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Heisenberg Inequalities for Wavelet States
APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS ARTICLE NO.
4, 119–146 (1997)
HA960207
Heisenberg Inequalities for Wavelet States Guy Battle Mathematics Department, Texas A&M University, College Station, Texas 77843
Communicated by Ingrid Daubechies Received September 27, 1995; revised December 12, 1996
We prove a number of uncertainty results for wavelet states, the simplest one being that if a wavelet state is real-valued or, more generally, has zero expected momentum, then the Heisenberg uncertainty is at least 32 instead of the universal 12. For wavelet states having a very mild nth-order decay property, we establish a similar result for the uncertainty based on the nth powers of position and momentum, with a lower bound that grows rapidly with n. The proof is extremely elementary. Other Heisenberg inequalities are proven which involve the deviations about the origin of phase space rather than about the mean position of the wavelet in phase space, and the scaling generator plays an even more direct role than in the result mentioned above. The proof is still very elementary, combining the interscale orthogonality property with an iterated application of Rolle’s Theorem. Naturally, the lower bounds are much greater for these deviations about the origin of phase space, but this yields consequences for how much ‘‘off-center’’ a wavelet must be if its uncertainty is approximately minimized. q 1997 Academic Press
1. INTRODUCTION
The Heisenberg inequality asserts that for a quantum-mechanical state of a particle, the product of the standard deviations of the position and momentum is always bounded below by 12 (in one dimension). Actually, the expected value of the scaling generator contributes to the lower bound, but it is usually discarded because it is zero for many states. On the other hand, wavelet decompositions—which are based on scaling—have been very useful in dealing with phase space localization problems that can be traced back to the Heisenberg Uncertainty Principle. This author began searching for a deeper connection between scaling and the position-momentum duality, and the collection of Heisenberg inequalities derived here for N-adic wavelets are a consequence of this search. Uncertainty principles in the wavelet context have been considered before [1, 2]. After all, the original motivation of wavelet decompositions was to localize phase space in the best possible way. Most recently, Chui and Wang have shown that for 119 1063-5203/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.
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a special but useful class of wavelets, the standard deviation in phase space increases without bound as the wavelets are made arbitrarily smooth [3, 4]. Obviously, such a result cannot hold for wavelets in general, because the Meyer wavelet [5] is a class C ` function with finite uncertainty. As one of our results in this paper we establish a somewhat different type of result. It applies to virtually all interscale-orthogonal wavelets that one could wish to construct, while it establishes lower bounds on higher order phase space deviations instead. Throughout this paper, we restrict our attention to one dimension only. We introduce the momentum operator P Å 0i(d/dx) and the position operator X, which is multiplication by x. Our starting point is the basic commutation relation [P, X] Å 0i,
(1.1)
which yields alternate expressions for the scaling generator: SÅ
1 (PX / XP) 2
Å PX /
1 i 2
Å XP 0
1 i. 2
(1.2)
The self-adjoint operator S generates the unitary group of scalings in the sense of Stone’s Theorem. Thus (eilSc)(x) Å el/2c(elx),
(1.3)
where c is a one-dimensional wave function normalized in L2(R). If »A…c Å (Ac, c) for observables A, then the Heisenberg inequality includes the expected value »S…c of this generator in the lower bound
Z
sc(P)sc(X) § 0
Z
1 i / »S…c 0 »P…c»X…c , 2
(1.4)
where sc denotes the standard deviation sc(A) Å »(A 0 »A…c)2…c.
(1.5)
The real contribution to the modulus in (1.4) is normally thrown away, but the connection to scaling is there. DEFINITION. A normalized square-integrable function c is an N-adic wavelet state if there is an integer r0 such that c(x) is orthogonal to N r/2c(N rx 0 N r0m) for m √ Z, r √ Z"{0}. N r0 is the length scale of the wavelet for the smallest such r.
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HEISENBERG INEQUALITIES FOR WAVELET STATES
Remark. We are assuming neither orthogonality of the discrete translates on the same scale, nor completeness of the set of functions generated—only interscale orthogonality. Thus, for any orthonormal wavelet basis [6], c can be an arbitrary normalized linear combination of basis elements from the same scale. In particular, an N-adic wavelet state c satisfies »eir(ln N)S…c Å (eir(ln N)Sc, c) Å 0
(1.6)
for r √ Z"{0}. This will play an important role in proving some of our results. THEOREM 1.1.
If c is an N-adic wavelet state, then 2 1/2 »P2…1/2 c »X …c §
q 1 (1 0 1/ N)01. 2
(1.7)
Remark. This lower bound is larger than the universal lower bound 1/2 for all choices of scale factor N, and it obviously approaches 1/2 as N r `. Moreover, in the N Å 2 case (dyadic scaling) the lower bound is É1.706. Proof.
The key is the identity d l/2 0ilS (e »e …c) Å 0iel/2»(S / i/2)e0ilS…c dl Å 0iel/2»XPe0ilS…c.
(1.8)
Combining this with eilSPe0ilS Å e0lP
(1.9)
and the fundamental theorem of calculus, we obtain q
N»e0i(ln N)S…c 0 1 Å 0i
*
ln N
e0l/2»Xe0ilSP…cdl.
(1.10)
0
On the other hand, »e0i(ln N)S…c Å 0
(1.11)
because c is an N-adic wavelet state. It follows from the Schwarz inequality that 1 £ \Pc\ \Xc\
*
ln N
e0l/2dl
0
q
Å 2(1 0 1/ N)\Pc\ \Xc\,
and so we have the desired inequality. j
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Remark.
This argument easily extends to a proof of the generalization r0 2 1/2 »P2…1/2 c »(X 0 N m) …c §
S
D
1 1 10q 2 N
01
for all m √ Z. The point is that S can be replaced by the operator S 0 N r0mP because N r0 is the length scale of the wavelet state c. We shall look at various generalizations of these inequalities to higher orders. A first approach in Section 2 leads to: THEOREM 1.2.
If c is an N-adic wavelet state, then 2 3 4 1/2 »P4…1/2 0 , c »X …c § 2 (ln N) 4 6 1/2 »P6…1/2 c »X … c §
(1.12)
9 15 0 . 2 (ln N) 8
(1.13)
We regard these inequalities as second and third order Heisenberg relations. Our method of estimation appears to break down for order n § 4, but Section 3 takes a different approach and proves: THEOREM 1.3.
If c is an N-adic wavelet state, then
2n 1/2 »P2n…1/2 c »X … c §
F*
1
N01
dt 1
*
t1
N02
dt2rrr
*
tn01
N0n
G
dq tn tn
01
.
(1.14)
,
(1.15)
In the second and third order cases, this inequality becomes 4 1/2 »P4…1/2 c »X … c §
»P6…1/2c»X6…1/2 c §
F
F
G
4 (1 0 N03/2) 0 2N01(1 0 N01) 3
S
D
01
G
8 4 (1 0 N021/2) / (1 0 N03) 2N07/2 0 N05 0 N03/2(1 0 N06) 15 3
01
,
(1.16)
respectively. In the dyadic case (N Å 2), the lower bounds are É2.752 and É3.298, respectively. Actually, these are not as good as the dyadic cases of (1.12) and (1.13), respectively. For N § 3, however, the situation changes. For N Å 3, the inequalities (1.15) and (1.16) produce lower bounds É1.580 and É5.590, respectively. The inequality (1.15) is already better than (1.12), but (1.16) is still worse than (1.13). For the N Å 4 case, (1.16) provides a lower bound É2.367, while (1.13) provides a lower bound slightly greater than 21/8. For N § 5, the lower bound given by (1.16) is finally the better one. If in addition, we assume the fairly mild condition ÉXÉn/1c √ L2(R),
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(1.17)
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then it is shown in Section 5 that (1.14) can be strengthened to a lower bound whose n Å 1, 2, 3 cases are 2 1/2 »P2…1/2 c »X …c §
4 1/2 »P4…1/2 c »X …c §
6 1/2 »P6…1/2 c »X …c §
3 , 2
(1.18)
S S
169 3 q 4 / 10 / 16 2 (ln N)4
D
1/2
,
(1.19)
D
9549 45 q 149 / 259 / 64 8 4(ln N)4
1/2
,
(1.20)
respectively. As in the 1st-order case, these higher order inequalities still hold with the replacement X ° X 0 N r0m, m √ Z. Now it can certainly be argued that all of these lower bounds are due, in part, to the fact that even when a wavelet state is translated as close as possible to the origin, using the discrete translates associated with its length scale, it is typically off center. As far as uncertainty principles are concerned, it is always more to the point to obtain lower bounds on deviations from the mean position (»X…c, »P…c) in phase space. We can obtain such bounds in the case where »P…c Å 0, and we do so in Section 4. This includes real-valued as well as wavelet states that are either symmetric or anti-symmetric about their barycenters in position space. We prove a fundamental result: THEOREM 1.4.
Let c be an N-adic wavelet state such that »P…c Å 0. Then sc(P)sc(X) §
3 . 2
(1.21)
We shall prove this theorem in the course of our discussion about wave functions whose zeroth-order moments vanish. First we observe the following result, implicit in [7]: THEOREM 1.5. Let c be an N-adic wavelet state in arbitrary dimension. If c is both integrable and continuous, then
* c(x )dx Å 0. û
û
(1.22)
In one dimension we have the refinement: THEOREM 1.6.
Let c be an N-adic wavelet state. If Xc and Pc both lie in L2(R),
then c is integrable and * c(x)dx Å 0. We prove this in the Appendix. The point here is that if the hypothesis fails, then sc(P)sc(X) Å ` anyway. It is interesting that we can prove (1.21) in a short, clever way if we make the hypothesis just a little stronger. Suppose the state c has the property
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w √ L2(R).
c Å Pw,
(1.23)
This stronger condition alone is enough, whether c is a wavelet state or not. To show that
2 1/2 »P2…1/2 c »(X 0 a) …c §
3 2
(1.24)
for all real a, we may assume without loss that a Å 0, because the property (1.23) is translation-invariant. On the other hand, \Xc\ Å \XPw\ Å
Å
ZZS ZZS
D ZZ D ZZ
S/i
1 w 2
S0i
1 w 2
.
(1.25)
It follows from the Schwarz inequality that
§
Å
ZZS
D ZZ ZS S D DZ ZS S D DZ S0i
1 w 2
c, P S 0 i
1 w 2
\Pc\ \Xc\ Å \Pc\
c, S 0 i
3 Pw 2
(1.26)
because [P, S] Å 0iP.
(1.27)
Hence
Z
\Pc\ \Xc\ § (c, Sc) 0 i §
Z
3 (c, c) 2
3 , 2
(1.28)
and so the estimation is complete. The stronger condition (1.23) follows from a very mild decay condition:
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HEISENBERG INEQUALITIES FOR WAVELET STATES
Let c be a state such that ÉXÉ1/1 c √ L2(R) for some 1 ú 0 and
LEMMA 1.7.
* c(x)dx Å 0.
(1.29)
Then c Å Pw for some w √ L2(R). We prove this technical lemma in the Appendix. In Section 4 we shall generalize the inequality (1.21)—for this kind of decay condition—to nth-order uncertainties. If we introduce the notation sc(n)(A) Å »(A 0 »A…c)2n…c1/2n
(1.30)
for an arbitrary observable A, we have: THEOREM 1.8. √ L2(R), then
Let c be an N-adic wavelet state such that »P…c Å 0. If ÉXÉn/1c
F∏ S D S ∑ ∏ S D D G n
sc(n)(P)sc(n)(X) §
k0
kÅ1
1 /n 2
n
k0
lÅ1 kxl
1 2
2
1/2
1/n
.
(1.31)
Note that we have here again the decay condition (1.17). In the 1st-order case, the result is just Theorem 1.4 weakened by this additional hypothesis. In Section 4 we prove Theorem 1.4, using a crucial observation by Fran Narcowich, and discuss the problem of extending to arbitrary n his strategy for avoiding this condition.
