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Uncertainty principles revisited.

Abstract. The Heisenberg uncertainty principle and the uncertainty principle for self-adjoint operators have been known and applied for decades. Both in quantum mechanics and in time-frequency analysis they play an important role. In this paper, the uncertainty principle is extended to symmetric operators and to normal operators. Further, different function spaces are studied in which we obtain a number of uncertainty principles of same type using various operators.

Key words. uncertainty principle, self-adjoint operators, symmetric operators, normal operators, periodic functions, ultraspherical polynomials, sphere.

AMS subject classifications. 26D10, 42C25, 47A05, 47A30, 47B47.

1. Introduction. The classical uncertainty principle (UP) was established by Heisenberg in [8] bringing a fundamental problem in quantum mechanics to the point: The position and the momentum of particles cannot be both determined explicitly but only in a probabilistic sense with a certain "uncertainty". The mathematical equivalent is that a vector in a Hilbert space and its Fourier transform cannot both be arbitrarily localized. This is the fundamental problem in time-frequency analysis, where one would like to have bases of vectors well-localized in both time and frequency.

Studying this problem on different domains and, hence, in different function spaces, the same question leads to a variety of answers and peculiarities puzzling at first sight. Recently, a number of papers has been published including UP's for periodic functions [9, 11, 15], functions on the interval [14] and on the sphere [7, 10]. These results provide qualitative and quantitative tools in order to determine the time-frequency localization of basis functions, e.g., wavelets, in different function spaces. On the search for the joint root and for the justification of the term "uncertainty", the author has been inspired by [1] to take the operator theoretical approach.

The present paper sheds some light behind the scene by extending the classical UP for self-adjoint operators to a wider class of operators, namely to symmetric operators and to normal operators. These issues may be known and, in fact, are not difficult to prove but were not found in the literature available to us. From these, almost all the UP's cited above follow (except for the sphere because of vector-valued operators). Further, this paper tries to widen the view by proposing the application of this UP to various pairs of operators. It is not conclusive, by far. Questions like the "natural" choice for the equivalent of the position and the momentum operators have not been considered here and will be subject of further investigations. Also, other concepts of UP's as, e.g., these in [4] could be studied from a more general point of view.

The paper is organized as follows. First, some notation is introduced. In Section 3, UP's are derived for symmetric and normal operators. Section 4 is devoted to higher moments and higher derivatives for both the real line and the circle. Section 5 deals with ultraspherical expansions on the interval and UP's for the related Dunkl and Laplace operators. Finally in Section 6, UP's on the sphere are discussed for the surface curl gradient and, for the first time, for the surface gradient which seems to be more appropriate.

2. Notation. Let H be a Hilbert space with inner product <*,*> and with norm [parallel] * [parallel] := [<*,*>.sup.1/2]. Further, let A,B be two (possibly unbounded) linear operators with domains D(A), D(B) [subset or equal to] H and ranges in H. Let

[[tau].sub.A](f) := <Af,f>/<f,f>

denote the (normalized) expectation value of the operator A with respect to f [member of] D(A), and

[[sigma].sub.A](f) := [parallel](A - [[tau].sub.A](f))f[parallel]

be the uncertainty or standard deviation of the operator A with respect to f [member of] D(A). Taking the operator as the lower index and f as variable shall suggest that the operator is usually fixed in our considerations whereas f varies over the whole domain of the operator. The variance is, as usual, the square of the uncertainty and can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

We will consider non-commuting operators A,B, i.e., those for which AB [not equal to] BA. We define the commutator and the anticommutator, respectively, as

[A,B] := AB - BA, [[A,B].sub.+] := AB + BA,

both with domain D(AB) [intersection] D(BA). For not-necessarily commuting operators, their covariance with respect to a function f is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

I being the identity operator.

A (not necessarily bounded) densely defined linear operator A in a Hilbert space H is said to be symmetric if D(A) [subset] D(A*) and Af = A*f for f [member of] D(A). Equivalently, A is symmetric if and only if <Af, g> = <f, Ag> for all f, g [member of] D(A). If D(A) = D(A*), then A is self-adjoint. We call A normal if A is closed and densely defined, and if A* A = AA*. Then we have D(A) = D(A*) and [parallel]Af[parallel] = [parallel]A*f[parallel] for every f [member of] D(A). Hence, an operator is self-adjoint if and only if it is both symmetric and normal.

