Approximation of bandlimited functions on a non-compact manifold by bandlimited functions on compact submanifolds.

Abstract

Although interesting results about sampling theory on manifolds have recently been obtained, there are still many results from classical sampling theory on the real line which have yet to be extended to this more general situation. Such generalized results would be of interest, in particular, in mathematical physics. Here, we consider the fact that the space of [OMEGA]-bandlimited functions is the strong limit (as L [right arrow] [infinity]) of the subspace of [OMEGA]--bandlimited trigonometric polynomials on [-L, L], when both subspaces are viewed as subspaces of [L.sup.2](R). We generalize this result to complete [C.sup.[infinity]] Riemannian manifolds of arbitrary dimension.

Key words and phrases : Bandlimited, Paley-Wiener space, Riemannian manifolds.

2000 AMS Mathematics Subject Classification--42A10,42A65.

1 Introduction

A bandlimit can be viewed as a cutoff on the spectrum of the Laplacian on the real line. The Laplacian [DELTA] on the real line is -[d.sup.2/[dx.sup.2], and if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the characteristic function of the interval [0, [[OMEGA].sup.2]], then the subspace of [OMEGA]-bandlimited functions, B(R, [OMEGA]), is the image of [L.sup.2](R) under the spectral projection, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([DELTA]). The projection operator, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([DELTA]), is defined using the functional calculus for self-adjoint operators. This operator projects, roughly speaking, onto the subspace spanned by the 'eigenvectors' to [DELTA] whose eigenvalues lie in the interval [0, [OMEGA].sup.2]].

This observation shows that a natural generalization of the space of bandlimited functions to a manifold M is B(M, [OMEGA]) := [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([DELTA])[L.sup.2](M) where [DELTA] is the Laplacian on M [10]. If M is a manifold that has compact closure, and a boundary, then there is no unique choice of a self-adjoint Laplacian on M. For example, by choosing different boundary conditions on the boundary of M, e.g., Dirichlet or Neumann boundary conditions, one can define different self-adjoint Laplacians on M. This means that one can define B(M, [OMEGA]) := [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([DELTA]')[L.sup.2](M) where [DELTA]' is any choice of a self-adjoint Laplacian on M. In other words, if M has compact closure, there is no unique natural choice for the subspace of [OMEGA]-bandlimited functions. This will be discussed in detail in section 2.

Sampling theory on Riemannian manifolds with compact closure is particularly simple. Indeed, if M is such a manifold, an appropriate choice of self-adjoint Laplacian on 114 yields a finite dimensional subspace, B(M, [OMEGA]). This will be discussed in detail in Section 4.

Now suppose that M is a [C.sup.[infinity]] complete Riemannian manifold. Let [K.sub.n] be a sequence of nested, oriented submanifolds of M ([K.sub.n] [subset] [K.sub.n+1]) with compact closures and smooth, [C.sup.[infinity]] boundaries whose union is all of M. A natural question we address in this paper is whether the projectors [P.sub.n,[OMEGA]]a onto the subspaees B([K.sub.n],[OMEGA]) converge in a suitable sense to the projector [P.sub.[OMEGA]] onto B(M,[OMEGA]) in the limit if these subspaces are viewed as subspaces of [L.sup.2](M). Here [P.sub.n,[OMEGA]] := [chi square]n[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([[DELTA].sub.K])[chi square]n, [P.sub.[OMEGA] := [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([DELTA]), [chi square]n denotes the projector of [L.sup.2](M) onto [L.sup.2] ([K.sub.n]), and [[DELTA].sub.K] denotes an arbitrary choice of self-adjoint Laplacian on K. The affirmative answer to this question will be given by the proof of the following proposition.

Proposition 1. If a, b [member of] R, a < b; a, b [not member of] [[sigma].sub.pp]([DELTA]), then [chi square]n[[chi square].sub.(a,b)]([[DELTA].sub.n])[chi square]n converges strongly to [[chi square].sub.[a,b]](DELTA).

Here [[DELTA].sub.n] is an arbitrary choice of a self-adjoint Laplacian on [K.sub.n] and [[sigma].sub.pp]([DELTA]) denotes the set of all eigenvalues of [DELTA]. In particular, this proposition claims that [P.sub.n] converges strongly to [P.sub.[OMEGA]] provided that 0 and [[OMEGA].sup.2] are not eigenvalues of [DELTA]. The proof of the above proposition is given in Section 3 and is the main result of this paper.

