. Am. A/Vol. 18, No. 10 /October 2001 Chong et al.

Fractional Fourier transform for quasi-periodicBloch functions

Chee Ching Chong and Apostolos Vourdas

Department of Electrical Engineering and Electronics, The University of Liverpool, Liverpool L69 3GJ, UK

Cherif Bendjaballah

Laboratoire des Signaux et Systemes, Ecole Superieure d’Electricite, Plateau de Moulon, 91192Gif-Sur-Yvette Cedex, France

Received November 27, 2000; revised manuscript received April 2, 2001; accepted April 17, 2001

The fractional Fourier transform (FRFT) for quasi-periodic Bloch functions is studied. An isomorphism be-tween square-integrable functions on the real line and quasi-periodic Bloch functions is used to extend existingwork on the fractional Fourier transform for the former functions to the latter. The properties of the FRFT forquasi-periodic Bloch functions are discussed, and various numerical examples are presented. © 2001 OpticalSociety of America

OCIS codes: 270.0270, 070.2590.

1. INTRODUCTIONThe Fourier transform plays an important role in variousbranches of science and engineering. An interesting gen-eralization in the context of phase-space methods is thefractional Fourier transform (FRFT), which was dis-cussed in a mathematical context in Refs. 1–3. Many ap-plications of the FRFT to optics4–11 and to signalprocessing12–16 have also been studied.

So far, all the work on the FRFT of which we are awareconsiders square-integrable functions on the real line,

E2`

1`

u f~x !u2dx , `. (1)

In many applications, periodic or quasi-periodic functionsplay an important role, and there is merit in studying theFRFTs of these functions. Such functions have beenused extensively in solid-state physics, where they areknown as Bloch functions. Solid-state systems are peri-odic, and the relevant Schrodinger equation is

@2]x2 1 V~x !#F~x ! 5 EF~x !, (2)

with a periodic potential V(x) 5 V(x 1 L). According tothe Bloch–Floquet theorem the solutions of this equationare quasi-periodic, of the type

F~x 1 L ! 5 exp~iu!F~x !. (3)

In this case

E0

L

uF~x !u2dx , `, (4)

and the integral *2`1`uF(x)u2dx diverges. We call these

functions quasi-periodic Bloch functions. Another physi-cal application of these functions is in quantum systemsin multiply connected spaces17 (e.g., electrons in meso-scopic rings). In this case u is the magnetic flux thread-ing of the ring times the electron charge.

0740-3232/2001/102478-08$15.00 ©

In this paper we utilize the isomorphism betweensquare-integrable and quasi-periodic Bloch functions tointroduce the FRFT for quasi-periodic Bloch functions.This is the first step toward a full phase-space formalism(Wigner functions, Weyl functions, etc.) for quasi-periodicBloch functions.

In Section 2 we explain the isomorphism and use it todefine the quasi-periodic analogs of the Gaussian func-tions and of the Hermite–Gaussian functions on the realline (coherent states and number states, respectively, inquantum-mechanical language). We also introduce cre-ation and annihilation operators for quasi-periodic Blochfunctions.

In Section 3 we introduce the FRFT for quasi-periodicBloch functions. We give three definitions and prove thatthey are equivalent to one another. We also present sev-eral examples, and discuss some properties of this FRFT.An important property of the FRFT is related to rescalingof the variables; in Section 4 we discuss this property inthe context of the present paper. We conclude in Section5 with a discussion of our results.

2. BASIC FORMALISMWe consider Hilbert space H of square-integrable func-tions f(x) in the interval (2`, 1`). In some examples be-low we shall use normalized functions so that

E2`

1`

u f~x !u2dx 5 1. (5)

We also consider Hilbert space H8(u) of quasi-periodicBloch functions F(f, u) such that

F~f 1 2p, u! 5 F~f, u!exp~iu!, (6)

where both the period of u and the quasi-period of f are2p. However, in Section 4 we shall consider a more gen-eral period. It is known18 that Hilbert space H is isomor-

2001 Optical Society of America

Chong et al. Vol. 18, No. 10 /October 2001 /J. Opt. Soc. Am. A 2479

phic to the direct integral of the Hilbert space H8(u),which we call H8 subsequently. The mapping betweenthese two isomorphic Hilbert spaces is as follows:

F~f, u! 5 (n52`

`

f~x 5 f 1 2pn !exp~2iun !, (7)

f~x 5 f 1 2pn ! 51

2pE

0

2p

F~f, u!exp~iun !du. (8)

We use lowercase and uppercase symbols for functions inHilbert spaces H and H8, respectively. We note that amapping similar to Eqs. (7) and (8) is known as a Zaktransform19–24 and has been used extensively in bothsolid-state physics and signal processing. Equation (3)can be viewed as a standard Fourier series analysis offunction F(f, u) in periodic variable u, in which case Eq.(4) simply gives the Fourier coefficients (in a notationsuitable for our purposes).