2. HIGHER ORDER HEISENBERG INEQUALITIES
Uncertainty relations involving powers of P and X are nothing new for arbitrary quantum-mechanical states, but they involve special combinations with an algebraic structure that is intimately related to the harmonic oscillator. One is often interested in the observables
HÅ
1 2 1 2 P / X , 2 2
(2.1)
LÅ
1 2 1 2 P 0 X . 2 2
(2.2)
H is the harmonic oscillator Hamiltonian and L is the harmonic oscillator Lagrangian. It is well known that a Lorentz group structure links the scaling generator S to the operators L and H. The harmonic oscillator time-evolution given by the one-parameter unitary group U(t) Å e0itH
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(2.4)
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interpolates the Fourier transform:
S D
U
1 p c Å cP . 2
(2.5)
Consider the relations U(t)01PU(t) Å P cos t / X sin t,
(2.6.1)
U(t)01XU(t) Å X cos t 0 P sin t,
(2.6.2)
from which it is easy to derive U(t)01LU(t) Å L cos 2t / S sin 2t,
(2.7.1)
U(t)01SU(t) Å S cos 2t 0 L sin 2t.
(2.7.2)
(Commutations turn out to be irrelevant in this case.) In particular, the scaling generator and the harmonic oscillator Lagrangian are related by the half-Fourier transform U(14p). Given (2.6), the relations (2.7) are certainly not surprising from a classical point of view, because the classical observables 12 p2 0 12 x2 and px are just hyperbolic coordinates in phase space. Now the observables H, L, and S are related almost (but not quite) as the components of angular momentum are related. It is easy to verify that 1 [S, H] Å 0iL, 2
(2.8)
1 [H, L] Å 0iS, 2
(2.9)
1 [L, S] Å iH, 2
(2.10)
and the signs make all the difference in the world. In contrast to the case of angular momentum components, two of our observables—namely L and S—have absolutely continuous spectra. This Lie algebra generates SO(1, 2) instead of SO(3) because eilSHe0ilS Å H cosh 2l / L sinh 2l, ilS
0ilS
(2.11.1)
Å L cosh 2l / H sinh 2l,
(2.11.2)
eimLHe0imL Å H cosh 2m / S sinh 2m,
(2.12.1)
eimLSe0imL Å S cosh 2m / H sinh 2m.
(2.12.2)
e Le
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HEISENBERG INEQUALITIES FOR WAVELET STATES
Now S and L are linked by a Heisenberg inequality as well. By (2.10) we have 2 1/2 »L2…1/2 c » S … c § »H … c
(2.13)
for an arbitrary state c, so if we recall the operator inequality H § 12 , it follows that 1 2 1/2 »L2…1/2 c »S … c § 2 .
(2.14)
This second-order Heisenberg inequality is part of a hierarchy of inequalities for arbitrary states, involving the creation and annihilation operators for the harmonic oscillator: 1 (X 0 iP), 2
(2.15)
1 (X / iP). 2
(2.16)
A* Å
q
AÅ
q
The basic commutation relations are: [A, A*] Å 1,
(2.17)
[H, A] Å 0A,
(2.18)
[H, A*] Å A*.
(2.19)
The relationship to the generators above are easily given: H Å A*A / Å AA* 0
SÅi LÅ
1 2 1 , 2
(2.20)
1 (A*2 0 A2), 2
(2.21)
1 (A*2 / A2). 2
(2.22)
Thus, a natural higher-order generalization of the pair {S, L} of observables is the pair {Bn , Cn} with Bn Å i Cn Å
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1 (A*n 0 An), 2
(2.23)
1 (A*n / An), 2
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(2.24)
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and by the Schwarz inequality, 2 1/2 »B2n…1/2 c »C n…c §
Å
1 É»[Bn , Cn]…cÉ 2 1 É»[An, A*n]…cÉ. 2
(2.25)
In this case, the relations of interest are
S ∏S n
AnA*n Å
∏
1 , 2
(2.26)
H0k/
1 . 2
(2.27)
kÅ1 n
A*nAn Å
D D
H/k0
kÅ1
Hence [An, A*n] Å Pn(H),
(2.28)
where al ú 0 in the polynomial n
S
Pn(z) Å ∏ z / k 0 kÅ1
D
1 0 2
n
∏ kÅ1
S
z0k/
D
1 2
[(1/2)(n01)]
å
∑
alzn02l01.
(2.29)
lÅ0
Therefore, the nth-order Heisenberg inequality in this hierarchy is obtained from applying H § 12 to the polynomial in H with positive coefficients. We get
SD
2 1/2 »B2n…1/2 c »Cn…c § Pn
1 , 2
(2.30)
and this concludes our review. 2n 1/2 There are universal lower bounds on the simpler nth-order products »P2n…1/2 c »X … c as well. Indeed, 2 n/2 »A2n…1/2 c § »A … c
(2.31)
for all self-adjoint operators A, so we obtain 2n 1/2 n »P2n…1/2 c »X …c § 1/2
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(2.32)
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if we apply the basic inequality 2 1/2 »P2…1/2 c »X …c § 1/2.
(2.33)
For the n Å 2 case, we can obtain another lower bound in terms of the scaling generator S. Since
S Å XP 0 i
1 , 2
(2.34)
we have
S2 Å XPXP 0 iXP 0
1 4
Å X 2P2 0 i2XP 0
1 4
Å X 2P2 0 i2S /
3 4
(2.35)
from which it follows that
»S2…c 0
3 / i2»S…c Å »P2c, X 2c… 4
(2.36)
D
(2.37)
for an arbitrary state c. Hence
S
»S2…c 0
3 4
2
/ 4»S…2c Å É(P2c, X 2c)É2,
and we can now apply the Schwarz inequality to obtain
»S2…c 0
3 £ \P2c\ \X 2c\ 4 4 1/2 Å »P4…1/2 c »X … c .
(2.38)
Combining this with the operator inequality
1 0 cos lS £
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1 2 2 lS, 2
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(2.39)
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where l is a real parameter, we get »cos lS…c /
S
D
1 2 3 4 1/2 l / »P4…1/2 § 1. c »X …c 2 4
(2.40)
This estimation does not even recover the universal lower bound (2.32). But now suppose c is an N-adic wavelet state. What restriction does this impose on the state? We proved in [7] that such a function c cannot have exponential decay in both position space and momentum space, but such a duality property is qualitative. A more quantitative restriction can now be derived by combining (2.40) with »e{i(ln N)S…c Å 0,
(2.41)
which is (1.6) with r Å {1. Thus, if we set l Å ln N in (2.40), the cosine term vanishes, and we obtain
S
D
3 1 4 1/2 § 1. (ln N)2 / »P 4…1/2 c »X …c 2 4
(2.42)
Therefore 4 1/2 »P 4…1/2 c »X … c §
2 3 0 (ln N)2 4
(2.43)
for such a state c. This is better than the universal lower bound of 1/4 only for N £ 4. A decimal approximation to this lower bound is 3.4145 in the case N Å 2. 2n 1/2 In the pursuit of lower bounds on »P2n…1/2 c »X …c for arbitrary order n, we will n n constantly use an operator identity for P X , introduced in the next section. In particular, P 3X 3 Å S3 0
S
D
23 9 15 , S 0 i S2 0 4 2 8
(2.44)
from which we can derive the inequality 6 1/2 »P 6…1/2 c »X … c §
9 15 0 2 (ln N) 8
(2.45)
for an N-adic wavelet state c. This concludes our proof of Theorem 1.2. This method of estimation appears to break down for order n § 4. We pursue another method in the next section. 3. AN nTH-ORDER HEISENBERG INEQUALITY FOR WAVELET STATES
How can we generalize the proof of Theorem 1.1 to obtain higher order Heisenberg inequalities? The key to the generalization is the identity
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HEISENBERG INEQUALITIES FOR WAVELET STATES n
P nX n Å
∏ (S 0 i(k 0 1/2)),
(3.1)
kÅ1
which can be verified by induction, using the obvious commutation relation [P, S] Å 0iP.
(3.2)
The simplest nth-order generalization of (1.8) appears to be
S D
e0l el
d n l/2 0ilS (e »e …c) Å (0i)ne(n01/2)l dl
K S S DD L n
∏
S/i k0
kÅ1
1 2
e0ilS
c
Å (0i)ne(n01/2)l»X nP ne0ilS…c.
(3.3)
eilSP ne0ilS Å e0nlP n,
(3.4)
On the other hand, (1.9) implies
so we have
S D
e0l el
d n l/2 0ilS (e »e …c) Å (0i)ne0l/2»X ne0ilSP n…c. dl
(3.5)
This formula is essential to our strategy in proving Theorem 1.3. Recall that el/2»e0ilS…c vanishes at the points l Å r ln N, r √ Z"{0}, by the wavelet property e0ir (ln N)Sc ⊥ c. Clearly we can apply Rolle’s Theorem to the real and imaginary parts of this expectation. To this end, we define u(l) Å el/2Re»e0ilS…c ,
(3.6.1)
£(l) Å el/2Im»e0ilS…c .
(3.6.2)
More generally, we define
S D S D
uk(l) Å e0l el
d k u(l), dl
(3.7.1)
£k(l) Å e0l el
d k £(l). dl
(3.7.2)
We concentrate on u(l) because the real and imaginary parts are decoupled in this iteration, and the real part will provide all of the lower bound; the imaginary part turns out to be useless. Applying Rolle’s Theorem to the property
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r √ Z"{0},
u(r ln N) Å 0,
(3.8)
we obtain a sequence (a1(1) , a2(1) , a3(1) , . . .) such that r ln N õ ar(1) õ (r / 1)ln N,
(3.9)
(1) r
u1(a ) Å 0.
(3.10)
Applying Rolle’s Theorem to these zeros of elu1(l), we now obtain another sequence (a1(2) , a2(2) , a3(2) , . . .) such that (1) ar(1) õ ar(2) õ ar/1 ,
(3.11)
u2(ar(2)) Å 0.
(3.12)
Obviously, we can iterate this procedure indefinitely. In general, we have a sequence (a(1k ) , a(2k ) , a(3k ) , . . .) satisfying the conditions k01) a(r k01) õ a(rk ) õ a(r/1 ,
(3.13)
(k) r
uk(a ) Å 0.
(3.14)
Now we are particularly interested in the smallest numbers in each sequence— namely, a1(1), . . . , a(1n01). It is easy to infer that a(1k ) õ (k / 1)ln N,
(3.15)
ln N õ a1(1) õ a1(2) õ rrr õ a(1n01).
(3.16)
Since the real-valued function £(l) also has the property (3.8), we can generate zeros b1(1) , . . . , b(1n01) of £1(l), . . . , £n01(l), respectively, which satisfy the conditions (3.15) and (3.16). LEMMA 3.1.
With the numbers a1(1) , . . . , a(1n01) derived as above,
*
(01)n Å
*
ln N
dl1e0l1
a1(1)
*
dl2e0l2rrr
l1
0
a1( n01)
dlnun(ln).
(3.17)
ln01
With the similarly derived numbers b11 , . . . , b1n01, 0Å
*
ln N 0l1
dl1e
dl2e
0l2
rrr
l1
0
Proof.
*
b1(1)
*
b1( n01)
dln£n(ln).
(3.18)
ln01
Iterate the formulas
*
a1( k01)
dlkuk(lk) Å 0elk01uk01(lk01),
(3.19.1)
lk01
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HEISENBERG INEQUALITIES FOR WAVELET STATES
*
b1( k01)
133
dlk£k(lk) Å 0elk01£k01(lk01),
(3.19.2)
dl1u1(l1) Å 0u(0) Å 01,
(3.20.1)
dl1£1(l1) Å 0£(0) Å 0
(3.20.2)
lk01
where
* *
ln N
0 ln N
0
are the last integrations. j Now by Equation (3.5), un(ln) Å e0(1/2)ln Re»(0i)nP ne0ilnSX n…/
(3.21)
Éun(ln)É £ e0(1/2)ln\P nc\ \X nc\
(3.22)
which implies
by the Schwarz inequality. Remark. Equation (3.17) supplies us with the lower bound, so we discard £n(ln) and use only un(ln). While b(1k ) does not coincide with a(1k ) in general, Eq. (3.18) shows that the imaginary part £n(ln) cannot add much to the lower bound. We included Eq. (3.18) for the sake of completeness. Combining Eq. (3.17) with (3.6.1), (3.7.1), and the bound (3.22), we obtain the inequality
* £ \P c\ \X c\ * Å \P c\ \X c\ *
ln N
1 £ \P c\ \X c\ n
n
dl1e
0l1
ln N
dl1e0l1
n
* rrr*
rrr
1
n
N01
dt1
*
2 ln N
02
n ln N
dl2e0l2
dlne01/2ln
ln01
t1
N
dlne01/2ln
ln01
l1
0
n
dl2e
a1(n01)
0l2
l1
0
n
* *
a1(1)
dt2rrr
*
tn01
N
1 dt n , tn
q
0n
(3.23)
where we have applied (3.15) and made the change of variable tk Å e0lk. Thus we have proved Theorem 1.3—i.e., we have the inequality (1.14) for N-adic wavelet states c. Remark.