EXAMPLE 1 (see [4, Corollary 1.35]) The standard operators in the context of uncertainty principles are the position operator A defined by Af(x) = xf(x) and the impulse operator B = -id/dx. Their commutator is [A,B] = iI , the "imaginary" identity operator. One can show that, with D(A) = {f [member of] [L.sup.2](R) : [integral] [x.sup.2][[absolute value of f(x)].sup.2] dx < [infinity]} and D(B) = {f [member of] [AC.sub.loc] (R) [intersection] [L.sup.2](R) : [integral] [[absolute value of f'(x)].sup.2] dx < [infinity]}, both operators are self-adjoint.

EXAMPLE 2 (see [9, Section 5]) On the torus, the choice Af(x) = xf(x) for periodic functions f [member of] [L.sup.2.sub.2[pi]] would yield an unexceptable dependence of [[tau].sub.A](f) and [[sigma].sub.A](f) on the choice of the integration bounds. Here, we have to take the position operator Af(x) = [e.sup.ix] f(x) instead. Then we have A*f(x) = [e.sup.-ix] f(x). Hence, A is a unitary operator, with D(A) = [L.sup.2.sub.2[pi]]. The commutator with B = -id/dx being self-adjoint on D(B) = {f [member of] [AC.sub.2[pi]] : f' [member of] [L.sup.2.sub.2[pi]]} is [A,B] = -A.

3. UP's for symmetric and normal operators. In the literature we usually find uncertainty principles only for self-adjoint operators (see e.g. [3, Theorem 1.34]). This is a restriction that is sometimes painful. On the one hand, unbounded symmetric operators can have different (or no) self-adjoint extensions where D(A) = D(A*), and these domains are often difficult to determine. On the other hand, certain function spaces require a normal but non-symmetric position operator.

Therefore, we start proving a more general theorem where the two operators in question can be either symmetric or normal (or both), also allowing a pair of one symmetric and one normal operator.

THEOREM 3.1. If A,B : H [right arrow] H are symmetric or normal then

[parallel](A - a)f[parallel] [parallel] (B - b)f[parallel] [greater than or equal to] 1/2 [absolute value of <[A,B]f, f>], (3.1)

[parallel](A - a)f[parallel] [parallel] (B - b)f[parallel] [greater than or equal to] 1/2 [absolute value of <[[A - aI, B - bI]+ f,f>].sub.+], (3.2)

for all f [member of] D(AB) [intersection] D(BA) and all a, b [member of] C. Equality is attained if and only if there exist constants [c.sub.1], [c.sub.2], [d.sub.1], [d.sub.2] [member of] C with ([absolute value of [c.sub.1]] + [absolute value of [d.sub.1]])( [absolute value of [c.sub.2]] + [absolute value of [d.sub.2]]) > 0 such that,

[c.sub.1](A* - [bar.a])f = [d.sub.1](B - b)f and [c.sub.2](A - a)f = [d.sub.2](B* - [bar.b])f, (3.3)

and, additionally, either at least one of the constants is zero, or [d.sub.1]/[c.sub.1] = [+ or -] [bar.[d.sub.2]]/[bar.[c.sub.2]] with the plus in case of (3.2) and the minus in case of (3.1), respectively.

Proof . For any linear operator A and any a [member of] C, the adjoint of A - aI is A* - [bar.a]I. Under the assumptions of the theorem, we have that

[parallel](A* - a)[parallel]f = [parallel](A - a)f[parallel], [parallel] (B* - [bar.b])f[parallel] = [parallel] (B - b)f[parallel]. (3.4)

Let us denote

[C.sub.-] = <[A - a, B - b]f, f> = <[A,B]f, f>, [C.sub.+] = <[[A - a, B - b].sub.+] f, f>.