For example, if M = R, then B(R, [OMEGA]) is the regular space of [OMEGA]-bandlimited functions. In this case one can choose a sequence of nested intervals (connected submanifolds of R with compact closure) [K.sub.n] := (-[L.sub.n], [L.sub.n]). One can further choose the particular self-adjoint Laplacian [[DELTA].sub.n] whose domain consists of functions which obey periodic boundary conditions. Then given any f [member of] [L.sup.2](R) it is not difficult to see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the characteristic function of the interval I. This image (1) of f under the projector [P.sub.n],[OMEGA] is just a partial sum of the Fourier series for f on [-[L.sub.n], [L.sub.n]]. In fact, it is the largest partial sum for f on that interval which contains no complex exponentials [e.sup.ikjx] whose frequencies [k.sub.j] are greater in magnitude than the bandlimit [OMEGA]. This shows that each [P.sub.n,[OMEGA]f is an [OMEGA]-bandlimited trigonometric polynomial of period 2[L.sub.n] truncated to the interval [[bar.K].sub.n] = [-[L.sub.n], [L.sub.n]]. Proposition 1 then implies that [P.sub.n,[OMEGA]]f converges to [P.sub.n]f as n [right arrow] [infinity]. In particular, if f = [P.sub.[OMEGA]]f is already bandlimited, we see that it is the [L.sup.2] limit of [OMEGA]-bandlimited trigonometric polynomials, a fact already proven using more elementary methods in [9]; see also [15].

2 Notation and assumptions

2.1 Symmetric operators and self-adjoint extensions

Throughout this paper we will use the concept of a symmetric operator and self-adjoint extensions of a symmetric operator which are densely defined in a separable Hilbert space. Let us briefy review the definitions of these concepts. Given a linear operator, T, defined on a dense linear subspace, D(T), of a separable Hilbert space, H, consider the domain of its adjoint:

D([T.sup.*] := {[phi] [member of] H | [??][[phi].sup.*] [member of] H s.t. (T [empty set], [phi]) = ([empty set], [[phi].sup.*] [??] [empty set] [member of] [??](T)}. (2)

Recall that the adjoint operator [T.sup.*] is then defined by [T.sup.*][phi] = [[phi].sup.*] for all [phi] [member of] D([T.sup.*]). Now a linear operator S with dense domain D(S) is called symmetric if <S[phi], [psi]> = <[phi], S[psi]) for all [psi], [phi] [member of] D(S). If A, B are linear operators, D(A) [subset] D(B) and [B.sub.|D(A)] = A, we write A [subset] B and say that B is an extension of A. By the definition of the domain of the adjoint of a symmetric operator S, it is clear that S [subset] [S.sup.*]. S is called self-adjoint if S = [S.sup.*]. If S is a bounded symmetric operator defined on all of H, then it is automatically self-adjoint. For a general unbounded symmetric operator, however, it can be that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that S [not equal] [S.sup.*]. For example, let D be the operator defined on the dense domain D(D) = {[phi] [member of] [L.sup.2][-[pi], [pi]] |[phi] [member of] AC[-[pi], [pi]]; [phi]([pi]) = 0 = [phi](-[pi]); [phi]' [member of] [L.sup.2][-[pi], [pi]]} [subset] [L.sup.2][-[pi], [pi]] by D[phi](t) = i[phi]'(t) where [phi]'(t) := d/dt [phi](t). Here, AC[-[pi], [pi]] denotes the set of all absolutely continuous functions on [-[pi], [pi]]. Using integration by parts, it is easy to verify that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

since functions in the domain of D vanish at the end points. This shows that D is symmetric. Furthermore, it can be shown that the domain of [D.sup.*] is D([D.sup.*]) = {[phi] [member of] [L.sup.2][-[pi], [pi]] | [phi] [member of] AC[-[pi], [pi]] ; [phi]' [member of] [L.sup.2][-[pi], [pi]]} ([1], section 49). This shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that D is not self-adjoint.

Now if A and B are densely defined linear operators and A [subset] B, then it is not difficult to verify that [B.sup.*] [subset] [A.sup.*]. One can show that in many cases, by enlarging the domain of a symmetric operator S and, hence, shrinking the domain of its adjoint, one can construct a self-adjoint operator. Such a selfadjoint operator S' [contains] S is called a self-adjoint extension of S. Returning to the above example, one can construct a one-parameter family of self-adjoint extensions of D, [{D([alpha])}.sub.[alpha][member of][[0,1)] by defining D([alpha]) = [D.sup.*][|.sub.D(D(alpha]))] where D(D([alpha])) = {[phi] [member of] D([D.sup.*]) | [phi] ([pi]) = [e.sup.i2[pi][alpha]][phi](-[pi])}. Using integration by parts, it is indeed easy to verify that D([alpha]) = [D([alpha]).sup.*] so that each D([alpha]) is self-adjoint.