The inner product of F1 and F2 in H8 is defined as

~F1 , F2! 51

2pE

0

2pE0

2p

F1~f, u!@F2~f, u!#* dfdu, (9)

where * denotes a complex conjugate. If f( y) is theFourier transform of f(x):

f~ y ! 5 E2`

1`

f~x !exp~2iyx !dx, (10)

then we can prove an alternative to Eq. (7) that givesF(f, u) in terms of f( y):

F~f, u! 51

2p (k52`

`

fS y 5 k 1u

2pD expF iS k 1

u

2pDfG .

(11)To prove this equation we start with the expansion

F~f, u! 5 (k52`

`

Fk~u!expF iS k 1u

2pDfG , (12)

which is consistent with Eq. (6). We calculate the coeffi-cients Fk(u) by using the inverse Fourier formula, withF(f, u) replaced by the expression given in Eq. (7).

As an example, we consider delta functions on a circle(or comb functions):

D~f 2 f0 , u 2 u0! 5 (n52`

`

d ~f 1 2pn 2 f0!

3 exp@2i~u 2 u0!n#, (13)

where f0 P R and fP[0,2p). They are the analogs, on acircle, of delta functions on the real line. Indeed, we caneasily prove that

F~f0 , u0! 51

2pE

0

2pE0

2p

F~f, u!D*

3 ~f 2 f0 , u 2 u0!dfdu. (14)

As a second example, we consider the Gaussian wavefunction in Hilbert space H:

s~x; A ! 5 p21/4 expS 2x2

21 A2Ax 2 AARD , (15)

where A 5 AR 1 iAI . In a quantum mechanical contextthis is the wave function of the coherent state uA&@s(x; A) 5 ^xuA&#. The corresponding function (Gauss-ian on a circle) in Hilbert space H8 can be found by sub-stitution of Eq. (15) into Eq. (7):

S~f, u; A ! 5 p21/4 expS 2f 2

21 A2Af 2 AARDU3

3 F2u

21 ip~f 2 A2A !; i2pG , (16)

where U3@u; t# is the theta function,25–27 defined as

U3@u; t# 5 (n52`

`

exp~iptn2 1 i2nu !. (17)

In Fig. 1 we present the real and imaginary parts ofS(f, u; A), with A 5 1. The figure provides intuitionfor the Gaussian on a circle. We present two periods off(22p < f < 2p) so that the quasi-periodic nature ofthe function [i.e., the effect of exp(iu)] can be seen.

The zero of the theta function is U3@umn ; t# 5 0,where

umn 5 ~2m 2 1 !p

21 ~2n 2 1 !

p

2t, (18)

where m, n are integers. Therefore

S~f0 , u0 ; Amn! 5 0, (19)

where

Amn 5 221/2H f0 2 p~2n 2 1 ! 1 iF S m 21

2 D 1u0

2pG J .

(20)

Using the general property of Eq. (14), we get

Fig. 1. Real and imaginary parts of function S(f, u; A 5 1).

2480 J. Opt. Soc. Am. A/Vol. 18, No. 10 /October 2001 Chong et al.

1

2pE

0

2pE0

2p

S~f, u; Amn!D* ~f 2 f0 , u 2 u0!dfdu 5 0.

(21)

This equation shows that all the S(f, u; Amn) are or-thogonal to D(f 2 f0 , u 2 u0).

As another example we consider the Hermite–Gaussian function

gm~x ! 5 ~Ap2mm! !21/2 expS 2x2

2 DHm~x !, (22)

where m is an integer and Hm(x) are Hermite polynomi-als. In a quantum-mechanical context the Hermite–Gaussian functions are the wave functions of the numberstates um&@ gm(x) 5 ^xum&#. Using Eq. (7), we find thecorresponding Gm(f, u):

Gm~f, u! 5 (n52`

`

gm~f 1 2pn !exp~2iun !. (23)

Note that G0(f, u) 5 S(f, u; A 5 0). The fact that thevalues of gm(x) form an orthonormal basis in Hilbertspace H together with the isomorphism between H andH8 implies that the Gm(f, u) form an orthonormal basisin Hilbert space H8. Therefore an arbitrary stateF(f, u) can be expanded as

F~f, u! 5 (m52`

`

cmGm~f, u!, (24)

where

cm 51

2pE

0

2pE0

2p

F~f, u!Gm* ~f, u!dfdu. (25)

For example, we can prove that

S~f, u; A ! 5 expS 2uAu2

2 D (m50

`

Am~m! !21/2Gm~f, u!.