For both of these, it follows that

[absolute value of [C.sub.[+ or -]]] [absolute value of <(B - b)f, (A* - [bar.a])f>] + [absolute value of <(A - a)f, (B* - [bar.b])f>] (3.5)

[less than or equal to] [parallel](A* - [bar.a])f[parallel] [parallel](B - b)f[parallel] + [parallel] (A - a) f[parallel] [parallel] (B* - [bar.b])f[parallel] (3.6)

using Cauchy-Schwarz-Bunjakovski inequality. Now, (3.4) yields (3.1) and (3.2).

Equality is attained for the second inequality (3.6) if and only if (3.3) holds for constants [c.sub.1], [c.sub.2], [d.sub.1], [d.sub.2] [member of] C with [absolute value of [c.sub.1]] + [absolute value of [d.sub.1]] > 0 and [absolute value of [c.sub.2]] + [absolute value of [d.sub.2]] > 0. If all these constants are nonzero then from (3.4) we obtain [absolute value of [d.sub.1]/[c.sub.1]] = [absolute value of [d.sub.2]/[c.sub.2]], and equality in (3.5) means that [absolute value of [bar.[d.sub.1]]/[c.sub.1] [+ or -] [d.sub.2]/[c.sub.2]] = [absolute value of [d.sub.1]/[c.sub.1]] + [absolute value of [d.sub.2]/[c.sub.2]] Hence, we have [d.sub.1]/[c.sub.1] = [+ or -] [bar.[d.sub.2]]/[bar.[c.sub.2]], where the plus stands in case of [C.sub.+] and the minus in case of [C.sub.-]. If at least one of the constants is zero, then equality in (3.5) follows trivially.

As far as equality is concerned, note that the case that (at least) one of the constants [c.sub.1], [c.sub.2], [d.sub.1], [d.sub.2] is zero means that f is in the kernel of at least one of the operators (A - aI), (A* - [bar.a]I), (B - bI), (B* - [bar.b]I). If (A - a)f = 0 then f is an eigenvector of A and a = [[tau].sub.A](f) is the corresponding eigenvalue.

For symmetric operators and real numbers a, b, the inequalities can be united thereby sharpening each other. A special case of this was given in [1, formula (15.87)].

THEOREM 3.2. If A and B are symmetric operators in a Hilbert space H, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

for all f [member of] D(AB) [intersection] D(BA) and all a, b [member of] R. Equality holds if and only if (A - a)f and (B - b)f are scalar multiples of one another.

Proof . In the proof of Theorem 3.1, we replace inequality (3.5) by the exact value

[absolute value of <(B - b)f, (A - a)f>] = [square root of ([([??]<(B - b)f, (A - a)f>).sup.2] + [(R<(B - b)f, (A - a)f>).sup.2])]

and realize that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So far, we have stated inequalities for arbitrary numbers a; b. They can be used if we consider the case of equality and wish to have a variety of solutions. Or, we can ask for which a; b the left-hand side of the functional inequalities (3.1), (3.2) and (3.7) is minimized. In our case, the minimum of [parallel] (A - a)f [parallel] is, for all f, the uncertainty of A, namely when a is the orthogonal projection of Af to f, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

COROLLARY 3.3. If A,B are symmetric or normal operators in a Hilbert space H, then

[[sigma].sub.A](f) [[sigma].sub.B](f) [greater than or equal to] 1/2 [absolute value of <[A,B]f, f>] (3.8)

and

[[sigma].sub.A](f) [[sigma].sub.B](f) [greater than or equal to] [cov.sub.AB](f), (3.9)

for all f [member of] D(AB) [intersection] D(BA); f [not equal to] 0.

Both (3.8) and (3.9) include uncertainties and, thus, can be called uncertainty principles. They state that the product of uncertainties of two (symmetric or normal) operators in a Hilbert space is bounded from below by the expectation values of their commutator (the "classical" UP) and their anticommutator. The other way round, we have an estimate for the latter ones from above in form of the uncertainty product including the simple fact that the covariance is bounded from above by the square root of the product of variances. Both directions can be improved for symmetric operators as follows from Theorem 3.2, squaring and subtracting the one or the other term from the right-hand side as the case may be.

