M. Ruzzi
Citation: Journal of Mathematical Physics 47, 063507 (2006); doi: 10.1063/1.2209770 View online: http://dx.doi.org/10.1063/1.2209770
View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/47/6?ver=pdfcov Published by the AIP Publishing
Jacobi
-functions and discrete Fourier transforms
M. Ruzzia兲
Instituto de Física Teórica, São Paulo State University 01405-900 Rua Pamplona, 145 São Paulo, SP, Brazil
共Received 18 April 2006; accepted 11 May 2006; published online 28 June 2006兲
Properties of the Jacobi3-function and its derivatives under discrete Fourier trans-forms are investigated, and several interesting results are obtained. The role of modulo N equivalence classes in the theory of-functions is stressed. An important conjecture is studied. © 2006 American Institute of Physics.
关DOI:10.1063/1.2209770兴
I. INTRODUCTION
Methods in mathematical physics usually provide an interface between quite different areas of physics, and it is not unusual that such areas advance in parallel, mostly ignoring each other’s steps. This is the case with finite dimensional inner product spaces共hereafter mentioned as the “discrete”兲, with its leading role in quantum mechanics 共hence quantum information theory兲 and in finite signal analysis. References 1–3 provide some links between those theories.
Both quantum mechanics of finite dimensional Hilbert spaces and finite signal analysis rely heavily on the discrete Fourier transform共DFT, sometimes mentioned finite or fractional Fourier transform兲, and, regarding quantum mechanics, after the seminal work of Weyl on finite dimen-sional systems,4it was Schwinger who observed and explored the fact that two physical observ-ables whose families of eigenstates are connected via DFT share a maximum degree of incompatibility.5
Although, at first glance, a finite system might look much simpler than anything defined on a nonenumerable infinite dimensional Hilbert space共hereafter referred to as the “continuum”兲, there is much more knowledge about the latter than the former. In one phrase, in the continuum we have one, and only one, harmonic oscillator, while in the discrete there are a lot of candidates for that role, each one surely with its virtues, but surely no undisputed champion.
The eigenstates associated to the harmonic oscillator, the Gaussian function and the Hermite polynomials, have a very distinguishable behavior under the action of the共usual兲 Fourier trans-form, so widely known that any comment on this regard is completely superfluous. Over such properties rests a huge amount of physical knowledge. On the other hand, however, although the discrete Fourier transform 共DFT兲 is a well known tool, there is nothing on this context which could claim for itself a role analogous to that of the Gaussian function/Fourier transform “duo.” A decisive step in an attempt to “regain,” in the discrete, all interpretative power derived from the qualitative behavior of the harmonic oscillator eigenfunctions, lost when one leaves the con-tinuum realm, was given in Ref. 6, where the eigenstates of the DFT are obtained. The purpose of this paper is to further explore this path, showing results which closely parallel those of the continuum. Those results are obtained in a strikingly simple fashion, exploring the technique of breaking infinite sums in modulo N equivalence classes. Pertinent research on the eigenstates of the DFT can also be found in Ref. 7.
A remark must be made about the orthogonality of the DFT’s eigenstates. Mehta has conjec-tured that those states are indeed orthogonal, what seems to be most reasonable. One may be led to believe that, just as in the continuum, the eigenstates of the DFT may be also共nondegenerate兲
a兲Email address: [email protected]
JOURNAL OF MATHEMATICAL PHYSICS 47, 063507共2006兲
47, 063507-1
eigenstates of some other共unitary or self-adjoint兲 operator, and thus orthogonal. Further evidence supporting such conjecture is that the continuous limit of the DFT eigenstates recovers, as ex-pected, the Gaussian times the Hermite polynomials. However, as it will be shown, quite surpris-ingly, the conjecture does not hold, giving another fine example of the peculiarities of the finite dimensional context.