The limit of the multiple integral as N r ` is
* dt * 1
t1
1
0
0
dt2rrr
*
tn01
0
1 dtn Å tn
q
n
1
∏ k 0 1/2 ,
kÅ1
so the limit of the lower bound is ∏nkÅ1 (k 0 12).
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(3.24)
134
GUY BATTLE
4. AN nTH-ORDER UNCERTAINTY PRINCIPLE FOR »P… Å 0 WAVELET STATES
In the Introduction we proved that
2 1/2 »P 2…1/2 c »X … c §
q 1 (1 0 1/ N)01 2
(4.1)
for such a wave function c. More generally, in the previous section we showed that an N-adic wavelet state c satisfies the Heisenberg inequality
2n 1/2 »P 2n…1/2 c »X … c §
F*
1
N01
dt1
*
t1
N02
dt2rrr
*
G
tn01
dtn q tn
N0n
01
(4.2)
for all positive integers n. However, these products are phase space deviations from the fixed point (0,0) for scaling rather than the phase space deviations from the mean coordinates (»X…c , »P…c) of the state. In this section we prove: THEOREM 4.1.
If c is an N-adic wavelet state with ÉXÉn/1c √ L2(R), then n
2n 1/2 »P 2n…1/2 c »(X 0 a) …c § n( ∑
n
∏ (k 0 1/2)2)1/2 / ∏
lÅ1 kxl
kÅ1
S D k0
1 2
(4.3)
for all real a and positive integers n. We have Theorem 1.8 as an immediate consequence. The key to proving the theorem is to combine a result for general states with a variation on a previous result [7] for wavelet states. Both results address the issue of vanishing moments for the wave function. THEOREM 4.2. and
Let c be a normalized element of L2(R) such that ÉXÉn/1c √ L2(R)
* x c(x)dx Å 0, m
m £ n 0 1.
(4.4)
Then for all real a, n
2n 1/2 »P2n…1/2 c »(X 0 a) …c § n( ∑
lÅ1 kxl
Proof.
n
∏ (k 0 1/2)2)1/2 / ∏ kÅ1
S D k0
1 . 2
(4.5)
By the triangle inequality and by convexity, we have Éx 0 aÉn/1 £ 2n/101(ÉxÉn/1 / ÉaÉn/1).
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(4.6)
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HEISENBERG INEQUALITIES FOR WAVELET STATES
Since we also have
*
SD
m
m (0a)m0m m
(x 0 a)mc(x)dx Å ∑ mÅ0
* x c(x)dx, m
(4.7)
the hypothesis is satisfied by arbitrary translates of c, so we need only establish (4.5) for a Å 0. By the vanishing moments condition, there is an element w √ L2(R) such that P nw Å c.
(4.8)
That w is indeed square-integrable follows from the condition ÉXÉn/1c √ L2, and the proof is an obvious generalization of the proof of Lemma 1.7. Thus X nc Å X nP nw n
Å
∏ (S / i(k 0 1/2))w,
(4.9)
kÅ1
where we are using the operator identity (3.1). On the other hand,
ZZ∏ n
ZZ S∏ S S D D D n
2
(S / i(k 0 1/2))w
kÅ1
2
1 2
S2 / k 0
Å
kÅ1
w, w
n
n
§ \w\2
∏ (k 0 1/2)2 / \Sw\2 ∑ ∏ lÅ1 kxl
kÅ1
Å
ZZS
S DD ZZ S∑ ∏ S D D n
anS { i
∏
kÅ1
n
an Å
k0
k0
lÅ1 kxl
1 2
2
1 2
w
S D k0
1 2
2
2
,
(4.10)
1/2
.
(4.11)
The choice of sign affects the lower bound obtained when we apply the Schwarz inequality. The better choice is clearly
ZZS
\P nc\
n
anS 0 i
Z Z Z
∏ kÅ1
S DD ZZ k0
1 2
w
S D Z S ∏ S DD S ∏ S DDZ
§ an(P nc, Sw) 0 i
n
∏
k0
kÅ1
1 (P nc, w) 2
n
Å an(c, SP nw) 0 i nan /
k0
kÅ1
n
Å an(c, Sc) 0 i nan /
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k0
kÅ1
1 2
1 2
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(c, P nw)
Z n
§ nan /
∏ kÅ1
achaa
S D k0
1 , 2
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(4.12)
136
GUY BATTLE
because [P n, S] Å 0inP n.
(4.13)
Combining (4.12) with (4.9) and (4.10), we obtain n
\P nc\ \X nc\ § nan /
∏ kÅ1
S D k0
1 , 2
(4.14)
which is the desired inequality. j On the other hand, we proved in [7] that: THEOREM 4.3. Let c be an N-adic wavelet state in arbitrary dimension such that û û û cP ( p ) is class C n/1 and (1 / É p É)n/1cO (p ) is integrable. Then
* x c(x )dx Å 0 a
û
û
(4.15)
for all multi-indices a such that ÉaÉ £ n. Remark. In [7] we assumed that N Å 2 and that discrete translates on the same scale were orthogonal, but neither assumption was necessary. Actually, this theorem is just a little too weak for our purpose, but we can prove a slightly stronger result in our one-dimensional case. THEOREM 4.4.
Let c be an N-adic wavelet state in one dimension such that cO (n/1)(p) √ L2(R),
(1 / ÉpÉ)n/1cO (p) √ L2(R). Then
* x c(x)dx Å 0, m
m £ n.
(4.16)
We prove this theorem in the Appendix, but the argument is similar in spirit to the proof of Theorem 4.3. Proof of Theorem 4.1. We assume without loss that P nc √ L2(R). Otherwise, the 2n 1/2 product »P 2n…1/2 c »(X 0 a) …c is infinite, in which case the lower bound is trivial. Now the Ho¨lder inequality 2n »A2m…n/m c £ »A … c
(4.17)
implies that the functions ÉPÉmc are elements of L2(R) for m £ n as well, so (1 / ÉPÉ)nc lies in L2(R). Since
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137
(1 / ÉXÉn/1)c √ L2 c Xnc √ L2 c cO (n)(p) √ L2,
(4.18)
we now apply Theorem 4.4 with n Å n 0 1 to infer that
* x c(x)dx Å 0, m
m £ n.
(4.19)
This, in turn, implies (4.5) because we now have the hypothesis of Theorem 4.2. j We conclude this section with a proof of Theorem 1.4 and a remark on the possibility of dropping the decay condition from the hypothesis of Theorem 1.8. Given Theorem 1.6, it is just a matter of using (1.29) instead of (1.23), and the idea is to simply 2 1/2 minimize »P…1/2 c »X …c over all states c satisfying the constraint (1.29). (As before, (1.24) reduces to the a Å 0 case because (1.29) is translation-invariant as well.) The variational problem is quite simple: we find the stationary points of the functional \Pc\2\Xc\2 / l(\c\2 0 1) / k
* c(x)dx,
where l and k are Lagrange multipliers. The result is an inhomogeneous equation: \Pc\2X 2c / \Xc\2P 2c / lc /
1 k Å 0. 2
(4.20)
Since any scaling (in x) of a solution is still a solution, we can choose that scaling such that \Pc\2 Å \Xc\2 å
1 a. 2
(4.21)
Thus—with 12 a identified as the uncertainty—the equation becomes aH / lc /
1 k Å 0, 2
(4.22)
where H is the harmonic oscillator Hamiltonian. Now in the case where k Å 0, the solutions are just those normalized eigenfunctions of H that happen to satisfy (1.29)— i.e., the odd Hermite functions. The first excited state is the function among these for which the uncertainty is smallest, and that value is indeed 32. The peculiarity of this variational problem arises in the case where k x 0, and we owe a crucial observation to Narcowich. All solutions c now have the form c(x) Å 0
1 k 2
* (aH / l)
01
(x, y)dy
`
Å0
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1 w2l(x) k ∑ 2 lÅ0 (2l / 12)a / l
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* w (y)dy,
(4.23)
2l
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GUY BATTLE
where wm denotes the mth Hermite function, because
*w
Å 0,
m
odd m.
(4.24)
On the other hand, if we integrate (4.22) against c, we obtain the relation 1 2 a / l Å 0, 2
(4.25)
so we may write k c(x) Å 0 2a
q
`
∑ lÅ0
w2l(x) (2l)! (01)l l , 1 1 2l / 2 0 2 a 2 l!
(4.26)
where we have inserted the value of * w2l . If we now apply (1.29), we obtain the condition `
∑ lÅ2
1 (2l)! Å 0. 2l / 12 0 12 a 4l(l!)2
(4.27)
However, this series [8] sums to the quotient
SD
1 1 1 1 G(4 0 4 a) G 2 2 G(34 0 14 a)
of gamma functions, and for a ú 0, this can vanish only where G(34 0 14 a) has poles. Hence, a Å 3, 7, 11, . . .
(4.28)
and a Å 3 clearly yields the smallest uncertainty, which is again 32 . This completes the proof of Theorem 1.4. Remark.
The value of k is determined by the constraint \c\2 Å 1, which reduces
to k2 4a2
`
∑ lÅ0
1 (2l)! Å 1. 1 2 l (2l / 0 2 a) 4 (l!)2
(4.29)
1 2
This series is essentially the a-derivative of the above series, so we have the condition
SD
1 1 k2 1 d G(4 0 4 a) G Å 1. 4a2 2 da G(34 0 14 a)
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(4.30)
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HEISENBERG INEQUALITIES FOR WAVELET STATES
139
Remark. It is reasonable to try this strategy on Theorem 1.8 with the idea of dropping the condition (1.17), but the problem is that H is now replaced with 1 2n 1 2n P / X , 2 2 and one would have to know a lot about the eigenfunctions.
5. AN n-TH ORDER HEISENBERG INEQUALITY FOR WAVELET STATES REVISITED
We proved in Section 4 that if c is an N-adic wavelet state, then (1.14) holds. The aim of this paper is to establish a different lower bound in the case where we have the condition (1.17) as well. This additional assumption is actually very mild in the sense that, without the 1 ú 0, the failure of the condition would mean »X 2n…c Å ` anyway. In Section 5 we derived a lower bound on the function 2n 1/2 a ° »P 2n…1/2 c »(X 0 a) …c
from the assumption (1.17). We now proceed to derive an a Å 0 lower bound from (1.17), and (4.4) plays an immediate role. The vanishing moments condition together with the decay condition implies c Å P nw
(5.1)
for some w √ L2(R). It follows from the identity n
X nP n Å
∏ (S / i(k 0 1/2))
(5.2)
kÅ1
that
ZZ∏ n
\X nc\2 Å
kÅ1
Å
S
ZZ
2
(S / i(k 0 1/2))w
n
∏ (S2 / (k 0 1/2)2)w, w kÅ1
D
n
Å
∑ al(n)(S2lw, w),
(5.3)
lÅ0
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GUY BATTLE
where the positive coefficients al(n) are defined by n
n
kÅ1
lÅ0
∏ (z / (k 0 1/2)2) Å ∑
al(n)z l.
(5.4)
Now separate out the first two terms q
q
a0(n)(w, w) / a1(n)(S2w, w) Å \( a1(n)S { i a0(n))w\2
(5.5)
and split the remaining sum into odd-l and even-l sums. We have \Xnc\2 q
q
[(1/2)n]
[(1/2)(n01)]
mÅ1
mÅ1
(n) (S 4mw, w) / ∑ a2m
Å \( a1(n)S / i a0(n))w\2 /
∑
(n) a2m/1 (S 4m/2w, w),
(5.6)
where [r] denotes the greatest integer function. Thus \P nc\2\X nc\2 has three contributions, each of which we shall underestimate in a different way. The first contribution is bounded below in exactly the same way as in Section 4: q
q
q
q
\P nc\ \( a1(n)S 0 i a0(n))w\ § n a1(n) / a0(n),
(5.7)
so we move immediately to the consideration of the other two contributions. Lower bound on the even contribution.