A densely defined linear operator T is called closed if its graph g(T) [subset] H [direct sum] H is closed ([1], section 46). Here G(T) = {([phi],T[phi]) [member of] H [direct sum] H|[phi] [member of] D(T)}. If T is not closed but there exists a T' [contains] T such that T' is closed, then T is called closable. Any closable operator T has a minimal closed extension [bar.T], which is called its closure ([1], section 38). In fact, one can show that if T is closable, then [bar.T] = [([T.sup.*]).sup.*] [1]. If r is a densely defined linear operator, then its adjoint [T.sup.*] is always closed. This is a simple consequence of the definition of the adjoint. A symmetric operator is always closable since S [subset] [S.sup.*] so that [S.sup.*] is a closed extension of S. A densely defined symmetric operator is called essentially self-adjoint if its closure is self-adjoint. We will use this terminology throughout the paper. If S is essentially self-adjoint, it is not hard to show that [bar.S] = [S.sup.*] ([13], pg. 256). This fact will be used in the proof of Proposition 2 in Section 3.

2.2 Complete Riemannian manifolds and compact submanifolds

Let M be a complete [C.sup.[infinity]] Riemannian manifold. Recall that a Riemannian manifold is a differentiable manifold that is equipped with a metric. Completeness of a Riemannian manifold is characterized by the following theorem ([6], pg. 18).

Theorem 1. (Hopf-Rinow) Let M be a Riemannian manifold. The following are equivalent:

(a) M is complete as a metric space.

(b) Closed and bounded subsets of M are compact.

(c) M is geodesically complete.

Given any point p [member of] M, and any tangent vector [xi] at that point, there is a maximal open interval [I.sub.[xi]] [subset] R and a unique geodesic [[gamma].sub.[xi]] which passes through p, [[gamma].sub.[xi]](0) = p, whose tangent vector at p is [xi]. M is said to be geodesically complete if [I.sub.[xi]] = ]R for any tangent vector [xi] ([6], pg.18).

The following theorem shows that the assumption of completeness ensures the essential self-adjointness of the Laplacian on the domain of infinitely differentiable functions with compact support [4, 5].

Theorem 2. Every power of the Laplacian - [DELTA] of a complete [[C.sup.[infinity]] Riemannian manifold is essentially self-adjoint on the dense domain [C.sup.[infinity].sub.0](M) [subset] [L.sup.2](M).

Since we are dealing with a complete [C.sup.[infinity]] Riemannian manifold, we will let [DELTA]' denote the essentially self-adjoint Laplacian whose domain is [C.sup.[infinity].sub.0](M) and [DELTA] : [bar.[DELTA]'] denote the unique self-adjoint Laplacian, which is the closure of [DELTA]'.

Now let {[K.sub.n]}[sub.n[member of]N] be a sequence of open, connected, and oriented submanifolds of M with compact closures and smooth, [C.sup.[infinity]] boundaries. That is, [[bar.K].sub.n] is compact for each n E N. Further assume that the [K.sub.n] are nested and form an open cover of M: [K.sub.n] [subset] [K.sub.n+l] and [U.sub.n][K.sub.n] = M.

On a [C.sup.[infinity]] open manifold K with compact closure and smooth boundary, the Laplacian defined on the domain [C.sup.[infinity]].sub.0](K) is a densely defined symmetric operator in [L.sup.2]([bar.K]). Here [C.sup.[infinity].sub.0](K) denotes the set of all infinitely differentiable functions with compact support on K, i.e., each function [phi] in this domain has support contained in a compact set which is contained in K. Since K is open, each such function [phi] vanishes before the boundary of K. Using this fact and Green's formula ([6], pg. 12), it is easy to see that the Laplacian [[DELTA]'.sub.K] defined on this domain is symmetric. Indeed, Green's formula states that if f, g [member of] [C.sup.[infinity]](K) and at least one of f, g [member of] [C.sup.[infinity].sub.0](K), then

[[integral].sub.K] f[[DELTA]'.sub.K][bar.g]dV = [[integral].sub.K]([DELTA]'.sub.K]f)[bar.g]dV. (4)

Here dV denotes the canonical volume measure of [L.sup.2](M), which is uniquely determined by the Riemannian metric on K. This Laplacian is not self-adjoint. For example, using the definition of the domain of the adjoint of a linear operator, equation (6) and Green's formula (4), we see that any f [member of] [C.sup.[infinity](K) which does not vanish on the boundary of K belongs to the domain of the adjoint of [[DELTA]'.sub.K] but does not belong to the domain of [[DELTA]'.sub.K]. As in the example of the derivative operator on an interval, one can construct a self-adjoint Laplacian [[DELTA].sub.K] by extending the domain of [[DELTA]'.sub.K] to include all those functions in the domain of the adjoint of [[DELTA]'.sub.K] that obey certain boundary conditions on the boundary of K. For example, let [[??].sub.K] be the Laplacian whose domain is all those [C.sup.[infinity]] functions on K which are also continuous on [bar.K] and which obey Dirichlet boundary conditions. Then the closure [[DELTA].sub.K] of [[??].sub.K] is a self-adjoint operator that is a self-adjoint extension of [[DELTA]'.sub.K] ([3], pg. 8). Similarly imposing other boundary conditions like Neumann boundary conditions yield different selfoadjoint extensions of [[DELTA]'.sub.K].