(26)

In Fig. 2 we present the real and imaginary parts ofG2(f, u). The figure provides intuition about the

Fig. 2. Real and imaginary parts of function G2(f, u).

Hermite–Gaussian functions (the analogs of numberstates in the present context).

It is easily seen that action of operators x and p5 2i]x on function f(x) corresponds to action of opera-tors f 1 i2p]u and 2i]f , respectively, on the corre-sponding function, F(f, u):

x → f 1 i2p]u , 2i]x → 2i]f . (27)

Consequently, action of the annihilation and creation op-erators a and a† on the function f(x) corresponds to actionof the operators 221/2(f 1 i2p]u 1 ]f) and 221/2(f1 i2p]u 2 ]f), respectively, on the corresponding func-tion F(f, u):

a 5 221/2~x 1 ip ! → b 5 221/2~f 1 i2p]u 1 ]f!,

a† 5 221/2~x 2 ip ! → b† 5 221/2~f 1 i2p]u 2 ]f!,

(28)

where @b, b†# 5 1. We can now verify that

bGm~f, u! 5 m1/2Gm 2 1~f, u!,(29)b†Gm~f, u! 5 ~m 1 1 !1/2Gm 1 1~f, u!;

b†b 512 @~f 1 i2p]u!2 2 ]f

2 2 1#, (30)

b†bGm~f, u! 5 mGm~f, u!, (31)

bS~f, u; A ! 5 AS~f, u; A !. (32)

3. FRACTIONAL FOURIER TRANSFORMFRFT FOR QUASI-PERIODIC BLOCHFUNCTIONSWe present here three definitions of the FRFT for quasi-periodic Bloch functions and prove that they are equiva-lent to one another. We thus have the flexibility of usingthe definition that is most suitable for a particular appli-cation.

A. Fractional Fourier Transform OperatorThe FRFT in Hilbert space H of square-integrable func-tions is introduced by the following equivalent of theother two definitions:

fa~ y ! 5 ~Fa f !~ y !, Fa [ exp~iaa†a !, (33)

fa~ y ! 5 E2`

1`

ka~ y, x !f~x !dx, (34)

where Fa is a fractional Fourier operator and

ka~ y, x ! 5 (m50

`

exp~ima!gm~ y !gm* ~x ! (35)

5 S 1 1 i cot a

2pD 1/2

3 expS 2ix2 1 y2

2 tan a1 i

xy

sin aD , (36)

where gm are Hermite–Gaussian functions of Eq. (22).For a5p/2 we get the ordinary Fourier transform, and inthis case we will sometimes omit the index for simplicityof notation.

Chong et al. Vol. 18, No. 10 /October 2001 /J. Opt. Soc. Am. A 2481

Analogously, we introduce the FRFT for quasi-periodicBloch functions as

Fa~f, u! 5 FaF~f, u!, Fa [ exp~iab†b !, (37)

where Fa is a unitary operator with eigenfunctionsGm(f, u) and eigenvalues exp(ima). The operators Fa

are isomorphic to the operators Fa .In the special case that a/2p is a rational number,

a/2p 5 p/q, (38)

where p and q are coprime integers, it is easily seen thatFp/q

q 5 1. In this case the operator Fp/q has only q differ-ent eigenvalues, exp(i2pmp/q), where m 5 0 ,..., (q2 1), and there is a degeneracy with many eigenfunc-tions corresponding to same eigenvalue:

Fp/qGm1nq~f, u! 5 expS i2pmp

q DGm1nq~f, u!. (39)

It is easily shown that

FaFb 5 Fa1b . (40)

B. Fractional Fourier Transform KernelWe next show that another equivalent definition of theFRFT is

Fa~f, u! 51

2pE

0

2pE0

2p

Ka~f, u; f 8, u8!

3 F~f8, u8!du8df 8, (41)

where

Ka~f, u; f 8, u8! 5 (m50

`

exp~ima!Gm~f, u!Gm* ~f8, u8!.