By the operator inequality
1 2 2 l S § 1 0 cos lS, 2
(5.8)
we can underestimate the even sum in (5.6) as follows: [(1/2)n]
[(1/2)n]
(n) (S 4mw, w) § ∑ ∑ a2m
mÅ1
mÅ1 [(1/2)n]
∑
Å
mÅ1
22m (n) a2m ((1 0 cos(S ln N))2mw, w) (ln N)4m 22m (n) a2m \(1 0 cos(S ln N))mw\2, (ln N)4m
(5.9)
where we have chosen l Å ln N to exploit (1.6). Since (4.13) and (5.1) yield \P nc\ \(1 0 cos(S ln N))mw\ § É(P nc, (1 0 cos(S ln N))mw)É Å É(c, P n(1 0 cos(S ln N))mw)É Å
ZS S D DZ ZS ∑ S DS D S∑ D DZ c, 1 0 m
Å
c,
mÅ0
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1 ∑ e{i(ln N)(S0in) 2 {
m m
0
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1 2
m
P nw
m
m
e{i(ln N)(S0in)
c
, (5.10)
{
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141
HEISENBERG INEQUALITIES FOR WAVELET STATES
it follows from (1.6) that [(1/2)m]
∑
\P nc\ \(1 0 cos(S ln N))mw\ §
nÅ0
S DS D
1 m 4n 2n
2n n
(5.11)
because (1.6) obviously implies
(c, (∑ e
HS D J 0, m odd, m , m even. 1 2m
{i(ln N)(S0in) m
) c) Å
{
(5.12)
Combining (5.11) with (5.9), we obtain [(1/2)n]
[(1/2)n]
mÅ0
mÅ1
\P nc\2
(n) (S 4mw, w) § ∑ ∑ a2m
H ∑ S DS DJ
4m (n) a2m (ln N)4m
Lower bound on the odd contribution. [(1/2)(n01)]
∑
∑
mÅ0
mÅ1 [(1/2)(n01)]
∑
Å
nÅ0
1 m 4n 2 n
2
2n n
.
(5.13)
We apply (5.8) as follows:
[(1/2)(n01)] (n) a2m/1 (S 4m/2w, w) §
[(1/2)m]
mÅ1
22m (n) a2m/1(S2(1 0 cos lmS)2mw, w) l4m m 22m (n) a2m/1\(1 0 cos lmS)mSw\2, l4m m
(5.14)
where our choice l Å lm for each m is yet to be made. Now if we apply the Schwarz inequality as before, (4.13) and (5.1) now yield \P nc\ \(1 0 cos lmS)mSw\ § É(P nc, (1 0 cos lmS)mSw)É Å
ZS S ∑ D ZS ∑ S DS D ∑ S D c, 1 0 m
Å
c,
mÅ0
1 2
m m
m
e{ilm(S0in)
(S 0 in)P nw
{
0
1 2
m
m
qÅ0
m
q
DZ
DZ
ei(m02q)lm(S0in)(S 0 in)c
.
(5.15)
On the other hand, we define useful functions wm(l) such that their derivatives at l Å lm are part of this inner product expression. If we set m
wm(l) Å ∑ qÅ0 2qxm
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SD m
q
i (c, ei(m02q)l(S0in)c), m 0 2q
05-08-97 13:53:01
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(5.16)
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142
GUY BATTLE
then
S SD m
c,
m
∑
q
qÅ0
ei(m02q)lm(S0in)(S 0 in)c
Å
H
D
w*m(lm), m odd, m w*m(lm) / 1 (c, (S 0 in)c), m even. 2m
SD
J
(5.17)
The usefulness of wm(l) lies in the wavelet property wm(ln N) Å wm(2 ln N) Å 0,
(5.18)
which is obviously shared by the function m
£m(l) Å
∑ mÅ0
S DS D m m
0
1 2
m
wm(l).
(5.19)
Applying Rolle’s Theorem to Im £m(l), we choose lm such that ln N õ lm õ 2 ln N,
(5.20)
Im £*m(lm) Å 0.
(5.21)
Thus (5.17) yields
S
m
Im c, ∑ mÅ0
S DS D S D m m
0
1 2
m
m
∑
qÅ0
m
D
ei(m02q)lm(S0in)(S 0 in)c
q
[(1/2)m]
Å
∑
nÅ0
S DS D
2n n, n
1 m 4n 2n
(5.22)
so if we combine this with (5.14) and (5.15), we finally have the lower bound [(1/2)(n01)]
\P nc\2
∑
(n) a2m/1 (S 4m/2w, w)
mÅ1 [(1/2)(n01)]
∑
§
mÅ1 [(1/2)(n01)]
§
∑
mÅ1
22m (n) a2m/1 l4m m
H
[(1/2)m]
∑
nÅ0
n2 (n) a2m/1 4m(ln N)4m
S DS D J H ∑ S DS DJ 2n n n
1 m 4n 2 n [(1/2)m]
nÅ0
1 m 4n 2n
2
2n n
2
,
where we have applied (5.20) in the last step.
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(5.23)
143
HEISENBERG INEQUALITIES FOR WAVELET STATES
Pulling all the estimations together, we now have the desired result: Let c be an N-adic wavelet state such that ÉXÉn/1c √ L2(R). Then
THEOREM 5.1. q
q
n
∑
\P nc\2\X nc\2 § ( a0(n) / n a1(n))2 /
lÅ2
1 (ln N)4[(1/2)l] 1 cl(n)
(n) l
c
Å
H
H
[(1/2)[(1/2)l]]
∑
nÅ0
1 4n
J
2lal(n),
l even,
(12)l01n2al(n),
l odd,
S DS DJ [12 l]
2n
2n
n
2
,
(5.24)
(5.25)
with the coefficients given by (5.4). This inequality survives the replacement X ° X 0 mN r0 for m √ Z, where N r0 is the length scale of c. APPENDIX
We now turn to the proof of Theorem 4.4, which is actually just a modification of the central idea of [7]. First, we claim that the 0th-order moment vanishes, and this follows from inspecting the proof of Lemma 1 of [7]. Indeed, (1 / ÉpÉ)cO (p) √ L2(R) c cO (p) √ L1(R),
(A.1)
and the continuity and integrability of cO (p) are the only regularity properties that are actually used in that argument. Suppose we have shown that all moments of order m £ k 0 1 vanish, where k is a positive integer £n, and assume
* x c(x)dx x 0.
(A.2)
cO (k)(0) x 0,
(A.3)
k
This means
and by the induction hypothesis, cO (p) Å
cO (k)(0) k p / Rk(p), k!
Rk(p) Å
1 (k / 1)!
* cO
(A.4)
p
(k/1)
(p 0 q)qkdq.
(A.5)
0
This version of Taylor’s Theorem makes sense because cO (k/1) is square-integrable and
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GUY BATTLE
therefore locally integrable. The same property guarantees the continuity of cO (k) in one dimension. For a given positive integer r—with value to be chosen sufficiently large later— pick an integer m0 such that c(k)(x0) x 0
(A.6)
with x0 Å N0rm0 , where the idea is that
*e
0ix0 p
cO (p)cO (N0rp)dp Å 0
(A.7)
(because c is an N-adic wavelet state) while N0rk (k) cO (0) k!
*
e0ix0 pcO (p)pkdp Å
i kN0rk (k) cO (0)c(k)(x0). k!
(A.8)
On the other hand, cO (k/1)(p) √ L2(R) c (1 / ÉxÉk/1)c(x) √ L2(R) c c(x) √ L1(R) c cO (p) √ L`(R),
(A.9)
which—together with (A.4)—implies ÉRk(p)É £ c(1 / ÉpÉk).
(A.10)
Now our basic strategy is the same as in [7]: since for arbitrarily large r we can find an m0 such that (A.8) does not vanish, we need only show that the integral Ik Å
* ÉcO (p)É ÉR (N
0r
k
p)Édp
(A.11)
decreases faster than N0rk as r r ` if we wish (A.4) to contradict (A.7). To this end, we proceed to estimate Ik in the same manner as in [7]—by decomposing the integral over the regions ÉpÉ £ 10N r and ÉpÉ ú 10N r —but our bound will be slightly different as a result of the slightly weaker hypothesis. First, (A.10) implies
*
ÉpÉú10N
r
ÉcO (p)É ÉRk(N 0rp)Édp
£c
*
ÉpÉú10N
£ c\c(k/1)\
r
ÉcO (p)Édp / cN0rk
FS*
ÉpÉú10N
r
*
ÉpÉú10N
D
ÉpÉ02k02dp
r
ÉpÉkÉcO (p)Édp
1/2
/ N0rk
S*
ÉpÉú10N
DG 1/2
r
ÉpÉ02dp
Å O(N0rk0(1/2)r ).
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(A.12)
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HEISENBERG INEQUALITIES FOR WAVELET STATES
Second, (A.5) yields the complementary estimate ÉRk(N0rp)É £ c\cO (k/1)\(N0rÉpÉ)k/1/2,
(A.13)
from which we infer that
*
ÉpÉ£10N
r
ÉcO (p)É ÉRk(N0rp)Édp £ cN0rk0(1/2)r \cO (k/1)\
*
ÉpÉ£10N
r
ÉcO (p)É ÉpÉk/1/2dp.
(A.14)
However,
*
ÉpÉ£10N r
ÉcO (p)É ÉpÉk/1/2dp £
S*
D S*
D
1/2
(1 / ÉpÉ)2k/2ÉcO (p)É2dp
ÉpÉ£10N r
1/2
(1 / ÉpÉ)01dp
q
Å O( ln N r ),
(A.15)
so this completes the proof of Theorem 4.4. Our other obligation is to prove Lemma 1.7. c is now an arbitrary state such that ÉXÉ1/1c √ L2(R) for some 1 ú 0 and
* c(x)dx Å 0.
(A.16)
In momentum space we have
* cO *(q)dq
(A.17)
1 cO (p) p
(A.18)
p
cO (p) Å
0
because cO (0) Å 0. The issue is whether wP (p) Å
is square-integrable in the region ÉpÉ £ 1 (as it is obviously square-integrable in the region ÉpÉ § 1). The Ho¨lder inequality implies ÉcO (p)É £
*
ÉqÉ£ÉpÉ
ÉcO *(q)Édq
£ (2ÉpÉ)1/2/d
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(A.19)
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with 1/r Å
1 2
0 d. By Young’s inequality,
S*
D S* 1/r
ÉcO *(q)Érdq
£c
D
1/2/d
Éxc(x)É20dH dx
(A.20)
with (12 / d)(2 0 dH ) Å 1. Now we estimate as follows:
* Éxc(x)É
20dH
* ((1 / ÉxÉ)Éc(x)É) dx Å * (1 / ÉxÉ) ((1 / ÉxÉ) Éc(x)É) dx £ S* (1 / ÉxÉ) dxD S* (1 / ÉxÉ) Éc(x)É dxD 20dH
dx £
0(20dH )1
20dH
1/1
02(20dH )1/dH
dH /2
2/21
2
10dH /2
Å c\(1 / ÉXÉ)1/1c\20dH ,
(A.21)
provided d ú 0 has been chosen such that 2(2 0 dH )1 ú dH . Thus
*
ÉpÉ£1
ÉwP (p)É2dp £ 21/2d
S*
£c
*
ÉpÉ01/2ddp
ÉpÉ£1
D
S*
D
2/r
ÉcP *(q)Érdq
1/2d
Éxc(x)É20dH dx
£ c\(1 / ÉXÉ)1/1c\2,
(A.22)
and this proves the square-integrability of w. ACKNOWLEDGMENT The author thanks F. Narcowich and R. Strichartz for useful conversations.
REFERENCES 1. S. Dalke and P. Maass, Computers Math. Appl. 30 (1995), 293. 2. T. Paul and K. Seip, Wavelets and Quantum Mechanics, in ‘‘Wavelets and Their Applications’’ (M. B. Ruskai et al., Eds.), p. 303, Jones & Bartlett, Boston, 1992. 3. C. Chui and J. Wang, ‘‘High-Order Orthonormal Scaling Functions and Wavelets Give Poor TimeFrequency Localization,’’ CAT Report No. 322. 4. C. Chui and J. Wang, ‘‘A Study of Compactly Supported Scaling Functions and Wavelets,’’ CAT Report No. 324. 5. Y. Meyer, Sem. Bourbaki 38 (1985/1986), 662. 6. I. Daubechies, Commun. Pure Appl. Math. 41 (1988), 909. 7. G. Battle, J. Math. Phys. 30, No. 10 (1989), 2195. 8. E. Whittaker and G. Watson, ‘‘A Course of Modern Analysis,’’ Cambridge Univ. Press, London, 1965.