Observe that [L.sup.2]([[bar.K].sub.n]) can be viewed as a subspace of [L.sup.2](M) in the same way that [L.sup.2][a, b] can be viewed as a subspace of [L.sup.2](R). Namely, we identify [L.sup.2]([[bar.K].sub.n]) with that subspace of square integrable functions on M which have support contained in [[bar.K].sub.n]. This identification makes sense as the measures of [L.sup.2]([[bar.K].sub.n]) and [L.sup.2](M) are defined by the metrics on [[bar.K].sub.n] and M, respectively, and the metric on [[bar.K].sub.n] is simply the restriction of the metric on M to [[bar.K].sub.n]. Hence, the measure of [L.sup.2]([[bar.K].sub.n]) is just the restriction of the measure on [L.sup.2](M) to Borel subsets of [[bar.K].sub.n]. On each submanifold [K.sub.n], let [[??].sub.n] be an arbitrary self-adjoint extension of the symmetric Laplacian [[DELTA]'.sub.n] which is defined on the dense domain [C.sup.[infinity].sub.0]([K.sub.n]) [subset] [L.sup.2]([[bar.K].sub.n]). Then [[??].sub.n] is a densely defined self-adjoint operator in [L.sup.2]([[bar.K].sub.n]). We view the operators [[??].sub.n] as operators acting on a dense domain in the larger Hilbert space [L.sup.2](M) in the following natural way. Given [??]([[DELTA].sub.n]) := [??]([??]) [direct sum] [L.sup.2][([[bar.K].sub.n]).sup.[perpendicular to]] in [L.sup.2](M) = [L.sup.2]([[bar.K].sub.n] [direct sum] [L.sup.2][([[bar.K].sub.n]).sup.[perpendicular to]] where [L.sup.2][([[bar.K].sub.n]).sup.[perpendicular to]] denotes the orthogonal complement of [L.sup.2]([[bar.K].sub.n]) in [L.sup.2](M), we define [[DELTA].sub.n] := [[??].sub.n] [direct sum] 0 on [??]([[DELTA].sub.n]). Then [[DELTA].sub.n] is a natural extension of [[??].sub.n] to a dense domain in [L.sup.2](M), and as a subspace of [L.sup.2](M) we define B([K.sub.n], [OMEGA]) := [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [chi square]n is the self-adjoint projection of [L.sup.2](M) onto [L.sup.2]([[bar.K].sub.n]). The operator [P.sub.n, [OMEGA]] := [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the self-adjoint projector of [L.sup.2](M) onto B([K.sub.n], [OMEGA]). Note that this definition of bandlimited functions on [K.sub.n] depends on the choice of boundary conditions (i.e., on the choice of self-adjoint extension [[??].sub.n] of the symmetric Laplacian [[DELTA]'.sub.n]) on [K.sub.n]. Finally observe that if [phi] [member of] [C.sup.[infinity].sub.0]([K.sub.n]) then [[DELTA].sub.n][phi] = [DELTA][phi].

3 Statement and proof of result

Proposition 1 will now be established using the assumptions of the previous section. This proposition is restated below for convenience.

Proposition 1. If a, b [member of] R, a < b; a, b [not member of] [[sigma].sub.pp]([DELTA]), then [chi square]n[[chi square].sub.(a,b)]([DELTA].sub.n])[chi square]n converges strongly to [[chi square].sub.[a,b]]([DELTA]).

Recall that [[sigma].sub.pp](A) denotes the set of eigenvalues of A. In particular, if O, [[OMEGA].sup.2] are not eigenvalues of [DELTA], then [P.sub.n,[OMEGA]] [??] [P.sub.[[OMEGA].

To prove Proposition 1, we show that the Laplacians [[DELTA].sub.n] on [K.sub.n] converge to the Laplacian [DELTA] on the full manifold using a suitable notion of convergence for unbounded self-adjoint operators. One says that a sequence of self-adjoint operators An converges to a self-adjoint operator A in the strong resolvent sense, [A.sub.n] [??] A, if there is a z [member of] C \ R for which [([A.sub.n] - z).sup.-1] converges strongly to [(A - z).sup.-1] [14]. It is known that if f is any bounded continuous function on R and [A.sub.n] [??] A, then f([A.sub.n]) [??] f(A). In particular, if [A.sub.n] [??] A then [([A.sub.n] - z).sup.-1] [??] [(A - z).sup.-1] for every z [member of] C \ IR [14].