(42)

The equivalence of this definition to that of Eq. (37) is eas-ily seen from the fact that the Fa has eigenfunctionsGm(f, u) and eigenvalues exp(ima).

To evaluate the sum, we first introduce the two-dimensional theta function U@ui ; t ij#, defined as

U~ui ; t ij! 5 (ni52`

`

expS ip( nit ijnj 1 2i( niui D ,

(43)

where t ij is a symmetric 2 3 2 matrix and ni are inte-gers. It is easily seen that, for any integers mi ,

U~ui 1 2pmi ; t ij! 5 U~ui ; t ij!. (44)

Substitution of Eq. (23) into Eq. (42) gives

Ka~f, u; f 8, u8! 5 S 1 1 i cot a

2pD 1/2

U~ui ; t ij!

3 expS 2if 2 1 f 82

2 tan a1 i

ff 8

sin aD ,

(45)

where

u1 5 2pf cot a 1pf 8

sin a2

u

2,

(46)

u2 5 2pf 8 cot a 1pf

sin a1

u8

2,

t11 5 t22 5 22p cot a, t12 5 t21 52p

sin a. (47)

In the special case a5p/2 the kernel Ka(f, u; f 8, u8) be-comes

Kp/2~f, u; f 8, u8! 5 ~2p!21/2U~ui ; t ij!exp~iff 8!, (48)

where

u1 5 pf 8 2u

2, u2 5 pf 1 u8/2, (49)

t11 5 t22 5 0, t12 5 t21 5 2p. (50)

C. Relationship of Fractional Fourier Transformfor Quasi-Periodic Bloch Functions to Square-Integrable FunctionsThere is a third way of defining the FRFT in Hilbert spaceH8. For a given quasi-periodic Bloch function F(f8, u8)we first find the corresponding function f(x) in Hilbertspace H, using Eq. (8), and then find its FRFT, fa( y), us-ing Eq. (34):

1

2p (n852`

` E0

2pE0

2p

ka~ y, f 8 1 2pn8!

3 exp~iu8n8!F~f8, u8!du8df 8. (51)

Integration over x 5 f 8 1 2pn8 has been replaced by in-tegration over f 8 and summation over n8. Finally, wefind its corresponding function in H8, using Eq. (7):

Fa~f, u! 51

2p (n,n852`

` E0

2pE0

2p

ka~f 1 2pn, f 8

1 2pn8!exp~iu8n8 2 iun !F~f8, u8!du8df 8

51

2pE

0

2pE0

2p

Ka~f, u; f 8, u8!

3 F~f8, u8!du8df 8. (52)

It can be seen that this approach leads to the same resultas the two previous ones.

D. ExamplesAs an example, we calculate analytically the FRFT of thefunction S(f, u; A) of Eq. (16). We first calculate theFRFT of the Gaussian function of Eq. (15), using Eq. (34):

sa~ y; A ! 5 p21/4 expS 2y2

21 A2Ay exp~ia!

2 iA2 exp~ia!sin a 2 AARD . (53)

We then use Eq. (7) and find that

2482 J. Opt. Soc. Am. A/Vol. 18, No. 10 /October 2001 Chong et al.

Sa~f, u; A !

5 p21/4 expF2f 2

21 A2A exp~ia!f 2 AAR 2 iA2

3 exp~ia!sin aGU3H 2u

21 ip@f 2 A2A

3 exp~ia!#; i2pJ . (54)

The FRFTs of the function Sa(f, u; A 5 1) for a5 p/2 and a 5 p/6 are shown in Figs. 3 and 4, respec-tively. It is known that the Fourier transform of aGaussian function in the real line is another Gaussianfunction. Here we consider the Gaussian function on acircle and show its Fourier transform in Fig. 3 and itsFRFT with a5p/6 in Fig. 4.

E. PropertiesThe FRFT for the square-integrable functions on the realline is known1 to obey the differentiation property:

Fa@]xf~x !# 5 cos a]yfa~ y ! 2 iy sin a fa~ y !. (55)

In Appendix A we prove the following analogous propertyfor quasi-periodic Bloch functions:

Fa@]fF~f8, u8!# 5 cos a]fFa~f, u! 2 i~f

1 i2p]u!sin aFa~f, u!. (56)

It can be seen that the two formulas are analogous, with]x in place of ]f and x in place of f 1 i2p]u [relations(27)].