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HA960207
Heisenberg Inequalities for Wavelet States Guy Battle Mathematics Department, Texas A&M University, College Station, Texas 77843
Communicated by Ingrid Daubechies Received September 27, 1995; revised December 12, 1996
We prove a number of uncertainty results for wavelet states, the simplest one being that if a wavelet state is real-valued or, more generally, has zero expected momentum, then the Heisenberg uncertainty is at least 32 instead of the universal 12. For wavelet states having a very mild nth-order decay property, we establish a similar result for the uncertainty based on the nth powers of position and momentum, with a lower bound that grows rapidly with n. The proof is extremely elementary. Other Heisenberg inequalities are proven which involve the deviations about the origin of phase space rather than about the mean position of the wavelet in phase space, and the scaling generator plays an even more direct role than in the result mentioned above. The proof is still very elementary, combining the interscale orthogonality property with an iterated application of Rolle’s Theorem. Naturally, the lower bounds are much greater for these deviations about the origin of phase space, but this yields consequences for how much ‘‘off-center’’ a wavelet must be if its uncertainty is approximately minimized. q 1997 Academic Press
1. INTRODUCTION
The Heisenberg inequality asserts that for a quantum-mechanical state of a particle, the product of the standard deviations of the position and momentum is always bounded below by 12 (in one dimension). Actually, the expected value of the scaling generator contributes to the lower bound, but it is usually discarded because it is zero for many states. On the other hand, wavelet decompositions—which are based on scaling—have been very useful in dealing with phase space localization problems that can be traced back to the Heisenberg Uncertainty Principle. This author began searching for a deeper connection between scaling and the position-momentum duality, and the collection of Heisenberg inequalities derived here for N-adic wavelets are a consequence of this search. Uncertainty principles in the wavelet context have been considered before [1, 2]. After all, the original motivation of wavelet decompositions was to localize phase space in the best possible way. Most recently, Chui and Wang have shown that for 119 1063-5203/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.
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GUY BATTLE
a special but useful class of wavelets, the standard deviation in phase space increases without bound as the wavelets are made arbitrarily smooth [3, 4]. Obviously, such a result cannot hold for wavelets in general, because the Meyer wavelet [5] is a class C ` function with finite uncertainty. As one of our results in this paper we establish a somewhat different type of result. It applies to virtually all interscale-orthogonal wavelets that one could wish to construct, while it establishes lower bounds on higher order phase space deviations instead. Throughout this paper, we restrict our attention to one dimension only. We introduce the momentum operator P Å 0i(d/dx) and the position operator X, which is multiplication by x. Our starting point is the basic commutation relation [P, X] Å 0i,
(1.1)
which yields alternate expressions for the scaling generator: SÅ
1 (PX / XP) 2
Å PX /
1 i 2
Å XP 0
1 i. 2
(1.2)
The self-adjoint operator S generates the unitary group of scalings in the sense of Stone’s Theorem. Thus (eilSc)(x) Å el/2c(elx),
(1.3)
where c is a one-dimensional wave function normalized in L2(R). If »A…c Å (Ac, c) for observables A, then the Heisenberg inequality includes the expected value »S…c of this generator in the lower bound
Z
sc(P)sc(X) § 0
Z
1 i / »S…c 0 »P…c»X…c , 2
(1.4)
where sc denotes the standard deviation sc(A) Å »(A 0 »A…c)2…c.
(1.5)
The real contribution to the modulus in (1.4) is normally thrown away, but the connection to scaling is there. DEFINITION. A normalized square-integrable function c is an N-adic wavelet state if there is an integer r0 such that c(x) is orthogonal to N r/2c(N rx 0 N r0m) for m √ Z, r √ Z"{0}. N r0 is the length scale of the wavelet for the smallest such r.
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Remark. We are assuming neither orthogonality of the discrete translates on the same scale, nor completeness of the set of functions generated—only interscale orthogonality. Thus, for any orthonormal wavelet basis [6], c can be an arbitrary normalized linear combination of basis elements from the same scale. In particular, an N-adic wavelet state c satisfies »eir(ln N)S…c Å (eir(ln N)Sc, c) Å 0
(1.6)
for r √ Z"{0}. This will play an important role in proving some of our results. THEOREM 1.1.
If c is an N-adic wavelet state, then 2 1/2 »P2…1/2 c »X …c §
q 1 (1 0 1/ N)01. 2
(1.7)
Remark. This lower bound is larger than the universal lower bound 1/2 for all choices of scale factor N, and it obviously approaches 1/2 as N r `. Moreover, in the N Å 2 case (dyadic scaling) the lower bound is É1.706. Proof.
The key is the identity d l/2 0ilS (e »e …c) Å 0iel/2»(S / i/2)e0ilS…c dl Å 0iel/2»XPe0ilS…c.
(1.8)
Combining this with eilSPe0ilS Å e0lP
(1.9)
and the fundamental theorem of calculus, we obtain q
N»e0i(ln N)S…c 0 1 Å 0i
*
ln N
e0l/2»Xe0ilSP…cdl.
(1.10)
0
On the other hand, »e0i(ln N)S…c Å 0
(1.11)
because c is an N-adic wavelet state. It follows from the Schwarz inequality that 1 £ \Pc\ \Xc\
*
ln N
e0l/2dl
0
q
Å 2(1 0 1/ N)\Pc\ \Xc\,
and so we have the desired inequality. j
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Remark.
This argument easily extends to a proof of the generalization r0 2 1/2 »P2…1/2 c »(X 0 N m) …c §
S
D
1 1 10q 2 N
01
for all m √ Z. The point is that S can be replaced by the operator S 0 N r0mP because N r0 is the length scale of the wavelet state c. We shall look at various generalizations of these inequalities to higher orders. A first approach in Section 2 leads to: THEOREM 1.2.
If c is an N-adic wavelet state, then 2 3 4 1/2 »P4…1/2 0 , c »X …c § 2 (ln N) 4 6 1/2 »P6…1/2 c »X … c §
(1.12)
9 15 0 . 2 (ln N) 8
(1.13)
We regard these inequalities as second and third order Heisenberg relations. Our method of estimation appears to break down for order n § 4, but Section 3 takes a different approach and proves: THEOREM 1.3.
If c is an N-adic wavelet state, then
2n 1/2 »P2n…1/2 c »X … c §
F*
1
N01
dt 1
*
t1
N02
dt2rrr
*
tn01
N0n
G
dq tn tn
01
.
(1.14)
,
(1.15)
In the second and third order cases, this inequality becomes 4 1/2 »P4…1/2 c »X … c §
»P6…1/2c»X6…1/2 c §
F
F
G
4 (1 0 N03/2) 0 2N01(1 0 N01) 3
S
D
01
G
8 4 (1 0 N021/2) / (1 0 N03) 2N07/2 0 N05 0 N03/2(1 0 N06) 15 3
01
,
(1.16)
respectively. In the dyadic case (N Å 2), the lower bounds are É2.752 and É3.298, respectively. Actually, these are not as good as the dyadic cases of (1.12) and (1.13), respectively. For N § 3, however, the situation changes. For N Å 3, the inequalities (1.15) and (1.16) produce lower bounds É1.580 and É5.590, respectively. The inequality (1.15) is already better than (1.12), but (1.16) is still worse than (1.13). For the N Å 4 case, (1.16) provides a lower bound É2.367, while (1.13) provides a lower bound slightly greater than 21/8. For N § 5, the lower bound given by (1.16) is finally the better one. If in addition, we assume the fairly mild condition ÉXÉn/1c √ L2(R),
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(1.17)
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then it is shown in Section 5 that (1.14) can be strengthened to a lower bound whose n Å 1, 2, 3 cases are 2 1/2 »P2…1/2 c »X …c §
4 1/2 »P4…1/2 c »X …c §
6 1/2 »P6…1/2 c »X …c §
3 , 2
(1.18)
S S
169 3 q 4 / 10 / 16 2 (ln N)4
D
1/2
,
(1.19)
D
9549 45 q 149 / 259 / 64 8 4(ln N)4
1/2
,
(1.20)
respectively. As in the 1st-order case, these higher order inequalities still hold with the replacement X ° X 0 N r0m, m √ Z. Now it can certainly be argued that all of these lower bounds are due, in part, to the fact that even when a wavelet state is translated as close as possible to the origin, using the discrete translates associated with its length scale, it is typically off center. As far as uncertainty principles are concerned, it is always more to the point to obtain lower bounds on deviations from the mean position (»X…c, »P…c) in phase space. We can obtain such bounds in the case where »P…c Å 0, and we do so in Section 4. This includes real-valued as well as wavelet states that are either symmetric or anti-symmetric about their barycenters in position space. We prove a fundamental result: THEOREM 1.4.
Let c be an N-adic wavelet state such that »P…c Å 0. Then sc(P)sc(X) §
3 . 2
(1.21)
We shall prove this theorem in the course of our discussion about wave functions whose zeroth-order moments vanish. First we observe the following result, implicit in [7]: THEOREM 1.5. Let c be an N-adic wavelet state in arbitrary dimension. If c is both integrable and continuous, then
* c(x )dx Å 0. û
û
(1.22)
In one dimension we have the refinement: THEOREM 1.6.
Let c be an N-adic wavelet state. If Xc and Pc both lie in L2(R),
then c is integrable and * c(x)dx Å 0. We prove this in the Appendix. The point here is that if the hypothesis fails, then sc(P)sc(X) Å ` anyway. It is interesting that we can prove (1.21) in a short, clever way if we make the hypothesis just a little stronger. Suppose the state c has the property
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w √ L2(R).
c Å Pw,
(1.23)
This stronger condition alone is enough, whether c is a wavelet state or not. To show that
2 1/2 »P2…1/2 c »(X 0 a) …c §
3 2
(1.24)
for all real a, we may assume without loss that a Å 0, because the property (1.23) is translation-invariant. On the other hand, \Xc\ Å \XPw\ Å
Å
ZZS ZZS
D ZZ D ZZ
S/i
1 w 2
S0i
1 w 2
.
(1.25)
It follows from the Schwarz inequality that
§
Å
ZZS
D ZZ ZS S D DZ ZS S D DZ S0i
1 w 2
c, P S 0 i
1 w 2
\Pc\ \Xc\ Å \Pc\
c, S 0 i
3 Pw 2
(1.26)
because [P, S] Å 0iP.
(1.27)
Hence
Z
\Pc\ \Xc\ § (c, Sc) 0 i §
Z
3 (c, c) 2
3 , 2
(1.28)
and so the estimation is complete. The stronger condition (1.23) follows from a very mild decay condition:
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HEISENBERG INEQUALITIES FOR WAVELET STATES
Let c be a state such that ÉXÉ1/1 c √ L2(R) for some 1 ú 0 and
LEMMA 1.7.
* c(x)dx Å 0.
(1.29)
Then c Å Pw for some w √ L2(R). We prove this technical lemma in the Appendix. In Section 4 we shall generalize the inequality (1.21)—for this kind of decay condition—to nth-order uncertainties. If we introduce the notation sc(n)(A) Å »(A 0 »A…c)2n…c1/2n
(1.30)
for an arbitrary observable A, we have: THEOREM 1.8. √ L2(R), then
Let c be an N-adic wavelet state such that »P…c Å 0. If ÉXÉn/1c
F∏ S D S ∑ ∏ S D D G n
sc(n)(P)sc(n)(X) §
k0
kÅ1
1 /n 2
n
k0
lÅ1 kxl
1 2
2
1/2
1/n
.
(1.31)
Note that we have here again the decay condition (1.17). In the 1st-order case, the result is just Theorem 1.4 weakened by this additional hypothesis. In Section 4 we prove Theorem 1.4, using a crucial observation by Fran Narcowich, and discuss the problem of extending to arbitrary n his strategy for avoiding this condition.