Even more useful for our purposes, the following theorem [14] shows that if [A.sub.n] [??] A, then certain spectral properties of A are related to those of the [A.sub.n].

Theorem 3. Suppose [A.sub.n] [right arrow] A in the strong resolvent sense then (a) if a, b [member of] R, a < b and (a, b) [intersection] [sigma]([A.sub.n]) = [empty set] for all n, then (a, b) [intersection] [sigma] (A) = [empty set]. That is, if [lambda] [member of] [sigma](A) then there are [[lambda].sub.n] [member of] [sigma]([A.sub.n]) such that [[lambda].sub.n] [right arrow] [lambda]. (b) if a, b [member of] R , a < b and a, b [not member of] [[sigma].sub.pp](A), then [P.sub.(a,b)]([A.sub.n]) converges strongly to [P.sub.[a,b]] (A).

The second part of the above theorem will imply Proposition 1 provided it can be shown that [[DELTA].sub.n] [??] [DELTA]. If this can be shown, it will follow that [[chi square].sub.[a,b}] ([[DELTA].sub.n]) [??} [[chi square].sub.[a,b]]([DELTA]). Using this and the fact that [[chi square].sub.n] [??] I, it is then elementary to prove Theorem 1. To establish that [[DELTA].sub.n] [??][DELTA], it will be easier to first show that the [[DELTA].sub.n] converge to [DELTA] in another sense which is in fact equivalent to strong resolvent convergence for self-adjoint operators.

Definition 1. Let [[DELTA].sub.n] be a sequence of operators on a separable Hilbert space H. The pair ([phi], [phi]) [member of] H x H is said to belong to the strong graph limit of [A.sub.n] if one can find a sequence [[psi].sub.n] [member of] D([A.sub.n]) such that [[psi].sub.n] [right arrow] [psi] and [A.sub.n] [[psi].sub.n] [right arrow] [phi]. The set of all pairs in this strong graph limit will be denoted [[GAMMA].sup.[infinity].sub.s]([A.sub.n]). If [[GAMMA].sup.[infinity].sub.s] ([A.sub.n]) is the graph of an operator A, [[GAMMA].sup.[infinity].sub.s]([A.sub.n] = [GAMMA](A) then A is called the strong graph limit of the [A.sub.n]. This will be written [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is a fact that for self adjoint operators [A.sub.n] and A strong graph convergence is equivalent to strong resolvent convergence [14]. It follows that Proposition 1 will be proven if we can show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which will be shown now.

Proposition 2. The Laplacian [DELTA] of any complete [C.sup.[infinity]] Riemannian manifold is the strong graph limit of the [[DELTA].sub.n].

In the literature this and related results are known for the special case M = [R.sup.d] [16, 17].

The proof of the following lemma is a simple application of the definition of a compact set, and is omitted.

Lemma 1. If B is a compact subset of M, then there exists N [member of] N such that B [subset] [K.sub.n] for all n [greater than or equal to] N.

Proof. (of proposition)

The proposition will be established if it can be shown that [GAMMA]([DELTA]) = [[GAMMA].sup.s.sub.[infinity]]([DELTA].sub.n]).

First it will be shown that [GAMMA]([DELTA]) [subset] [[GAMMA].sup.[infinity].sub.s]([DELTA].sub.n]). Let ([phi], [DELTA][phi]) [member of] [GAMMA]([DELTA]). Since, by Theorem (2), [DELTA]' is essentially self-adjoint on [C.sup.[infinity].sub.0](M), there exists a sequence of elements ([[phi].sub.n], [DELTA][[phi].sub.n]) [member of] [GAMMA]([DELTA]') such that [[phi].sub.n] [member of] [C.sup.[infinity].sub.0](M) and [[phi].sub.n] [right arrow] [phi], [DELTA][[phi].sub.n] [right arrow] [DELTA][phi].

Each [[phi].sub.n] and, hence, [DELTA][[phi].sub.n] have support on a compact subset [B.sub.n] [subset] M. Since [K.sub.n+l] [contains] [K.sub.n] and [U.sub.n][K.sub.n] = M, Lemma (1) implies that for each n there is a [N.sub.n] [member of] N such that if m [greater than or equal to] [N.sub.n], [B.sub.n] [subset] [K.sub.m].