COROLLARY 3.4. If A and B are symmetric operators in a Hilbert space H, then

[[sigma].sub.A](f) [[sigma].sub.B](f) [greater than or equal to] 1/2 [square root of ([[absolute value of <[A,B]f,f>][.sup.2]] + [4cov.sup.2.sub.AB](f))] (3.10)

for all f [member of] D(AB) [intersection] D(BA); f [not equal to] 0. Equality holds if and only if (A - [[tau].sub.A](f))f and (B - [[tau].sub.B](f))f are scalar multiples of one another.

Note that the right-hand side in (3.10) is greater or equal to

[absolute value of <(A - [[tau].sub.A](f))f, (B - [[tau].sub.B] (f))f>] = [absolute value of <Af,Bf> - [[tau].sub.A](f)[[tau].sub.B](f) [[parallel]f[parallel].sup.2]].

4. UP's on the real line and on the circle. Applying (3.8) to the operators from Example 1 we get the classical Heisenberg UP on the real line (see [2, Section 2.8] for the two common versions). Example 2 yields an UP for 2[pi]-periodic functions (see [11, Theorem 2.1]; for an asymptotic connection between the two see [12]). In both cases, in principle, operator A yields an expectation value which is known as first (trigonometric) moment, whereas B stands for the first derivative. An obvious generalization gives uncertainty principles for higher moments and higher derivatives.

First, we look at H = [L.sup.2](R) and take

[A.sub.k]f(x) = [x.sup.k]f(x) and [B.sub.l]f(x) = [(-id/dx).sup.l]f(x)

for arbitrary fixed k, l [member of] N, with D([A.sub.k]) = {f [member of] [L.sup.2](R) : [integral] [x.sup.2k][[absolute value of f(x)].sup.2] dx < [infinity]} and D([B.sub.l]) = {f [member of] [L.sup.2](R) [intersection] [C.sup.l](R) : [f.sup.(l)] [member of] [L.sup.2](R}} Their commutator is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Obviously, [A.sub.k] and [B.sub.l] are symmetric operators. (Clearly, we could specify the domains of self-adjointness for both of them.) So, from Corollary 3.3 we obtain

COROLLARY 4.1. For f [member of] D([A.sub.k][B.sub.l]) [intersection] D([B.sub.l][A.sub.k]), the uncertainty principle

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

holds.

In H = [L.sup.2.sub.2[pi]], we consider

[A.sub.m]f(x) = [e.sup.imx] f(x) and [B.sub.l]f(x) = [(-id/dx)'f(x).sup.l] f(x)

for any fixed m [member of] Z, l [member of] N, with D([A.sub.m]) = [L.sup.2.sub.2[pi]] and D([B.sub.l]) = [C.sup.l.sub.2[pi]]. The commutator is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [A.sub.m] is a unitary operator and [B.sub.l] is symmetric, from Corollary 3.3 we obtain a similar result as for [L.sup.2](R).

COROLLARY 4.2. For f [member of] [C.sup.l.sub.2[pi]], the uncertainty principle

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

holds.

Whether these inequalities are known or useful is not clear to the author at the moment. There might be applications for the one or the other special commutator to be estimated from above, or to compute a lower bound for one variance by dividing the inequality by the other variance, respectively.

5. UP's on the interval. For ultraspherical expansions, an UP has been proved by Rosler and Voit in [14] making use of the Dunkl operator as the angular momentum operator. Here we give a more general UP for a whole class of "position" operators. Furthermore, we obtain a similar UP for the Laplace operator. The question of which operators can be considered as the "natural" position and momentum operators on the interval is not answered, yet.

We consider the weighted Hilbert space H = [L.sup.2([0,[pi]], [[omega].sub.[alpha]]), for fixed [alpha] [greater than or equal to] -1/2, with probability measure

d[[omega].sub.[alpha]] (t) = [c.sub.[alpha]] [(sin t).sup.2[alpha]+1] dt, [c.sub.[alpha]] = [GAMMA](2[alpha] + 2) / [GAMMA][([alpha] + 1).sup.2] [2.sup.2[alpha]+1].

The polynomials orthogonal with respect to this measure are the ultraspherical (or Gegenbauer) polynomials

[p.sup.([alpha]).sub.n] (t) := [sub.2][F.sub.1] (-n, n + 2[alpha] + 1; [alpha] + 1; [sin.sup.2] t/2) = [P.sup.[alpha][alpha].sub.n] (cos t) (n [member of] [N.sub.0]).