2. HIGHER ORDER HEISENBERG INEQUALITIES
Uncertainty relations involving powers of P and X are nothing new for arbitrary quantum-mechanical states, but they involve special combinations with an algebraic structure that is intimately related to the harmonic oscillator. One is often interested in the observables
HÅ
1 2 1 2 P / X , 2 2
(2.1)
LÅ
1 2 1 2 P 0 X . 2 2
(2.2)
H is the harmonic oscillator Hamiltonian and L is the harmonic oscillator Lagrangian. It is well known that a Lorentz group structure links the scaling generator S to the operators L and H. The harmonic oscillator time-evolution given by the one-parameter unitary group U(t) Å e0itH
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GUY BATTLE
interpolates the Fourier transform:
S D
U
1 p c Å cP . 2
(2.5)
Consider the relations U(t)01PU(t) Å P cos t / X sin t,
(2.6.1)
U(t)01XU(t) Å X cos t 0 P sin t,
(2.6.2)
from which it is easy to derive U(t)01LU(t) Å L cos 2t / S sin 2t,
(2.7.1)
U(t)01SU(t) Å S cos 2t 0 L sin 2t.
(2.7.2)
(Commutations turn out to be irrelevant in this case.) In particular, the scaling generator and the harmonic oscillator Lagrangian are related by the half-Fourier transform U(14p). Given (2.6), the relations (2.7) are certainly not surprising from a classical point of view, because the classical observables 12 p2 0 12 x2 and px are just hyperbolic coordinates in phase space. Now the observables H, L, and S are related almost (but not quite) as the components of angular momentum are related. It is easy to verify that 1 [S, H] Å 0iL, 2
(2.8)
1 [H, L] Å 0iS, 2
(2.9)
1 [L, S] Å iH, 2
(2.10)
and the signs make all the difference in the world. In contrast to the case of angular momentum components, two of our observables—namely L and S—have absolutely continuous spectra. This Lie algebra generates SO(1, 2) instead of SO(3) because eilSHe0ilS Å H cosh 2l / L sinh 2l, ilS
0ilS
(2.11.1)
Å L cosh 2l / H sinh 2l,
(2.11.2)
eimLHe0imL Å H cosh 2m / S sinh 2m,
(2.12.1)
eimLSe0imL Å S cosh 2m / H sinh 2m.
(2.12.2)
e Le
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Now S and L are linked by a Heisenberg inequality as well. By (2.10) we have 2 1/2 »L2…1/2 c » S … c § »H … c
(2.13)
for an arbitrary state c, so if we recall the operator inequality H § 12 , it follows that 1 2 1/2 »L2…1/2 c »S … c § 2 .
(2.14)
This second-order Heisenberg inequality is part of a hierarchy of inequalities for arbitrary states, involving the creation and annihilation operators for the harmonic oscillator: 1 (X 0 iP), 2
(2.15)
1 (X / iP). 2
(2.16)
A* Å
q
AÅ
q
The basic commutation relations are: [A, A*] Å 1,
(2.17)
[H, A] Å 0A,
(2.18)
[H, A*] Å A*.
(2.19)
The relationship to the generators above are easily given: H Å A*A / Å AA* 0
SÅi LÅ
1 2 1 , 2
(2.20)
1 (A*2 0 A2), 2
(2.21)
1 (A*2 / A2). 2
(2.22)
Thus, a natural higher-order generalization of the pair {S, L} of observables is the pair {Bn , Cn} with Bn Å i Cn Å
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1 (A*n / An), 2
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and by the Schwarz inequality, 2 1/2 »B2n…1/2 c »C n…c §
Å
1 É»[Bn , Cn]…cÉ 2 1 É»[An, A*n]…cÉ. 2
(2.25)
In this case, the relations of interest are
S ∏S n
AnA*n Å
∏
1 , 2
(2.26)
H0k/
1 . 2
(2.27)
kÅ1 n
A*nAn Å
D D
H/k0
kÅ1
Hence [An, A*n] Å Pn(H),
(2.28)
where al ú 0 in the polynomial n
S
Pn(z) Å ∏ z / k 0 kÅ1
D
1 0 2
n
∏ kÅ1
S
z0k/
D
1 2
[(1/2)(n01)]
å
∑
alzn02l01.
(2.29)
lÅ0
Therefore, the nth-order Heisenberg inequality in this hierarchy is obtained from applying H § 12 to the polynomial in H with positive coefficients. We get
SD
2 1/2 »B2n…1/2 c »Cn…c § Pn
1 , 2
(2.30)
and this concludes our review. 2n 1/2 There are universal lower bounds on the simpler nth-order products »P2n…1/2 c »X … c as well. Indeed, 2 n/2 »A2n…1/2 c § »A … c
(2.31)
for all self-adjoint operators A, so we obtain 2n 1/2 n »P2n…1/2 c »X …c § 1/2
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(2.32)
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if we apply the basic inequality 2 1/2 »P2…1/2 c »X …c § 1/2.
(2.33)
For the n Å 2 case, we can obtain another lower bound in terms of the scaling generator S. Since
S Å XP 0 i
1 , 2
(2.34)
we have
S2 Å XPXP 0 iXP 0
1 4
Å X 2P2 0 i2XP 0
1 4
Å X 2P2 0 i2S /
3 4
(2.35)
from which it follows that
»S2…c 0
3 / i2»S…c Å »P2c, X 2c… 4
(2.36)
D
(2.37)
for an arbitrary state c. Hence
S
»S2…c 0
3 4
2
/ 4»S…2c Å É(P2c, X 2c)É2,
and we can now apply the Schwarz inequality to obtain
»S2…c 0
3 £ \P2c\ \X 2c\ 4 4 1/2 Å »P4…1/2 c »X … c .
(2.38)
Combining this with the operator inequality
1 0 cos lS £
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where l is a real parameter, we get »cos lS…c /
S
D
1 2 3 4 1/2 l / »P4…1/2 § 1. c »X …c 2 4
(2.40)
This estimation does not even recover the universal lower bound (2.32). But now suppose c is an N-adic wavelet state. What restriction does this impose on the state? We proved in [7] that such a function c cannot have exponential decay in both position space and momentum space, but such a duality property is qualitative. A more quantitative restriction can now be derived by combining (2.40) with »e{i(ln N)S…c Å 0,
(2.41)
which is (1.6) with r Å {1. Thus, if we set l Å ln N in (2.40), the cosine term vanishes, and we obtain
S
D
3 1 4 1/2 § 1. (ln N)2 / »P 4…1/2 c »X …c 2 4
(2.42)
Therefore 4 1/2 »P 4…1/2 c »X … c §
2 3 0 (ln N)2 4
(2.43)
for such a state c. This is better than the universal lower bound of 1/4 only for N £ 4. A decimal approximation to this lower bound is 3.4145 in the case N Å 2. 2n 1/2 In the pursuit of lower bounds on »P2n…1/2 c »X …c for arbitrary order n, we will n n constantly use an operator identity for P X , introduced in the next section. In particular, P 3X 3 Å S3 0
S
D
23 9 15 , S 0 i S2 0 4 2 8
(2.44)
from which we can derive the inequality 6 1/2 »P 6…1/2 c »X … c §
9 15 0 2 (ln N) 8
(2.45)
for an N-adic wavelet state c. This concludes our proof of Theorem 1.2. This method of estimation appears to break down for order n § 4. We pursue another method in the next section. 3. AN nTH-ORDER HEISENBERG INEQUALITY FOR WAVELET STATES
How can we generalize the proof of Theorem 1.1 to obtain higher order Heisenberg inequalities? The key to the generalization is the identity
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P nX n Å
∏ (S 0 i(k 0 1/2)),
(3.1)
kÅ1
which can be verified by induction, using the obvious commutation relation [P, S] Å 0iP.
(3.2)
The simplest nth-order generalization of (1.8) appears to be
S D
e0l el
d n l/2 0ilS (e »e …c) Å (0i)ne(n01/2)l dl
K S S DD L n
∏
S/i k0
kÅ1
1 2
e0ilS
c
Å (0i)ne(n01/2)l»X nP ne0ilS…c.
(3.3)
eilSP ne0ilS Å e0nlP n,
(3.4)
On the other hand, (1.9) implies
so we have
S D
e0l el
d n l/2 0ilS (e »e …c) Å (0i)ne0l/2»X ne0ilSP n…c. dl
(3.5)
This formula is essential to our strategy in proving Theorem 1.3. Recall that el/2»e0ilS…c vanishes at the points l Å r ln N, r √ Z"{0}, by the wavelet property e0ir (ln N)Sc ⊥ c. Clearly we can apply Rolle’s Theorem to the real and imaginary parts of this expectation. To this end, we define u(l) Å el/2Re»e0ilS…c ,
(3.6.1)
£(l) Å el/2Im»e0ilS…c .
(3.6.2)
More generally, we define
S D S D
uk(l) Å e0l el
d k u(l), dl
(3.7.1)
£k(l) Å e0l el
d k £(l). dl
(3.7.2)
We concentrate on u(l) because the real and imaginary parts are decoupled in this iteration, and the real part will provide all of the lower bound; the imaginary part turns out to be useless. Applying Rolle’s Theorem to the property
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GUY BATTLE
r √ Z"{0},
u(r ln N) Å 0,
(3.8)
we obtain a sequence (a1(1) , a2(1) , a3(1) , . . .) such that r ln N õ ar(1) õ (r / 1)ln N,
(3.9)
(1) r
u1(a ) Å 0.
(3.10)
Applying Rolle’s Theorem to these zeros of elu1(l), we now obtain another sequence (a1(2) , a2(2) , a3(2) , . . .) such that (1) ar(1) õ ar(2) õ ar/1 ,
(3.11)
u2(ar(2)) Å 0.
(3.12)
Obviously, we can iterate this procedure indefinitely. In general, we have a sequence (a(1k ) , a(2k ) , a(3k ) , . . .) satisfying the conditions k01) a(r k01) õ a(rk ) õ a(r/1 ,
(3.13)
(k) r
uk(a ) Å 0.
(3.14)
Now we are particularly interested in the smallest numbers in each sequence— namely, a1(1), . . . , a(1n01). It is easy to infer that a(1k ) õ (k / 1)ln N,
(3.15)
ln N õ a1(1) õ a1(2) õ rrr õ a(1n01).
(3.16)
Since the real-valued function £(l) also has the property (3.8), we can generate zeros b1(1) , . . . , b(1n01) of £1(l), . . . , £n01(l), respectively, which satisfy the conditions (3.15) and (3.16). LEMMA 3.1.
With the numbers a1(1) , . . . , a(1n01) derived as above,
*
(01)n Å
*
ln N
dl1e0l1
a1(1)
*
dl2e0l2rrr
l1
0
a1( n01)
dlnun(ln).
(3.17)
ln01
With the similarly derived numbers b11 , . . . , b1n01, 0Å
*
ln N 0l1
dl1e
dl2e
0l2
rrr
l1
0
Proof.
*
b1(1)
*
b1( n01)
dln£n(ln).
(3.18)
ln01
Iterate the formulas
*
a1( k01)
dlkuk(lk) Å 0elk01uk01(lk01),
(3.19.1)
lk01
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*
b1( k01)
133
dlk£k(lk) Å 0elk01£k01(lk01),
(3.19.2)
dl1u1(l1) Å 0u(0) Å 01,
(3.20.1)
dl1£1(l1) Å 0£(0) Å 0
(3.20.2)
lk01
where
* *
ln N
0 ln N
0
are the last integrations. j Now by Equation (3.5), un(ln) Å e0(1/2)ln Re»(0i)nP ne0ilnSX n…/
(3.21)
Éun(ln)É £ e0(1/2)ln\P nc\ \X nc\
(3.22)
which implies
by the Schwarz inequality. Remark. Equation (3.17) supplies us with the lower bound, so we discard £n(ln) and use only un(ln). While b(1k ) does not coincide with a(1k ) in general, Eq. (3.18) shows that the imaginary part £n(ln) cannot add much to the lower bound. We included Eq. (3.18) for the sake of completeness. Combining Eq. (3.17) with (3.6.1), (3.7.1), and the bound (3.22), we obtain the inequality
* £ \P c\ \X c\ * Å \P c\ \X c\ *
ln N
1 £ \P c\ \X c\ n
n
dl1e
0l1
ln N
dl1e0l1
n
* rrr*
rrr
1
n
N01
dt1
*
2 ln N
02
n ln N
dl2e0l2
dlne01/2ln
ln01
t1
N
dlne01/2ln
ln01
l1
0
n
dl2e
a1(n01)
0l2
l1
0
n
* *
a1(1)
dt2rrr
*
tn01
N
1 dt n , tn
q
0n
(3.23)
where we have applied (3.15) and made the change of variable tk Å e0lk. Thus we have proved Theorem 1.3—i.e., we have the inequality (1.14) for N-adic wavelet states c. Remark.