The eigenstates of the DFT are seen to be the Jacobi3-function and its derivatives.8Interest
in Jacobi-functions, by their own turn, may come from a variety of directions. First, its math-ematical interest goes without saying共see, for example, Ref. 9 and references therein兲. To cite relatively recent examples in physics, in quantum physics it is deeply related to coherent states associated to both circle10and finite lattice topology.11Its modular properties have proven to be of fundamental importance in superstring theory, as it is shown by standard literature in this field.12 The basic notation adopted in this paper and some preliminary results are presented in the next section. Following, orthogonality of the DFT’s eigenstates is discussed. Section IV contains the main results, for which a two variable generalization is verified in the subsequent section. Further relations among3-functions are obtained in Sec. VI, which precedes the concluding section.
II. PRELIMINARY RESULTS
In Ref. 6 it is shown that there is a set of functions with the following remarkable property
fn共j兲 = in
冑
N兺
k=0 N−1 fn共k兲exp冋
2i N kj册
, 共1兲where N is a natural number. The functions
fn共j兲 =
兺
␣=−⬁ ⬁ exp冋
− N共␣N + j兲 2册
H n共
⑀共␣N + j兲兲
, ⑀=冑
2 N 共2兲are defined making use of the Hermite polynomial Hn. Writing Hn共x兲 in terms of its generating
function, Hn共x兲=
n
tnexp关2xt−t 2兴兩
t=0, it is possible to write this state共to use a quantum mechanical
terminology兲 as8 fn共j兲 = 1
冑
N冏
n tn3冉
j N− ⑀ t, i N冊
exp关t 2兴冏
t=0 , 共3兲 where 3共z,兲 =兺
␣=−⬁ ⬁exp关i␣2兴exp关2i␣z兴, Im共兲 ⬎ 0, 共4兲
is the Jacobi3-function, following Vilenkin’s notation.13In this notation the basic properties of
this even function read as
3共z + m + n,兲 = exp关− in2兴exp关− 2inz兴3共z,兲, 共5兲
3共z,i兲 =−1/2exp
冋
− z2 册
3冉
z i, i 冊
, 共6兲emphasizing its period 1 and quasiperiod. A beautiful consequence of共6兲 is that this function can be written as a sum of Gaussians,
3
冉
z L, i 2冊
=兺
␣=−⬁ ⬁ exp冋
−冉
L冊
2 共␣L + z兲2册
, 共7兲a form in which the width L /becomes apparent. Property共6兲 also provides an easy way to obtain the additional identity共also given by Ref. 6兲
fn共j兲 =⑀共− i兲n
兺
␣=−⬁ ⬁ exp冋
− N␣ 2+2i N j␣册
Hn共⑀␣兲, 共8兲which is in fact a generalization of Eq.共7兲 共if one compares it to Eq. 共2兲兲.
III. ORTHOGONALITY OF THE fn’S
According to Eq. 共1兲, the functions 兵fn共j兲其 are eigenstates of the DFT with associated
eigen-value in. Mehta has conjectured that兵f
n共j兲其n=0N−1is an orthogonal set, and thus complete, over a finite
set of N points 共for odd N. For even N one must replace fN−1共j兲 by fN共j兲兲. This reasonable
conjecture, quite surprisingly indeed, does not hold for arbitrary N共it holds for large N兲. As, in the following, evidence will be collected against the original conjecture, details shall be kept to a level higher than usual.
Let共fn, fm兲 denote the inner product
共fn, fm兲 =
兺
j=0 N−1 fn *共j兲f m共j兲 =⑀2共− i兲n+m兺
j=0 N−1兺
␣,=−⬁ ⬁ exp冋
− N共␣ 2+2兲 +2i N j共␣−兲册
Hn共⑀␣兲Hm共⑀兲.The sum over兵j其 is a realization of the modulo N Kroenecker delta,
␦␣,关N兴=
再
1 ␣= 共mod N兲, 0 ␣⫽ 共mod N兲, thus 共fn, fm兲 = 2共− i兲n+m兺
␣,=−⬁ ⬁ ␦␣,关N兴 exp冋
−N共␣2+2兲册
Hn共⑀␣兲Hm共⑀兲. 共9兲The well-known identity, exp
冋
−1 2x 2册
H k共x兲 = ik冑
2冕
−⬁ ⬁ dy exp冋
−1 2y 2+ ixy册
H k共y兲,together with the sum over兵其 leads to 共fn, fm兲 =
兺
␣,␥=−⬁ ⬁冕
−⬁ ⬁ dy dz exp冋
− 1 2共y 2+ z2兲 + iy⑀␣+ iz⑀共␣+␥N兲册
H n共y兲Hm共z兲,where the infinite sum on兵␥其 covers the equivalence class present in␦␣,关N兴. Now, the sum over兵␣其 by its turn is a realization of a modulo 2 Dirac delta, thus, with the integration over 兵z其 and convenient changes of variables,
共fn, fm兲 = 2
兺
␥,=−⬁ ⬁冕
−⬁ ⬁ dy exp冋
− y2− 2v2 ⑀2 + i共y⑀N −vN兲␥册
Hn冉
y − v ⑀冊
Hm冉
y + ⑀冊
,where again an infinite sum is introduced due to the modulo 2 delta.