In order to define appropriate operators A, B in H, no general theory seems to be available, at present. In the Heisenberg UP, operator B was a differential operator of first kind being a root of the Laplace operator, at the same time. Ultraspherical polynomials are solutions of the differential equations [L.sub.[alpha]] [p.sup.([alpha]).sub.n] = n(n + 2[alpha] + 1)[p.sup.([alpha]).sub.n], for n [member of] [N.sub.0] , where the Laplace operator is defined as

[L.sub.[alpha]] f(t) = - (f"(t) + (2[alpha] + 1) cos t / sin t f'(t))

with domain D([L.sub.[alpha]]) = {f [member of] [C.sup.2][0, [pi]] : f'(0) = f'([pi]) = 0}. A root (in a generalized meaning) is given in [14] by the corresponding Dunkl operator

([T.sub.[alpha]]f)(t) := f'(t) + ([alpha] + 1 / 2) cos t / sin t (f(t) - f(-t))

with domain D([T.sub.[alpha]) = {f [member of] AC[-[pi],[pi]] : f' [member of] [L.sup.2] [-[pi],[pi]], f(-[pi]) = f([pi])}. Then, I[T.sub.[alpha]] is symmetric (see [14, Lemma 3.1]). This differential-difference operator requires the extension of the original interval to [-[pi],[pi]]. As in [14], we define the probability measure

d[[??].sub.[alpha]](t) = [c.sub.[alpha]] / 2 [[absolute value of sin t].sup.2[alpha]+1] dt

as well as the even extension and the restriction operators

e : [L.sup.2]([0,[pi]], [[omega].sub.[alpha]]) [right arrow] [L.sup.2]([-[pi], [pi]],[[??].sub.[alpha]]), e(f)(t) := f([absolute value of t]),

r : [L.sup.2]([-[pi],[pi]], [[??].sub.[alpha]]) [right arrow] [L.sup.2] ([0,[pi]],[[omega].sub.[alpha]]), f [??] f[|.sub.[0,[pi]],

which are isometric isomorphisms between [L.sup.2]([0,[pi]], [[omega].sub.[alpha]]) and [L.sup.2.sub.e] ([-[pi],[pi]], [[??].sub.[alpha]]) being the subspace of even functions. Then, the Dunkl and the Laplace operators are related by

[L.sub.[alpha]]f = -r([T.sup.2.sub.[alpha]] (e(f))) for f [member of] D([L.sub.[[alpha]).

5.1. UP's for the Dunkl operator. We will work in H' = [L.sup.2]([-[pi],[pi]], [[??].sub.[alpha]]) for a moment and require f [member of] H' to be even. Let h [member of AC[-[pi],[pi]] be fixed, and define the operators A,B : H' [right arrow] H',

Af = hf , Bf = -I[T.sub.[alpha]]f.

Their commutator turns out to be

[A,B]f(t) = i(h'(t)f(t) + ([alpha] + 1 / 2) cos t / sin t (h(t) - h(-t))f(-t)).

So, for g [member of] [L.sup.2.sub.e] ([-[pi],[pi]], [[??].sub.[alpha]]) , we simply have

[A,B]g = ig[T.sub.[alpha]]h.

In order to guarantee h [member of] D([T.sub.[alpha]]), we have to require h([pi]) = h(-[pi]) ; and then it holds that D(AB) = D(BA) = D([T.sub.[alpha]]). Clearly, A is a normal operator (allowing complex-valued "positioning" functions h) and B is symmetric. Due to (3.8) for these operators an UP follows easily.

THEOREM 5.1. Let [alpha] [greater than or equal to] -1/2. In H' = L.sup.2]([-[pi],[pi]], [[??].sub.[alpha]]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all f; h [member of] D([T.sub.[alpha]]), f even.

Proof . For even f [member of] D([T.sub.[alpha]]), we have that [T.sub.[alpha]] f = f' is odd. Hence <Bf, f> = 0 and [[sigma].sub.B](f) = [parallel]f'[parallel]. Then the assertion follows from (3.8).