The limit of the multiple integral as N r ` is
* dt * 1
t1
1
0
0
dt2rrr
*
tn01
0
1 dtn Å tn
q
n
1
∏ k 0 1/2 ,
kÅ1
so the limit of the lower bound is ∏nkÅ1 (k 0 12).
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(3.24)
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GUY BATTLE
4. AN nTH-ORDER UNCERTAINTY PRINCIPLE FOR »P… Å 0 WAVELET STATES
In the Introduction we proved that
2 1/2 »P 2…1/2 c »X … c §
q 1 (1 0 1/ N)01 2
(4.1)
for such a wave function c. More generally, in the previous section we showed that an N-adic wavelet state c satisfies the Heisenberg inequality
2n 1/2 »P 2n…1/2 c »X … c §
F*
1
N01
dt1
*
t1
N02
dt2rrr
*
G
tn01
dtn q tn
N0n
01
(4.2)
for all positive integers n. However, these products are phase space deviations from the fixed point (0,0) for scaling rather than the phase space deviations from the mean coordinates (»X…c , »P…c) of the state. In this section we prove: THEOREM 4.1.
If c is an N-adic wavelet state with ÉXÉn/1c √ L2(R), then n
2n 1/2 »P 2n…1/2 c »(X 0 a) …c § n( ∑
n
∏ (k 0 1/2)2)1/2 / ∏
lÅ1 kxl
kÅ1
S D k0
1 2
(4.3)
for all real a and positive integers n. We have Theorem 1.8 as an immediate consequence. The key to proving the theorem is to combine a result for general states with a variation on a previous result [7] for wavelet states. Both results address the issue of vanishing moments for the wave function. THEOREM 4.2. and
Let c be a normalized element of L2(R) such that ÉXÉn/1c √ L2(R)
* x c(x)dx Å 0, m
m £ n 0 1.
(4.4)
Then for all real a, n
2n 1/2 »P2n…1/2 c »(X 0 a) …c § n( ∑
lÅ1 kxl
Proof.
n
∏ (k 0 1/2)2)1/2 / ∏ kÅ1
S D k0
1 . 2
(4.5)
By the triangle inequality and by convexity, we have Éx 0 aÉn/1 £ 2n/101(ÉxÉn/1 / ÉaÉn/1).
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(4.6)
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HEISENBERG INEQUALITIES FOR WAVELET STATES
Since we also have
*
SD
m
m (0a)m0m m
(x 0 a)mc(x)dx Å ∑ mÅ0
* x c(x)dx, m
(4.7)
the hypothesis is satisfied by arbitrary translates of c, so we need only establish (4.5) for a Å 0. By the vanishing moments condition, there is an element w √ L2(R) such that P nw Å c.
(4.8)
That w is indeed square-integrable follows from the condition ÉXÉn/1c √ L2, and the proof is an obvious generalization of the proof of Lemma 1.7. Thus X nc Å X nP nw n
Å
∏ (S / i(k 0 1/2))w,
(4.9)
kÅ1
where we are using the operator identity (3.1). On the other hand,
ZZ∏ n
ZZ S∏ S S D D D n
2
(S / i(k 0 1/2))w
kÅ1
2
1 2
S2 / k 0
Å
kÅ1
w, w
n
n
§ \w\2
∏ (k 0 1/2)2 / \Sw\2 ∑ ∏ lÅ1 kxl
kÅ1
Å
ZZS
S DD ZZ S∑ ∏ S D D n
anS { i
∏
kÅ1
n
an Å
k0
k0
lÅ1 kxl
1 2
2
1 2
w
S D k0
1 2
2
2
,
(4.10)
1/2
.
(4.11)
The choice of sign affects the lower bound obtained when we apply the Schwarz inequality. The better choice is clearly
ZZS
\P nc\
n
anS 0 i
Z Z Z
∏ kÅ1
S DD ZZ k0
1 2
w
S D Z S ∏ S DD S ∏ S DDZ
§ an(P nc, Sw) 0 i
n
∏
k0
kÅ1
1 (P nc, w) 2
n
Å an(c, SP nw) 0 i nan /
k0
kÅ1
n
Å an(c, Sc) 0 i nan /
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kÅ1
1 2
1 2
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Z n
§ nan /
∏ kÅ1
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(4.12)
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GUY BATTLE
because [P n, S] Å 0inP n.
(4.13)
Combining (4.12) with (4.9) and (4.10), we obtain n
\P nc\ \X nc\ § nan /
∏ kÅ1
S D k0
1 , 2
(4.14)
which is the desired inequality. j On the other hand, we proved in [7] that: THEOREM 4.3. Let c be an N-adic wavelet state in arbitrary dimension such that û û û cP ( p ) is class C n/1 and (1 / É p É)n/1cO (p ) is integrable. Then
* x c(x )dx Å 0 a
û
û
(4.15)
for all multi-indices a such that ÉaÉ £ n. Remark. In [7] we assumed that N Å 2 and that discrete translates on the same scale were orthogonal, but neither assumption was necessary. Actually, this theorem is just a little too weak for our purpose, but we can prove a slightly stronger result in our one-dimensional case. THEOREM 4.4.
Let c be an N-adic wavelet state in one dimension such that cO (n/1)(p) √ L2(R),
(1 / ÉpÉ)n/1cO (p) √ L2(R). Then
* x c(x)dx Å 0, m
m £ n.
(4.16)
We prove this theorem in the Appendix, but the argument is similar in spirit to the proof of Theorem 4.3. Proof of Theorem 4.1. We assume without loss that P nc √ L2(R). Otherwise, the 2n 1/2 product »P 2n…1/2 c »(X 0 a) …c is infinite, in which case the lower bound is trivial. Now the Ho¨lder inequality 2n »A2m…n/m c £ »A … c
(4.17)
implies that the functions ÉPÉmc are elements of L2(R) for m £ n as well, so (1 / ÉPÉ)nc lies in L2(R). Since
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137
(1 / ÉXÉn/1)c √ L2 c Xnc √ L2 c cO (n)(p) √ L2,
(4.18)
we now apply Theorem 4.4 with n Å n 0 1 to infer that
* x c(x)dx Å 0, m
m £ n.
(4.19)
This, in turn, implies (4.5) because we now have the hypothesis of Theorem 4.2. j We conclude this section with a proof of Theorem 1.4 and a remark on the possibility of dropping the decay condition from the hypothesis of Theorem 1.8. Given Theorem 1.6, it is just a matter of using (1.29) instead of (1.23), and the idea is to simply 2 1/2 minimize »P…1/2 c »X …c over all states c satisfying the constraint (1.29). (As before, (1.24) reduces to the a Å 0 case because (1.29) is translation-invariant as well.) The variational problem is quite simple: we find the stationary points of the functional \Pc\2\Xc\2 / l(\c\2 0 1) / k
* c(x)dx,
where l and k are Lagrange multipliers. The result is an inhomogeneous equation: \Pc\2X 2c / \Xc\2P 2c / lc /
1 k Å 0. 2
(4.20)
Since any scaling (in x) of a solution is still a solution, we can choose that scaling such that \Pc\2 Å \Xc\2 å
1 a. 2
(4.21)
Thus—with 12 a identified as the uncertainty—the equation becomes aH / lc /
1 k Å 0, 2
(4.22)
where H is the harmonic oscillator Hamiltonian. Now in the case where k Å 0, the solutions are just those normalized eigenfunctions of H that happen to satisfy (1.29)— i.e., the odd Hermite functions. The first excited state is the function among these for which the uncertainty is smallest, and that value is indeed 32. The peculiarity of this variational problem arises in the case where k x 0, and we owe a crucial observation to Narcowich. All solutions c now have the form c(x) Å 0
1 k 2
* (aH / l)
01
(x, y)dy
`
Å0
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1 w2l(x) k ∑ 2 lÅ0 (2l / 12)a / l
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* w (y)dy,
(4.23)
2l
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GUY BATTLE
where wm denotes the mth Hermite function, because
*w
Å 0,
m
odd m.
(4.24)
On the other hand, if we integrate (4.22) against c, we obtain the relation 1 2 a / l Å 0, 2
(4.25)
so we may write k c(x) Å 0 2a
q
`
∑ lÅ0
w2l(x) (2l)! (01)l l , 1 1 2l / 2 0 2 a 2 l!
(4.26)
where we have inserted the value of * w2l . If we now apply (1.29), we obtain the condition `
∑ lÅ2
1 (2l)! Å 0. 2l / 12 0 12 a 4l(l!)2
(4.27)
However, this series [8] sums to the quotient
SD
1 1 1 1 G(4 0 4 a) G 2 2 G(34 0 14 a)
of gamma functions, and for a ú 0, this can vanish only where G(34 0 14 a) has poles. Hence, a Å 3, 7, 11, . . .
(4.28)
and a Å 3 clearly yields the smallest uncertainty, which is again 32 . This completes the proof of Theorem 1.4. Remark.
The value of k is determined by the constraint \c\2 Å 1, which reduces
to k2 4a2
`
∑ lÅ0
1 (2l)! Å 1. 1 2 l (2l / 0 2 a) 4 (l!)2
(4.29)
1 2
This series is essentially the a-derivative of the above series, so we have the condition
SD
1 1 k2 1 d G(4 0 4 a) G Å 1. 4a2 2 da G(34 0 14 a)
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(4.30)
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HEISENBERG INEQUALITIES FOR WAVELET STATES
139
Remark. It is reasonable to try this strategy on Theorem 1.8 with the idea of dropping the condition (1.17), but the problem is that H is now replaced with 1 2n 1 2n P / X , 2 2 and one would have to know a lot about the eigenfunctions.
5. AN n-TH ORDER HEISENBERG INEQUALITY FOR WAVELET STATES REVISITED
We proved in Section 4 that if c is an N-adic wavelet state, then (1.14) holds. The aim of this paper is to establish a different lower bound in the case where we have the condition (1.17) as well. This additional assumption is actually very mild in the sense that, without the 1 ú 0, the failure of the condition would mean »X 2n…c Å ` anyway. In Section 5 we derived a lower bound on the function 2n 1/2 a ° »P 2n…1/2 c »(X 0 a) …c
from the assumption (1.17). We now proceed to derive an a Å 0 lower bound from (1.17), and (4.4) plays an immediate role. The vanishing moments condition together with the decay condition implies c Å P nw
(5.1)
for some w √ L2(R). It follows from the identity n
X nP n Å
∏ (S / i(k 0 1/2))
(5.2)
kÅ1
that
ZZ∏ n
\X nc\2 Å
kÅ1
Å
S
ZZ
2
(S / i(k 0 1/2))w
n
∏ (S2 / (k 0 1/2)2)w, w kÅ1
D
n
Å
∑ al(n)(S2lw, w),
(5.3)
lÅ0
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GUY BATTLE
where the positive coefficients al(n) are defined by n
n
kÅ1
lÅ0
∏ (z / (k 0 1/2)2) Å ∑
al(n)z l.
(5.4)
Now separate out the first two terms q
q
a0(n)(w, w) / a1(n)(S2w, w) Å \( a1(n)S { i a0(n))w\2
(5.5)
and split the remaining sum into odd-l and even-l sums. We have \Xnc\2 q
q
[(1/2)n]
[(1/2)(n01)]
mÅ1
mÅ1
(n) (S 4mw, w) / ∑ a2m
Å \( a1(n)S / i a0(n))w\2 /
∑
(n) a2m/1 (S 4m/2w, w),
(5.6)
where [r] denotes the greatest integer function. Thus \P nc\2\X nc\2 has three contributions, each of which we shall underestimate in a different way. The first contribution is bounded below in exactly the same way as in Section 4: q
q
q
q
\P nc\ \( a1(n)S 0 i a0(n))w\ § n a1(n) / a0(n),
(5.7)
so we move immediately to the consideration of the other two contributions. Lower bound on the even contribution.