Each [[phi].sub.n] has support on a compact subset of [K.sub.m] and, hence, vanishes before the boundary of [K.sub.m]. It follows that each [[phi].sub.n] belongs to [C.sup.[infinity].sub.0]([K.sub.m]). It follows that each [[phi].sub.n] [member of] [??]([[DELTA].sub.m]), and that [DELTA][[phi].sub.n] = [[DELTA].sub.][[phi].sub.n] for all m [greater than or equal to] [N.sub.n].

This shows that, with suitable relabelling, one can find a sequence [[??].sub.n] [member of] D([[DELTA].sub.n]) [intersection] D([DELTA]) such that ([[??].sub.n], [[DELTA].sub.n][[??].sub.n]) = ([[??].sub.n], [DELTA][[??]s.ub.n]) [right arrow] ([phi], [DELTA][phi]).

Such a sequence can be constructed explicitly as follows. If [max.sub.k[member of]N [N.sub.k] < [infinity], then M = [N.sub.i] for some i [member of] N. In this case let [[??].sub.j] = 0 for all j < [N.sub.i] and let [[??].sub.j] = [[phi].sub.j] for all j [greater than or equal to] [N.sub.i]. Then for j [greater than or equal to] [N.sub.i] the support of [[??].sub.j] = [[phi].sub.j] is contained in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [N.sub.j] [less than or equal to] [N.sub.i] [less than or equal to] j, it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that [B.sub.j] [subset] [K.sub.j] and [[??].sub.j] [member of] D[[DELTA].sub.j]).

Conversely, if there is no upper bound on the [N.sub.k], let [m.sub.0] := 1 and for each k [member of] N let [m.sub.k] be the smallest integer for which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is not difficult to see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a positive, strictly increasing sequence of integers. In this case let [[??].sub.j] = 0 for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and then for each k [member of] N[union]{0} and for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the support of [[??].sub.j] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is contained in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This again shows that [[??].sub.j] [member of] D([[DELTA].sub.j]) for each j.

In both of the above two cases it is straightforward to check that ([[??].sub.n], [[DELTA].sub.n][[??].sub.n]) = ([[??].sub.n], [[DELTA][[??].sub.n]) [right arrow] ([phi], [[DELTA][phi]). It follows that ([phi], [DELTA][phi]) [member of] [[GAMMA].sup.[infinity].sub.s]([[DELTA].sub.n]) so that [GAMMA]([DELTA]) [subset] [[GAMMA].sup.[infinity].sub.s] ([[DELTA].sub.n]).

It remains to show that [[GAMMA].sup.[infinity].sub.s]([[DELTA].sub.n]) [subset] [GAMMA]([DELTA]). Suppose ([phi], [psi]) [member of] [[GAMMA].sup.[infinity].sub.s]([[DELTA].sub.n]). By definition this means that there exists a sequence ([[phi].sub.n], [[DELTA].sub.n][[phi].sub.n]) [member of] [GAMMA]([[DELTA].sub.n]) such that [[phi].sub.n] [right arrow] [phi] and [[DELTA].sub.n][[phi].sub.n] [right arrow] [psi]. Let [[DELTA]' be the restriction of [DELTA] to the domain [C.sup.[infinity].sub.0](M). For any given f [member of] D([DELTA]') f has support on B where B is some compact subset of M. Choose N [member of] N such that for all n > N, B [subset] [K.sub.n]. It is clear that f [member of] D([[DELTA].sub.n]) for all n > N, and that [[DELTA].sub.n]f = [[DELTA]' f for all n > N. Therefore, since [[DELTA].sub.n] is self-adjoint,

(f, [[DELTA].sub.n][[phi].sub.n) = ([[DELTA].sub.n] f, [[phi].sub.n]) = [DELTA]' f, [[phi].sub.n]). (5)

Taking the limit as n [right arrow] [infinity] of both sides of the above equation yields

<f, [psi]) = [DELTA]' f, [phi]>. (6)

Since the choice of f [member of] D ([DELTA]') was arbitrary, equation (6) holds for all f [member of] D([DELTA]'). By definition of the domain of the adjoint, [phi] [member of] D(([[DELTA]').sup.*]) and [([DELTA]').sup.*][phi] = [psi]. But since [DELTA]' is essentially self-adjoint, [DELTA] = [([DELTA]').sup.*]. Therefore, ([phi], [psi]) [member of] [GAMMA]([DELTA]), [[GAMMA.sup.[infinity].sub.s]([[DELTA].sub.n]) [subset] [GAMMA]([DELTA]) and the proof is complete.