Another property of the square-integrable functions onthe real line is the shifting property1:

Fa@ f~x 1 g!# 5 fa~ y 1 g cos a!

3 expF2ig2

4sin~2a! 2 iyg sin aG .

(57)

Fig. 3. Real and imaginary parts of the FRFT of functionS(f, u; A 5 1) for a5p/2.

In Appendix B we prove the following analogous propertyfor quasi-periodic Bloch functions:

Fa@F~f8 1 g, u8!# 5 expF2ig2

4sin~2a! 2 igf sin aG

3 Fa~f 1 g cos a, u 1 2pg sin a!.

(58)

These properties are helpful in practical calculations.

4. RESCALING OF THE VARIABLESIn the previous sections, for simplicity we considered thecase where u has period 2p and f has quasi-period 2p.Here we consider the more general case where u has pe-riod 2p/R and f has quasi-period 2pR. Now Eqs. (7)and (8) become

F~f, u; R !

5 R1/2 (n52`

`

f~x 5 f 1 2pnR !exp~2iunR !, (59)

f~x 5 f 1 2pnR !

5R1/2

2pE

0

2p/R

F~f, u; R !exp~iunR !du. (60)

Note that when we omit the variable R from the notation,it means that R 5 1. The functions F(f, u; R) obey therelation

F~f 1 2pR, u! 5 F~f, u!exp~iRu!. (61)

We discuss below some cases in which this rescalingleads to important relations. As we explained in Subsec-tion 3.C, the Fourier transform of F(f, u; R) can befound from Eq. (59) with f replaced by f:

F~a, b; R ! 5 R1/2 (n52`

`

f~ y 5 a 1 2pnR !exp~2ibnR !.

(62)

Fig. 4. Real and imaginary parts of the FRFT of functionS(f, u; A 5 1) for a5p/6.

Chong et al. Vol. 18, No. 10 /October 2001 /J. Opt. Soc. Am. A 2483

On the other hand, Eq. (11) can be written as

F~f, u! 51

2pexpS iuf

2pD (

n52`

`

fS y 5u 1 2pn

2pD

3 exp~inf !. (63)

Comparison of Eqs. (62) and (63) shows that

F~f, u! 51

2pR1/2 expS iuf

2pD FS u

2p, 2 2pf; R 5

1

2pD .

(64)Here F(f, u) is expressed in terms of its Fourier trans-form with interchange and rescaling of the variables andis multiplied by an exponential term.

The FRFT for square-integrable functions on the realline obeys the following scaling property2:

Fa@ f~cx !# 5 S 1 1 i cot a

c2 1 i cot aD 1/2

3 expF iy2 cot a

2 S cos2 a

cos2 b2 1 D G fbS y sin b

c sin aD ,

(65)

where b 5 arctan(c2 tan a). The change in angle from ato b has a nice physical interpretation: Contraction inthe x axis by a factor c (x8 5 x/c) is related to dilation inthe y axis ( y8 5 yc) in the ordinary Fourier transform.Therefore a FRFT by an angle a (with tan a 5 y/x) afterrescaling will be related to a FRFT by an angle b (withtan b 5 y8/x8 5 c2y/x 5 c2 tan a).

In this paper we extend this property for quasi-periodicBloch functions, and prove in Appendix C that

Fa@F~cf 8, u8!#

5 R21/2S 1 1 i cot a

c2 1 i cot aD 1/2

3 expF i~f 1 i2p]u!2

2cot aS cos2 a

cos2 b2 1 D G

3 FbS sin b

c sin af,

c sin a

sin bu; R 5

sin b

c sin aD . (66)

5. DISCUSSIONWe have extended the recent work on the FRFT of square-integrable functions on the real line to quasi-periodicBloch functions by using the isomorphism of Eqs. (7) and(8), which is similar to the Zak transform in solid-statephysics. In Eqs. (28) we defined creation and annihila-tion operators for quasi-periodic Bloch functions. In Eqs.(37), (41), and (52) we introduced different definitions ofthe FRFT and proved their equivalence. We also studiedthe differentiation, shifting, and rescaling properties.The general theory has been exemplified with various nu-merical results. The present work is the first step to-ward a full phase-space formalism for quasi-periodicBloch functions (which will involve the study of the corre-sponding Wigner functions, etc.). Such a formalism

could be applied to solid-state systems, quantum systemsin multiply connected spaces (e.g., electrons in mesoscopicrings), Bloch electrons in magnetic fields, etc. There aremany important phenomena in these areas, and we be-lieve that a phase-space formalism in this context willprovide a deeper insight into these phenomena.