By the operator inequality
1 2 2 l S § 1 0 cos lS, 2
(5.8)
we can underestimate the even sum in (5.6) as follows: [(1/2)n]
[(1/2)n]
(n) (S 4mw, w) § ∑ ∑ a2m
mÅ1
mÅ1 [(1/2)n]
∑
Å
mÅ1
22m (n) a2m ((1 0 cos(S ln N))2mw, w) (ln N)4m 22m (n) a2m \(1 0 cos(S ln N))mw\2, (ln N)4m
(5.9)
where we have chosen l Å ln N to exploit (1.6). Since (4.13) and (5.1) yield \P nc\ \(1 0 cos(S ln N))mw\ § É(P nc, (1 0 cos(S ln N))mw)É Å É(c, P n(1 0 cos(S ln N))mw)É Å
ZS S D DZ ZS ∑ S DS D S∑ D DZ c, 1 0 m
Å
c,
mÅ0
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1 ∑ e{i(ln N)(S0in) 2 {
m m
0
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m
P nw
m
m
e{i(ln N)(S0in)
c
, (5.10)
{
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HEISENBERG INEQUALITIES FOR WAVELET STATES
it follows from (1.6) that [(1/2)m]
∑
\P nc\ \(1 0 cos(S ln N))mw\ §
nÅ0
S DS D
1 m 4n 2n
2n n
(5.11)
because (1.6) obviously implies
(c, (∑ e
HS D J 0, m odd, m , m even. 1 2m
{i(ln N)(S0in) m
) c) Å
{
(5.12)
Combining (5.11) with (5.9), we obtain [(1/2)n]
[(1/2)n]
mÅ0
mÅ1
\P nc\2
(n) (S 4mw, w) § ∑ ∑ a2m
H ∑ S DS DJ
4m (n) a2m (ln N)4m
Lower bound on the odd contribution. [(1/2)(n01)]
∑
∑
mÅ0
mÅ1 [(1/2)(n01)]
∑
Å
nÅ0
1 m 4n 2 n
2
2n n
.
(5.13)
We apply (5.8) as follows:
[(1/2)(n01)] (n) a2m/1 (S 4m/2w, w) §
[(1/2)m]
mÅ1
22m (n) a2m/1(S2(1 0 cos lmS)2mw, w) l4m m 22m (n) a2m/1\(1 0 cos lmS)mSw\2, l4m m
(5.14)
where our choice l Å lm for each m is yet to be made. Now if we apply the Schwarz inequality as before, (4.13) and (5.1) now yield \P nc\ \(1 0 cos lmS)mSw\ § É(P nc, (1 0 cos lmS)mSw)É Å
ZS S ∑ D ZS ∑ S DS D ∑ S D c, 1 0 m
Å
c,
mÅ0
1 2
m m
m
e{ilm(S0in)
(S 0 in)P nw
{
0
1 2
m
m
qÅ0
m
q
DZ
DZ
ei(m02q)lm(S0in)(S 0 in)c
.
(5.15)
On the other hand, we define useful functions wm(l) such that their derivatives at l Å lm are part of this inner product expression. If we set m
wm(l) Å ∑ qÅ0 2qxm
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SD m
q
i (c, ei(m02q)l(S0in)c), m 0 2q
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(5.16)
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GUY BATTLE
then
S SD m
c,
m
∑
q
qÅ0
ei(m02q)lm(S0in)(S 0 in)c
Å
H
D
w*m(lm), m odd, m w*m(lm) / 1 (c, (S 0 in)c), m even. 2m
SD
J
(5.17)
The usefulness of wm(l) lies in the wavelet property wm(ln N) Å wm(2 ln N) Å 0,
(5.18)
which is obviously shared by the function m
£m(l) Å
∑ mÅ0
S DS D m m
0
1 2
m
wm(l).
(5.19)
Applying Rolle’s Theorem to Im £m(l), we choose lm such that ln N õ lm õ 2 ln N,
(5.20)
Im £*m(lm) Å 0.
(5.21)
Thus (5.17) yields
S
m
Im c, ∑ mÅ0
S DS D S D m m
0
1 2
m
m
∑
qÅ0
m
D
ei(m02q)lm(S0in)(S 0 in)c
q
[(1/2)m]
Å
∑
nÅ0
S DS D
2n n, n
1 m 4n 2n
(5.22)
so if we combine this with (5.14) and (5.15), we finally have the lower bound [(1/2)(n01)]
\P nc\2
∑
(n) a2m/1 (S 4m/2w, w)
mÅ1 [(1/2)(n01)]
∑
§
mÅ1 [(1/2)(n01)]
§
∑
mÅ1
22m (n) a2m/1 l4m m
H
[(1/2)m]
∑
nÅ0
n2 (n) a2m/1 4m(ln N)4m
S DS D J H ∑ S DS DJ 2n n n
1 m 4n 2 n [(1/2)m]
nÅ0
1 m 4n 2n
2
2n n
2
,
where we have applied (5.20) in the last step.
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(5.23)
143
HEISENBERG INEQUALITIES FOR WAVELET STATES
Pulling all the estimations together, we now have the desired result: Let c be an N-adic wavelet state such that ÉXÉn/1c √ L2(R). Then
THEOREM 5.1. q
q
n
∑
\P nc\2\X nc\2 § ( a0(n) / n a1(n))2 /
lÅ2
1 (ln N)4[(1/2)l] 1 cl(n)
(n) l
c
Å
H
H
[(1/2)[(1/2)l]]
∑
nÅ0
1 4n
J
2lal(n),
l even,
(12)l01n2al(n),
l odd,
S DS DJ [12 l]
2n
2n
n
2
,
(5.24)
(5.25)
with the coefficients given by (5.4). This inequality survives the replacement X ° X 0 mN r0 for m √ Z, where N r0 is the length scale of c. APPENDIX
We now turn to the proof of Theorem 4.4, which is actually just a modification of the central idea of [7]. First, we claim that the 0th-order moment vanishes, and this follows from inspecting the proof of Lemma 1 of [7]. Indeed, (1 / ÉpÉ)cO (p) √ L2(R) c cO (p) √ L1(R),
(A.1)
and the continuity and integrability of cO (p) are the only regularity properties that are actually used in that argument. Suppose we have shown that all moments of order m £ k 0 1 vanish, where k is a positive integer £n, and assume
* x c(x)dx x 0.
(A.2)
cO (k)(0) x 0,
(A.3)
k
This means
and by the induction hypothesis, cO (p) Å
cO (k)(0) k p / Rk(p), k!
Rk(p) Å
1 (k / 1)!
* cO
(A.4)
p
(k/1)
(p 0 q)qkdq.
(A.5)
0
This version of Taylor’s Theorem makes sense because cO (k/1) is square-integrable and
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GUY BATTLE
therefore locally integrable. The same property guarantees the continuity of cO (k) in one dimension. For a given positive integer r—with value to be chosen sufficiently large later— pick an integer m0 such that c(k)(x0) x 0
(A.6)
with x0 Å N0rm0 , where the idea is that
*e
0ix0 p
cO (p)cO (N0rp)dp Å 0
(A.7)
(because c is an N-adic wavelet state) while N0rk (k) cO (0) k!
*
e0ix0 pcO (p)pkdp Å
i kN0rk (k) cO (0)c(k)(x0). k!
(A.8)
On the other hand, cO (k/1)(p) √ L2(R) c (1 / ÉxÉk/1)c(x) √ L2(R) c c(x) √ L1(R) c cO (p) √ L`(R),
(A.9)
which—together with (A.4)—implies ÉRk(p)É £ c(1 / ÉpÉk).
(A.10)
Now our basic strategy is the same as in [7]: since for arbitrarily large r we can find an m0 such that (A.8) does not vanish, we need only show that the integral Ik Å
* ÉcO (p)É ÉR (N
0r
k
p)Édp
(A.11)
decreases faster than N0rk as r r ` if we wish (A.4) to contradict (A.7). To this end, we proceed to estimate Ik in the same manner as in [7]—by decomposing the integral over the regions ÉpÉ £ 10N r and ÉpÉ ú 10N r —but our bound will be slightly different as a result of the slightly weaker hypothesis. First, (A.10) implies
*
ÉpÉú10N
r
ÉcO (p)É ÉRk(N 0rp)Édp
£c
*
ÉpÉú10N
£ c\c(k/1)\
r
ÉcO (p)Édp / cN0rk
FS*
ÉpÉú10N
r
*
ÉpÉú10N
D
ÉpÉ02k02dp
r
ÉpÉkÉcO (p)Édp
1/2
/ N0rk
S*
ÉpÉú10N
DG 1/2
r
ÉpÉ02dp
Å O(N0rk0(1/2)r ).
610e$$samp
(A.12)
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145
HEISENBERG INEQUALITIES FOR WAVELET STATES
Second, (A.5) yields the complementary estimate ÉRk(N0rp)É £ c\cO (k/1)\(N0rÉpÉ)k/1/2,
(A.13)
from which we infer that
*
ÉpÉ£10N
r
ÉcO (p)É ÉRk(N0rp)Édp £ cN0rk0(1/2)r \cO (k/1)\
*
ÉpÉ£10N
r
ÉcO (p)É ÉpÉk/1/2dp.
(A.14)
However,
*
ÉpÉ£10N r
ÉcO (p)É ÉpÉk/1/2dp £
S*
D S*
D
1/2
(1 / ÉpÉ)2k/2ÉcO (p)É2dp
ÉpÉ£10N r
1/2
(1 / ÉpÉ)01dp
q
Å O( ln N r ),
(A.15)
so this completes the proof of Theorem 4.4. Our other obligation is to prove Lemma 1.7. c is now an arbitrary state such that ÉXÉ1/1c √ L2(R) for some 1 ú 0 and
* c(x)dx Å 0.
(A.16)
In momentum space we have
* cO *(q)dq
(A.17)
1 cO (p) p
(A.18)
p
cO (p) Å
0
because cO (0) Å 0. The issue is whether wP (p) Å
is square-integrable in the region ÉpÉ £ 1 (as it is obviously square-integrable in the region ÉpÉ § 1). The Ho¨lder inequality implies ÉcO (p)É £
*
ÉqÉ£ÉpÉ
ÉcO *(q)Édq
£ (2ÉpÉ)1/2/d
610e$$samp
S*
D
1/r
ÉcO *(q)Érdq
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(A.19)
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146
GUY BATTLE
with 1/r Å
1 2
0 d. By Young’s inequality,
S*
D S* 1/r
ÉcO *(q)Érdq
£c
D
1/2/d
Éxc(x)É20dH dx
(A.20)
with (12 / d)(2 0 dH ) Å 1. Now we estimate as follows:
* Éxc(x)É
20dH
* ((1 / ÉxÉ)Éc(x)É) dx Å * (1 / ÉxÉ) ((1 / ÉxÉ) Éc(x)É) dx £ S* (1 / ÉxÉ) dxD S* (1 / ÉxÉ) Éc(x)É dxD 20dH
dx £
0(20dH )1
20dH
1/1
02(20dH )1/dH
dH /2
2/21
2
10dH /2
Å c\(1 / ÉXÉ)1/1c\20dH ,
(A.21)
provided d ú 0 has been chosen such that 2(2 0 dH )1 ú dH . Thus
*
ÉpÉ£1
ÉwP (p)É2dp £ 21/2d
S*
£c
*
ÉpÉ01/2ddp
ÉpÉ£1
D
S*
D
2/r
ÉcP *(q)Érdq
1/2d
Éxc(x)É20dH dx
£ c\(1 / ÉXÉ)1/1c\2,
(A.22)
and this proves the square-integrability of w. ACKNOWLEDGMENT The author thanks F. Narcowich and R. Strichartz for useful conversations.
REFERENCES 1. S. Dalke and P. Maass, Computers Math. Appl. 30 (1995), 293. 2. T. Paul and K. Seip, Wavelets and Quantum Mechanics, in ‘‘Wavelets and Their Applications’’ (M. B. Ruskai et al., Eds.), p. 303, Jones & Bartlett, Boston, 1992. 3. C. Chui and J. Wang, ‘‘High-Order Orthonormal Scaling Functions and Wavelets Give Poor TimeFrequency Localization,’’ CAT Report No. 322. 4. C. Chui and J. Wang, ‘‘A Study of Compactly Supported Scaling Functions and Wavelets,’’ CAT Report No. 324. 5. Y. Meyer, Sem. Bourbaki 38 (1985/1986), 662. 6. I. Daubechies, Commun. Pure Appl. Math. 41 (1988), 909. 7. G. Battle, J. Math. Phys. 30, No. 10 (1989), 2195. 8. E. Whittaker and G. Watson, ‘‘A Course of Modern Analysis,’’ Cambridge Univ. Press, London, 1965.
610e$$samp
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