The above expression is rather elucidative. It is not hard to realize that the infinite sums over 兵␥,其 are a direct consequence of the equivalence classes brought in by the modulo N Kroenecker
delta present in Eq. 共9兲. For large N, the term corresponding to ␥== 0 becomes increasingly important, and a simple check shows that this term is exactly␦n,m. Thus, as expected, the limit
N→⬁ recovers the usual harmonic oscillator results. For finite 共and small兲 N, however, all terms
in the above summation must be taken into account.
Following then, the sum on ␥ is seen to be a realization of the modulo 2 Dirac delta,
␦关2兴共y⑀N −vN兲, and after a change of variables one has
共fn, fm兲 = 2⑀
兺
,=−⬁ ⬁ exp冋
−2 N冉
+ Nv 2冊
2 −Nv 2 2册
Hn共⑀兲Hm冉
2v ⑀ +⑀冊
.Again, summation over兵其 must be included to account for the 2periodicity of the Dirac delta. Splitting the sum onin two sums, over the odd and even integers and shifting the sum onby
N results in 共fn, fm兲 = 2⑀
兺
,=−⬁ ⬁ exp冋
−2 N 2− 2Nv2册
H n共⑀−⑀Nv兲Hm共⑀+⑀Nv兲 + 2⑀兺
,=−⬁ ⬁ exp冋
−2 N冉
+ N 2冊
2 − 2N共v + 1/2兲2册
Hn共⑀−⑀Nv兲Hm共⑀共+vN + N兲兲.Denoting the second term above by共fn, fm兲odd, if N = 2h + k, where the binary variable k controls
the parity of N, then 共fn, fm兲odd= 2⑀
兺
,=−⬁ ⬁ exp冋
−2 N 共+ h + k/2兲 2− 2N共v + 1/2兲2册
H n共⑀−⑀Nv兲Hm共⑀共+vN + N兲兲and yet again shifting the sum onby h + k / 2 and the one on by 1 / 2, 共fn, fm兲odd= 2⑀
兺
=−⬁ ⬁ 共k兲兺
=−⬁ ⬁ 共1兲exp冋
−2 N 2− 2Nv2册
H n共⑀共− Nv兲兲Hm共⑀共+vN兲兲,where now兺=−⬁⬁ 共k兲 denotes a sum over the integers 共half-integers兲 if k=0 共k=1兲, so that back to the general expression,
共fn, fm兲 = 2⑀
兺
,=−⬁ ⬁ exp冋
−2 N 2− 2Nv2册
H n共⑀共− Nv兲兲Hm共⑀共+vN兲兲 + 2⑀兺
=−⬁ ⬁ 共k兲兺
=−⬁ ⬁ 共1兲exp冋
−2 N 2− 2Nv2册
H n共⑀共− Nv兲兲Hm共⑀共+vN兲兲.Now, recourse to the Hermite polynomial’s generating function gives 共fn, fm兲 = 2⑀ n tn m sm
再
,=−⬁兺
⬁ exp冋
−2 N 2+ 2⑀共t + s兲 − 2⑀N共t − s兲 − t2− s2册
+兺
=−⬁ ⬁ 共k兲兺
=−⬁ ⬁ 共1兲exp冋
−2 N 2+ 2⑀共t + s兲 − 2⑀N共t − s兲 − t2− s2册
冎
t=s=0 . The sum on兵其 results in a3-function in the first term, and a3for k = 0 or a2for k = 1 in the second. The sum on兵其, by its turn, gives3-function in the first term, and a2in the second, as共fn, fm兲 = 2⑀ n tn m sm
再
冋
3冉
i⑀共t + s兲 , 2i N冊
3冉
i⑀N共t − s兲 ,2Ni冊
冋
+3−k冉
i⑀共t + s兲 , 2i N冊
2冉
i⑀N共t − s兲 ,2Ni冊
册
exp关− t2− s2兴兩冎
冏