The work can be applied to optical signal processing,solid-state systems, quantum systems in multiply con-nected spaces, quantum phase space methods,28,29 quan-tum communications,30 etc.

APPENDIX AIn this appendix we prove the differentiation property ofEq. (56). The proof is based on the differentiation prop-erty of square-integrable functions in conjunction withthe isomorphism between square-integrable and quasi-periodic Bloch functions. We start with Eq. (52) and re-place F(f8, u8) by ]f 8F(f8, u8):

1

2p (n,n852`

` E0

2pE0

2p

ka~f 1 2pn, f 8 1 2pn8!

3 exp@i~u8n8 2 un !#]f 8F~f8, u8!du8df 8

5 (n,n852`

` E0

2p

exp~2iun !ka~f 1 2pn,

f 8 1 2pn8!]f 8f~f8 1 2pn8!df 8

5 (n52`

`

exp~ 2 iun !E2`

1`

ka~f 1 2pn, x !]xf~x !dx.

(A1)

We next use the differentiation property of square-integrable functions [Eq. (55)] and get

(n52`

`

exp~2iun !@cos a]f fa~f 1 2pn !

2 i~f 1 2pn !sin a fa~f 1 2pn !]. (A2)

Now use Eq. (7), and thus Eq. (56) follows.

APPENDIX BIn this appendix we prove the shifting property of Eq.(58). The full proof is tedious; hence only some key stepsare presented. We start with Eq. (52) and replaceF(f8, u8) with F(f8 1 g, u8):

1

2p (n,n852`

` E0

2pE0

2p

ka~f 1 2pn, f 8 1 2pn8!

3 exp@i~u8n8 2 un !#F~f8 1 g, u8!du8df 8

2484 J. Opt. Soc. Am. A/Vol. 18, No. 10 /October 2001 Chong et al.

5 (n,n852`

` E0

2p

exp~2iun !ka~f 1 2pn, f 8 1 2pn8!

3 f~f8 1 g 1 2pn8!df 8

5 (n,n852`

` E0

2p

exp~2iun !

3 expF i2g ~f 8 1 g 1 2pn8! 2 g2

2

3 cot a 2ig ~f 1 2pn !

sin aGka~f 1 2pn, f 8 1 g

1 2pn8!f~f8 1 g 1 2pn8!d~f8 1 g!

5 (n52`

`

exp~2iun !expF2ig2

2cot a 2

ig ~f 1 2pn !

sin aG

3 E2`

1`

exp~igx cot a!ka~f 1 2pn, x !f~x !dx.

(B1)

We then use the shifting property of square-integrablefunctions1:

Fa@exp~igx !f~x !#

5 fa~ y 1 g sin a!expF ig2

4sin~2a! 1 iyg cos aG . (B2)

Hence Eq. (B1) becomes

(n52`

`

exp~2iun !expF2ig2

2cot a 2

ig ~f 1 2pn !

sin aG

3 expF ig2

4cos2 a sin~2a! 1 i~f 1 2pn !g cot a cos aG

3 fa~f 1 2pn 1 g cot a sin a!. (B3)

Using Eq. (7), we prove Eq. (58).

APPENDIX CIn this appendix we prove the scaling property of Eq. (66).We start with Eq. (52) and replace F(f8, u8) withF(cf 8, u8):

1

2p (n,n852`

` E0

2pE0

2p

ka~f 1 2pn, f 8 1 2pn8!

3 exp@i~u8n8 2 un !#F~cf 8, u8!du8df 8

5 (n,n852`

` E0

2p

exp~2iun !ka~f 1 2pn, f 8 1 2pn8!

3 f~cf 8 1 2pcn8!df 8

5 (n52`

`

exp~2iun !E2`

1`

ka~f 1 2pn, x !f~cx !dx.

(C1)

Next we use the scaling property of square-integrablefunctions given in Eq. (65); we have

(n52`

`

exp~2iun !S 1 1 i cot a

c2 1 i cot aD 1/2

3 expF i~f 1 2pn !2

2cot aS cos2 a

cos2 b2 1 D G

3 fbF ~f 1 2pn !sin b

c sin aG . (C2)

Comparing Eq. (C2) with Eq. (62), we get Eq. (66).

Corresponding author A. Vourdas can be reached bye-mail at ee21@liverpool.ac.uk.

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