Let us consider special cases for h [member of] D([T.sub.[alpha]]). If h is even, then the expectation value can be reduced to <hf, f> = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], but

<[A,B]f, f> = i<f[T.sub.[alpha]]h, f> = i<h'f, f> = 0,

and so the inequality becomes trivial. For odd h, we have <hf, f> = 0. In general, h = [h.sub.e] + [h.sub.o], where [h.sub.e] is the even and [h.sub.o] the odd part of h, respectively, and (omitting r when restricting the integration to [0, [pi]], for simplicity)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.3)

Now, let us look at the result by Rosler and Voit. They defined the generalized mean

[[tau].sub.[alpha]](f) := Z [[integral].sup.[pi].sub.0] cos t [[absolute value of f(t)].sup.2] [d[omega].sub.[alpha]](t)

and proved the following

THEOREM 5.2. ([14, Theorem 2.2]) Let [alpha] [greater than or equal to] 1/2 and f [member of] [L.sup.2]([0, [pi]], [[omega].sub.[alpha]]) with [parallel]f[parallel] =

1. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the constant [([alpha] + 1).sup.2] is optimal.

In fact, we need f [member of] D([L.sub.[alpha]]). Then, this theorem is a special case of Theorem 5.1 for h(t) = [e.sup.it] which yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Moreover, since

[T.sup.2.sub.[alpha]]g(t) = g"(t) + (2[alpha] + 1) cos t / sin t g'(t) (g [member of] D([T.sup.2.sub.[alpha]]))

is even, for even g, and since I[T.sub.[alpha]] is symmetric we have, for f [member of] D([L.sub.[alpha]]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we use h(t) = sin t instead, then Theorem 5.1 yields

COROLLARY 5.3. Let [alpha] [greater than or equal to] -1/2 and f [member of] D([L.sub.[alpha]]) . Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the latter being valid also for f [member of] AC([0, [pi]]).

For real-valued functions f, these inequalities are sharper than the one in Theorem 5.2 due to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

But in general, this is just another example of uncertainty principles in [L.sup.2]([0, [pi]], [[omega].sub.[alpha]]).

Two other similar UP's in view of higher moments (cp. Section 4) follow from Theorem 5.1 taking h(t) = sin 2t and h(t) = [e.sup.2it], respectively. Among other interpretations they provide further lower bounds for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

COROLLARY 5.4. Let [alpha] [greater than or equal to] -1/2 and f [member of] AC([0, [pi]]). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof . Due to (5.3), for both h(t) = sin 2t = 2 sin t cos t and h(t) = [e.sup.2it], we have on the right-hand side of (5.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

from which the two forms in the assertion can be easily deduced. The rest follows from (5.2) and Theorem 5.1.

Last but not least, let us look at B as an operator in H = [L.sup.2]([0, [pi]] [[omega].sub.[alpha]]) itself. When dealing with even functions f [member of] D([T.sub.[alpha]), we had [T.sub.[alpha]]f = f'. Hence, let Bf = if'. For f, g [member of] D(B) = {f [member of] [C.sup.1][0,[pi]] : f(0) = f([pi]) = 0}, integration by parts gives us

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and so the adjoint is B*f(t) = i(f'(t) + (2[alpha] + 1) cos t /sin t f(t)). Thus, except for [alpha] = -1/2, operator B is not normal since (BB* - B*B)f = (2[alpha] + 1)f(t)/ [sin.sup.2] t . Hence, we cannot directly gain our results for the first derivative in H without the "detour" via H'.

5.2. UP's for the Laplace operator. In view of higher derivatives as in Section 4, we want to consider the Laplace operator [L.sub.[alpha]] in H = [L.sup.2]([0, [pi]], [[omega].sub.[alpha]]). Let

Af = hf, Bf = [L.sub.[alpha]]f,

where h [member of] D([L.sub.[alpha]]) = {f [member of] [C.sup.2][0, [pi]]: f'(0) = f'([pi]) = 0} is fixed. The commutator is in this case

[A,B]f = [hL.sub.[alpha]]f - [L.sub.[alpha]](hf) = 2f'h' - [fL.sub.[alpha]]h.