t=s=0 . Using the basic properties,3共z,i兲 =−1/2exp
冋
− z2 册
3冉
z i, i 冊
, 2共z,i兲 =−1/2exp冋
− z2 册
4冉
z i, i 冊
, one gets 共fn, fm兲 = 23/2 N n tn m sm再
冋
3冉
i⑀共t + s兲 , 2i N冊
3冉
⑀共t − s兲 2 , i 2N冊
+3−k冉
i⑀共t + s兲 , 2i N冊
4冉
⑀共t − s兲 2 , i 2N冊
册
exp关− 2ts兴兩冎
t=s=0 . Finally, compact expressions can be achieved with3共z,兲 = 1 2
冋
3冉
z 2, 4冊
+4冉
z 2, 4冊
册
, 2共z,兲 = 1 2冋
3冉
z 2, 4冊
−4冉
z 2, 4冊
册
, thus for k = 0 共fn, fm兲 = 3/2 N n tn m sm冏
3冉
i⑀共t + s兲 , 2i N冊
3冉
⑀共t − s兲 , 2i N冊
exp关− 2ts兴冏
t=s=0 and for k = 1 共fn, fm兲 = 3/2 N n tn m sm再
3冉
i⑀共t + s兲 , 2i N冊
3冉
⑀共t − s兲 , 2i N冊
− 24冉
i⑀共t + s兲 2 , i 2N冊
4冉
⑀共t − s兲 2 , i 2N冊
exp关− 2ts兴冎
t=s=0.Again, the limit N→⬁ easily recovers the usual results, as the i factor inside the -functions guarantees that, in this limit, only a term proportional to共n/tn兲共m/sm兲exp关−4ts兴兩
t=s=0survives.
Anyhow, with the above expressions any term共fn, fm兲 can be calculated as a sum of-function
derivatives evaluated at zero. The particular situation m = 0, for example, for N even, is quite instructive. In this case
共fn, f0兲 = 3/2 N n tn3
冉
i⑀t , 2i N冊
3冉
⑀t , 2i N冊
t=0,共fn, f0兲 =
冏
3/2 N兺
j=0 n冉
n j冊
i j j tj3冉
⑀t , 2i N冊
冏
t=0冉
冉
n−j tn−j3冉
⑀t , 2i N冊
冏
t=0冊
,and it is immediate to see that all n=odd terms are zero. For n = 2共and for all even numbers not multipliers of 4兲, the symmetry of the binomial term and the multiplicity of the powers of i lead to a pairwise cancellation of all non-zero terms. For n = 4 共and its multipliers兲, the situation is different. The simplest case is n = 4,
共f4, f0兲 =
冏
3/2 N兺
j=0 4冉
4 j冊
i j j tj3冉
⑀t , 2i N冊
冏
t=0冋
冋
4−j t4−j3冉
⑀t , 2i N冊
冏
t=0册
共f4, f0兲 = 3/2 N再
23冉
0, 2i N冊
3⬙⬙
冉
0, 2i N冊
− 6冋
3⬙
冉
0, 2i N冊
册
2冎
.This term 共with proper normalization兲 goes to zero quite fast with increasing N. In fact, for N = 10 it is already of order of 10−6. On the other hand, it is immaterial to discuss the case N = 4共or
smaller兲, as in this situation the distinct eigenvalues of the Fourier operator are enough to guar-antee orthogonality of the whole set. Considering all this, it comes down to, literally, one-half a dozen different values of the dimensionality N共the range 关5,10兴兲 for which a significant deviation from the “expected” results共that is, orthogonality兲 can be observed.