Observe that the [K.sub.n] do not need to have compact closures in order for the above proof to work. However, for our purposes it is convenient to choose the [K.sub.n] so that their closures are compact, since in this case the subspaces B([K.sub.n], [OMEGA]) can be chosen to be finite dimensional. Further, observe that the above proof does not depend on properties unique to the Laplacian. The same proof strategy could be used to prove an analogous result about other differential operators on a manifold M which are essentially self-adjoint on [C.sup.[infinity].sub.0](M) and have a symmetric restriction to a dense domain in [L.sup.2] of any submanifold of M. For example, the same proof strategy could be used to prove an analogous proposition for higher powers of the Laplacian. We would also like to thank one of the referees for pointing out that this proposition could probably be refined by considering more general sequences of submanifolds [K.sub.n], which converge to the full manifold M with respect to a certain topology, for example, the Gromov-Hausdorff topology [12]. These refinements of Proposition 2 will appear as part of a future paper.

4 Outlook

Choose [[DELTA].sub.n] to be that Laplacian with either Neumann or Dirichlet boundary conditions on [K.sub.n]. A convenient property of the subspaces B([K.sub.n], [OMEGA]) is that they are finite dimensional. This follows from the known fact that the self-adjoint Laplacian on a compact manifold K obeying Dirichlet or Neumann boundary conditions has a purely discrete spectrum consisting of eigenvalues of finite multiplicity with no finite accumulation point ([3], pg. 8). Hence, [P.sub.n], [OMEGA] will project onto a finite dimensional subspace of eigenfunctions to [[DELTA].sub.n], and the dimension of this subspace B([K.sub.n], [omega]) is equal to N([K.sub.n], [OMEGA]) where N([K.sub.n], [OMEGA]) is the number of eigenvalues to [[DELTA].sub.n] that lie in the interval [0, [[OMEGA].sup.2]].

Such a finite dimensional function space always trivially obeys a sampling theorem. For example, consider an N-dimensional function space, F, spanned by some generic basis functions [{[b.sub.i](x)}.sub.i=1 ... N], i.e., all f [member of] F can be written f(x) = [[summation].sup.N.sub.i=1] [[lambda].sub.i] [b.sub.i](x) for some [{[[lambda].sub.i]}.sup.N.sub.i=1] [subset] C. There automatically holds a sampling theorem for this function space: assume we know of a function f [member of] F only its amplitudes [a.sub.n] = f([x.sub.n]), for n = 1 ... N at some N generically chosen points [x.sub.n], i.e.,

f([x.sub.n]) = [a.sub.n] = [N.summation over (i=1)] [[lambda].sub.i] [b.sub.i]([x.sub.n]). (7)

Then, (7) generally allows us to determine the coefficients [[lambda].sub.i] and, therefore, f(x) for all x. This is because for generic basis functions [b.sub.i], if the sample points [x.sub.n] are chosen such that the N x N matrix B = [([b.sub.i]([x.sub.n])).sub.i,n=1...N] has a nonvanishing determinant, then: [[lambda].sub.i] = [[summation].sup.N.sub.j=1] [B.sup.-1].sub.ij] [a.sub.j] and, therefore,

f(x) = [N.summation over (n=1)] f([x.sub.n])G([x.sub.n], x) for all x

where the reconstruction kernel G reads: G([x.sub.n], x) = [[summation].sup.N.sub.i=1][B.sup.-1.sub.ni] [b.sub.i](x). Since the [b.sub.i] are linearly independent functions, such a set of N points [x.sub.n] must exist. It follows that the minimum number of points needed for a set of sampling is N, the dimension of F.

There are various results from the field of spectral geometry that can be used to estimate N([K.sub.n], [OMEGA]), the number of points needed for a set of sampling for B([K.sub.n], [OMEGA]). For example Weyl's asymptotic formula [6] states that

N([K.sub.n], [OMEGA]) ~ [[OMEGA].sup.d]V(K)V([B.sub.d])/[(2[pi]).sup.d] (8)

for large [OMEGA], where V(K) is the volume of K and V([B.sub.d]) is the volume of the d-dimensional unit ball in [R.sup.d].

We remark here that. in [11] Pesenson calculates the eigenvalues and eigenvectors of the Laplacian on a compact manifold in a limiting procedure using polyharmonic spline functions centered about a finite number of [??] points on the manifold. Interestingly, as Pesenson has observed, Weyl's asymptotic formula provides a number proportional to this necessary number, [??], of points.

Now an interesting question to ask is whether the density N([K.sub.n], [OMEGA])/V([K.sub.n]) has a finite limit [rho] as n [right arrow] [infinity]. Since N([K.sub.n], [OMEGA]) is the number of points needed for a set of sampling for B([K.sub.n], [OMEGA]), and since B([K.sub.n], [OMEGA]) converges strongly to B(M, [OMEGA]), this would then seem to suggest that the necessary density a discrete set of points [LAMBDA] := {[[lambda].sub.n]} [subset] M must have in order to be a set of sampling for B(M, [OMEGA]) is [rho]. Here, the density of the countable set of points [LAMBDA] [subset] M may be defined analogously to Beurling-Landau density of a discrete set of points in [R.sup.n]. Namely, one may define n_(r) to be the minimum number of the points of [LAMBDA] in any dim(M) dimensional ball of proper radius r in M, and then define the lower Beurling-Landau density of [LAMBDA] to be D_([LAMBDA]):= [lim.sub.r [right arrow] [infinity] n -(r)/r.