Again, Corollary 3.3 provides an uncertainty principle for these operators. THEOREM 5.5. Let [alpha] [greater than or equal to] -1/2. In H = [L.sup.2]([0, [pi]], [[omega].sub.[alpha]]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all f, h [member of] D([L.sub.[alpha]).

Proof . Since A is normal and B is symmetric, we can apply (3.8) to the operators A;B defined above.

This choice of the operators is questionable. Let us look at the following example. For real-valued f and h(t) = [e.sup.ikt] or h(t) = [p.sup.[alpha].sub.n](t) ; k [member of] Z, n [member of] [N.sub.0], the right-hand side vanishes, i.e.,

<2f'h' - [fL.sub.[alpha]]h, f> = 0,

with the exception <[2ike.sup.ik.]f' - [fL.sub.-1] [e.sup.ik.], f> = ik([(-1).sup.k][f.sup.2]([pi]) - [f.sup.2](0))/[pi]. So, for these operators A and real-valued functions the UP's above become trivial.

6. UP's on the sphere. We consider the unit sphere [S.sup.2] [subset] [R.sup.3] and the Hilbert space H = [L.sup.2]([S.sup.2]) . In [10], Narcowich and Ward gave an UP on the sphere using the multiplication with the surface variable [eta] [member of] [S.sup.2] as position operator and the angular momentum operator [OMEGA] -iL* = -I[eta] x [nabla]* as momentum operator where [nabla]* denotes the surface gradient and L* the surface curl gradient. Note that both are roots of the Laplace-Beltrami operator [nabla]* of the unit sphere in the sense that

[DELTA]* = L*L* = [nabla]* [nabla]*

which is the tangential part of the Laplace operator in [R.sup.3]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

See [6] for more details. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denote the Euclidean norm in [R.sup.3].

THEOREM 6.1. ([10, Theorem 5.1]) If f([eta]) is a twice-continuously differentiable complex-valued function on [S.sup.2], and if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For real-valued functions, another proof can be found in [5, Theorem 1.3.1].

If we set

Af([eta]) = [eta]f([eta]), Bf([eta]) = -iL*f([eta]) = - i([eta] x [[nabla]*)f([eta]), (6.2)

with D(A) = H and D(B) = {f [member of] AC([S.sup.2]) : [nabla]*f [member of] H [cross product] H [cross product] H}, we notice that A,B are vector-valued operators which do not fit in our scheme in Section 3. In particular, the commutator is not easily definable. It will have to be studied because it does not suffice to consider the single operator components [A.sub.k], [B.sub.l] : H [right arrow] H, for k, l = 1, 2, 3 , as we will see in the sequel.

We define the (normalized) expectation vector of any vector-valued operator B by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, the variance of such a vector-valued operator can be written, for any f [member of] D(B), as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.3)

Furthermore, for our choice of A in (6.2) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can easily prove the following:

THEOREM 6.2. If A;B are vector-valued operators with symmetric or normal components [A.sub.k] [B.sub.l] acting in a Hilbert space H, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all f [member of] ([[intersection].sub.k,l] D([A.sub.k] [B.sub.l])) [intersection] ([[intersection].sub.k,l] D([B.sub.l][A.sub.k])).

Proof . Apply (3.8) to each summand of the right-hand side, and then, in the first inequality, use (6.3).

Our operators A and B defined in (6.2) obviously have symmetric components. We compute

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, from Theorem 6.2, we obtain in this case

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.4)

where the constant in the lower bound is only half of the constant in (6.1). The second inequality of Theorem 6.2 gives us another inequality, now for uncertainties instead of variances. Here, as in (6.1), the constant should be optimal.

COROLLARY 6.3. The operators A,B defined in (6.2) satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all f [member of] D(B) .

Note that [[tau].sub.A](f) is the center of mass of the density [[absolute value of f].sup.2][[parallel]f[parallel].sup.2] distributed around the sphere and, hence, is located in the unit ball. So, there are a lot of functions f with [[tau].sub.A](f) = 0 for which the inequalities above become trivial. The second problem is that, for real-valued f,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which raises interpretation problems in terms of frequency localization on the sphere (mentioned in [7]).

Alternatively, we propose to consider

Af([eta]) = [eta]f([eta]), Bf([eta]) = - I[nabla]f([eta]) (6.5)

the components of which are also symmetric and have commutators

[[A.sub.k],[B.sub.l]]f = if [[delta].sub.k,l].