IV. DFT AND WIDTH INVERSION
Starting from the own definition of the3-function, Eq.共4兲, with苸R, a fractional shift of
the3 function can be calculated,
3
冉
z + k N, i2 N冊
=␣=−⬁兺
⬁ exp冋
− N 2␣2册
exp冋
2i␣冉
z +k N冊
册
,where k is an integer. The sum over兵␣其 can be broken into modulo N equivalence classes as
3
冉
z + k N, i2 N冊
=兺
j=0 N−1兺
=−⬁ ⬁ exp冋
− N 2共j +N兲2册
exp冋
2i共j +N兲冉
z + k N冊
册
.Conveniently regrouping the terms one gets
3
冉
z + k N, i2 N冊
=兺
j=0 N−1冉
=−⬁兺
⬁ exp关
−N22兴
exp关
2i共i2j + Nz兲兴
冊
⫻exp
冋
−N
2j2+ 2ijz +2i N jk
册
,where the term inside the brackets can be identified as3-function,
3
冉
z + k N, i2 N冊
=兺
j=0 N−13共i2j + Nz,iN2兲exp
冋
−
N
2j2+ 2ijz +2i N jk
册
.Use of property共6兲 leads to
3
冉
z + k N, i2 N冊
= 1冑
N2兺
j=0 N−1 3冉
iz 2− j N, i N2冊
exp冋
− N 2 z 2+2i N jk册
共10兲3
冉
iz 2− k N, i N2冊
=冑
N 2兺
j=0 N−1 3冉
z + j N, i2 N冊
exp冋
N 2 z 2− 2i N jk册
. 共11兲Particular cases of these equations are most interesting, and a lot of peculiar relations can be obtained with the different possible choices of z , k and. Two straightforward examples are: First, setting z = 0 in共10兲, 3
冉
k N, i2 N冊
= 1冑
N兺
j=0 N−1 3冉
j N, i N2冊
exp冋
2i N jk册
, 共12兲and, according to Eq.共7兲, the3-function on the left-hand side has width, while the one under
the action of the DFT has width −1. This property is the obvious discrete counterpart of the well-known behavior of the Gaussian function under the usual Fourier transform.
The case k = 0, by its turn, after some manipulation gives
3共Nz,iN2兲 =
冑
N 2兺
j=0 N−1 3冉
z + j N, i2 N冊
. A. ApplicationWith the above results it is possible to generalize the result of Ref. 6 in a straightforward way. Introducing fn共j,兲 =
冑
N 冏
n tn3冉
j N− ⑀ t, i2 N冊
exp关t 2兴冏
t=0 , its DFT can be directly calculated,fn共k,兲 = 1
冑
N兺
j=0 N−1 exp冋
2i N jk册
fn共j,兲, fn共k,兲 = 1冑
n tn兺
j=0 N−1 exp冋
2i N jk册
冏
3冉
j N− ⑀ t, i2 N冊
exp关t 2兴冏
t=0 , and use of Eq.共11兲 together with change of variables from t to it leads tofn共k,兲 =
冑
Nin n tn冏
3冉
k N− ⑀t , i N2冊
exp关t 2兴冏
t=0 , thus fn共k,−1兲 = in兺
j=0 N−1 exp冋
2i N jk册
fn共j,兲, 共13兲which reproduces Eq.共1兲 for= 1. From this relation, most identities obtained in Ref. 6 may also be generalized.