If these ideas can be made rigorous, this could then provide an approach for generalizing H.J. Landau's theorem [8] on necessary density for sets of sampling for bandlimited functions to manifolds. Such a generalized result would be of great interest in particular to mathematical physics [7].

Pesenson [10] has already shown that functions in B(M, [OMEGA]) are stably re-constructible from their values taken on certain sets of points [DELTA], which have a finite proper 'density', in the case where M has bounded geometry. It, therefore, seems reasonable that the numbers [N([K.sub.n], [OMEGA])/V([K.sub.n]) should have a finite limit, or at least be bounded above for such manifolds. Methods from the field of spectral geometry, including those used to calculate Weyl's asymptotic formula, should be useful for investigating these ideas. In particular, the fact that correction terms to Weyl's asymptotic formula can be calculated in terms of integrals of scalars formed from the curvature tensor and its covariant derivatives, which is also used in non-commutative geometry, should be useful here [2].

Results from classical sampling theory, such as Landau's theorem on the necessary density for sets of sampling and its generalization to curved space, will be of interest in mathematical physics, in particular for quantum gravity. The usefulness of sampling theory in quantum gravity, as motivated in [7], is that sampling theory could provide a crucial link between quantum theory and general relativity. This is because quantum theory appears to require that space be discrete, while general relativity requires space-time to be a continuous, smooth manifold, i.e., not discrete. The ansatz proposed in [7] is that space-time could be a continuous manifold, and yet at, the same time effectively discrete for the physical fields which describe particles in quantum theory provided that these physical fields are bandlimited functions. Since physical laws cannot depend on the choice of co-ordinate system, these physical fields should be bandlimited in a co-ordinate independent way, such as that described in this paper and in [10]. In conclusion, many results of classical sampling theory, once generalized to manifolds, could find application to physics on curved space.

References

[1] N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space, Two volumes bound as one. Dover Publications, New York, NY, 1993.

[2] A. Chamseddine and A. Connes, The spectral action principle, Commun. Math. Phys., 186:731-750, 1997.

[3] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press Inc., New York, NY, 1984.

[4] P.R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Fun. Anal., 12:401-414, 1973.

[5] H.O. Cordes, Self-adjointness of powers of elliptic operators on non-compact manifolds, Math. Ann., 195:257-272, 1972.

[6] B. Davies and Y. Safarov, editors, Spectral Theory and Geometry, volume 273, Cambridge University Press, 1999.

[7] A. Kempf, Covariant information-density cutoff in curved space-time, Phys. Rev. Lett., 92(22):221301, 2004.

[8] H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math., 117:37-52, 1967.

[9] R. Martin, Approximation of bandlimited functions by bandlimited trigonometric polynomials, Sampl. Theory Signal Image Process., 6:273-296, 2007.

[10] I. Pesenson, A sampling theorem on homogenous manifolds, Trans. Am. Math. Soc., 352(9):4257-4269, 2000.

[11] I. Pesenson, An approach to spectral problems on Riemannian manifolds, Pac. J. Math., 215(1):183-199, 2004.

[12] P. Petersen, Riemannian Geometry, Springer, New York, NY, 1998.

[13] M. Reed and B.Simon, Methods of Modern Mathematical Physics v.2: Fourier Analysis, Self-adjointness, Academic Press, Cambridge, UK, 1999.

[14] M. Reed and B. Simon, Methods of Modern Mathematical Physics v.l: Functional Analysis, Academic Press, New York, NY, 1972.

[15] G. Schmeisser, Approximation of entire functions of exponential type by trigonometric polynomials, Sampl. Theory Signal Image Process., 6:297-306, 20O7.

[16] P. Stollman, A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains, Math. Z., 219:275, 1995.

[17] J. Weidmann, Stetige Abhangigkeit der Eigenwerte und Eigenfunktionen elliptischer Differentialoperatoren vom Gebiet. [Continuous dependence of eigenvalues and eigenfunctions of elliptic differential operators on the domain], Math. Scand., 54:51, 1984.

R. Martin

Applied Mathematics, University of Waterloo,

200 University Ave. W., Waterloo, N2L 3G1, ON, Canada

[e-mail]

A. Kempf

Applied Mathematics, University of Waterloo,

200 University Ave. W., Waterloo, N2L 3G1, ON, Canada

[e-mail]