From Theorem 6.2 we conclude

COROLLARY 6.4. For A;B defined in (6.5), we have

[var.sub.A](f) [var.sub.B](f) [greater than or equal to] 3 / 4 [[parallel]f[parallel].sup.4]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all f [member of] D(B) .

These inequalities using [nabla]* are obviously nontrivial, and their right-hand sides remember us of the Heisenberg UP. The author suggests studying [nabla]* as operator for frequency analysis. From the discrepancy between (6.1) and (6.4), we can guess that it should be possible to prove that [var.sub.A](f) [var.sub.i[nabla]*] (f) [greater than or equal to] 9 / 4 [[parallel]f[parallel].sup.4], and that probably this bound is sharp. An appropriate generalization of the UP's for vector-valued operators will likely resolve this issue.

On the other hand, we have Theorem 6.1 and Corollary 6.3 using the angular momentum operator L? as momentum operator. The role of this operator should be reconsidered within the framework of time-frequency analysis.

Note that for B in both (6.2) and (6.5) and for f [member of] [C.sup.2]([S.sup.2]), we have

[3.summation over (l=1]] [[parallel][B.sub.l]f[parallel].sup.2] = [3.summation over (l=1]] <[B.sup.2.sub.l]f,f> = - <[DELTA]*f,f>

holding two different lower bounds by Theorem 6.1 and by Corollary 6.4.

Acknowledgement. The author would like to thank Professor C. A. Micchelli for fruitful discussions in 1996 and the anonymous referee for insightful comments.

REFERENCES

[1] L. COHEN, Time-frequency analysis, Prentice Hall PTR, Princeton 1989.

[2] H. DYM, H. P. MCKEAN, Fourier series and integrals, Academic Press, San Diego 1972.

[3] G. B. FOLLAND, Harmonic analysis in phase space, Princeton University Press, 1995.

[4] G. B. FOLLAND, A. SITARAM, The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl. 3(3) (1997), 207-238 .

[5] W. FREEDEN, Multiscale modelling of spaceborne geodata, Teubner, Stuttgart 1999.

[6] W. FREEDEN, T. GERVENS, M. SCHREINER, Constructive approximation on the sphere, Clarendon Press, Oxford 1998.

[7] W. FREEDEN, U. WINDHEUSER, Combined spherical harmonic and wavelet expansion - a future concept in earth's gravitational determination, Appl. Comput. Harmon. Anal. 4 (1997), 1-37.

[8] W. HEISENBERG, Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Physik 43 (1927), 172-198.

[9] F. J. NARCOWICH, J. D. WARD,Wavelets associated with periodic basis functions, Appl. Comput. Harmon. Anal. 3(1) (1996), 40-56.

[10] F. J. NARCOWICH, J. D. WARD, Nonstationary wavelets on the m-sphere for scattered data, Appl. Comput. Harmon. Anal. 3(4) (1996), 324-336.

[11] J. PRESTIN, E. QUAK, Optimal functions for a periodic uncertainty principle and multiresolution analysis, Proc. Edinb. Math. Soc. 42 (1999), 225-242.

[12] J. PRESTIN, E. QUAK, H. RAUHUT, K. SELIG, On the connection of uncertainty principles for functions on the circle and on the real line, Preprint, 2001.

[13] H. RAUHUT, Best time-localized trigonometric polynomials and wavelets, in preperation.

[14] M. R OSLER, M. VOIT, An uncertainty principle for ultraspherical expansions, J. Math. Anal. Appl. 209 (1997), 624-634.

[15] K. SELIG, Trigonometric wavelets and the uncertainty principle, in: Approximation theory (M.W.Muller, M. Felten, D. H. Mache, eds.), Akademie Verlag, Berlin 1995, 293-304.

KATHI K. SELIG ([dagger])

* This work was partly supported by the European Union, contract number HPRN-CT-1999-00117. Received December 21, 2000. Accepted for publication May 7, 2001. Communicated by Sven Ehrich.

([dagger]) TU Munchen, Zentrum Mathematik, D-80290 Munchen (selig@ma.tum.de)
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