V. TWO VARIABLE’S DFT
Yet another generalization of the main result of Ref. 6 regards a two variable DFT, which, for the sake of briefness, here it will be merely verified. Apart from the obvious product solution
fm共j兲fn共l兲, if one considers the quantity
Fm,n共j,l兲 =
兺
k=0 N−1 fm共k兲fn共k − l兲exp冋
2i N jk册
, which obeys 共Fm,n共j,l兲兲*= Fm,n共j,l兲exp冋
2i N jl册
,use of Eq.共1兲, and some simple manipulations lead to the nontrivial result 兩Fm,n共j,l兲兩2= 共− i兲m+n N a,b=0
兺
N−1 兩Fm,n共a,b兲兩2exp冋
2i N 共ma + nb兲册
.As in the one variable case, these states obey
兺
j,l=0 N−1
兩Fm,n共j,l兲兩2兩Fm⬘,n⬘共j,l兲兩2=␦m,m⬘␦n,n⬘, m + n⫽ m
⬘
+ n⬘
共mod 4兲,which imply a multitude of relations involving derivatives of the 3-functions 共or the Hermite
polynomials兲. Motivated by the preceding section, it should be investigated whether this relation holds for m + n = m
⬘
+ n⬘
共mod 4兲.VI. FURTHER RELATIONS INVOLVING THE WIDTH
So far it has been seen that to break up the infinite sum present in the definition of the Jacobi
3-function leads to interesting properties of this very function. In order to further explore this
technique, from Eq.共7兲 it is straightforward to write
3
冉
z L, i2 L冊
=冑
L ␣=−⬁兺
⬁ exp冋
− L冉
z L+ ␣ 冊
2册
, 共14兲with L a positive real number. Choosing integer, it is possible to break the sum over 兵␣其 into moduloequivalence classes
3
冉
z L, i2 L冊
=冑
L 兺
j=0 −1兺
=−⬁ ⬁ exp冋
− L冉
z L+ j + 冊
2册
, 3冉
z L, i2 L冊
=冑
L 兺
j=0 −1兺
=−⬁ ⬁ exp冋
− L冉
z + jL L +冊
2册
, and the infinite sum can be identified as a3,3
冉
z L, i2 L冊
= 1 兺
j=0 −1 3冉
z + jL L , i L冊
. 共15兲It is quite interesting to set z = zabove and observe that
3
冉
z L, i2 L冊
= 1 兺
j=0 −1 3冉
z L+ j , i L冊
,3
冉
2z L, 4i L冊
= 1 2冋
3冉
z L, i L冊
+3冉
z L+ 1 2, i L冊
册
, 3冉
2z L, 4i L冊
= 1 2冋
3冉
z L, i L冊
+4冉
z L, i L冊
册
.Similar reasoning would lead to the complementary relation
3
冉
z L, i L冊
= 1 兺
j=0 −1 3冉
z L+ j , i L2冊
. 共16兲And again, the particular case= 2 gives
3
冉
z L, i L冊
= 1 2冋
3冉
z 2L, i 4L冊
+4冉
z 2L, i 4L冊
册
.Equations共15兲 and 共16兲 can be combined to provide an alternative width inversion relation
3
冉
z L, i2 L冊
= 1 2兺
j,j⬘=0 −1 3冉
z L+ j⬘
+ j 2, i L2冊
. VII. CONCLUSIONSThe results here presented seem to argue in favor of one basic point: The Jacobi3-function,
together with the DFT, plays, in finite dimensional spaces, the same role played by the Gaussian function in conjunction with the usual Fourier transform. Concerning quantum mechanics, Schwinger has already noted that, if the families of eigenstates of two different observables are connected via DFT, then those observables share a maximum degree of incompatibility.5In this connection, the width inversion relation obeyed by the fn共j,兲 functions strongly suggests that one
may be able to construct, for finite dimensional spaces, states which behavior resembles that of the continuous minimum uncertainty states.
However, such a reasoning meets an important hindrance if one considers that the orthogo-nality of the DFT’s eigenstates ultimately fails. It is a fact, however, that with increasing N it becomes, in a numerical sense, true, and in this case the N→⬁ limit is reached, as witty as it may sound, somewhere near one dozen. This fact may illustrate a true finite dimensional idiosyncrasy, or it might lead one to look for the possibility of finding different sets of DFT’s eigenstates, an issue which is a matter of current research.
ACKNOWLEDGMENT
This work is supported by FAPESP under Contract No. 03/13488-0. The author is grateful to E.C. da Silva and D. Galetti for reading the paper and to D. Nedel and A. Gadelha for pertinent suggestions.
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