# Sampling theory in abstract reproducing Kernel Hilbert space.

Abstract

Let H be a separable Hilbert space and k(t) an H-valued function on a subset [OMEGA] of tHe real line R such that {k(t) | T [member of] [OMEGA]} is total in H. Then

{[f.sub.x] := [<x, k(t)>.sub.H] | x [OMEGA] H}

becomes a reproducing kernel Hilbert space (RKHS) in a natural way. Here, we develop a sampling tbrmula for functions in this RKHS, which generalizes the well-known celebrated Whittaker-Shamton-Kotel'nikov sampling formula in tim Paley-Wiener space of band-limited signals. To be more precise, we develop a multi-channel sampling formula in which each channel is given a rather arbitrary sampling rate. We also discuss stability and oversampling.

Key words and phrases : Sampling, Reproducing Kernel Hilbert space, oversampling

1 Introduction

Let f(t) be a band-limited signal with band region [-[pi], [pi]], that is, a squre-integrable function on R of which the Fourier transform [??] vanishes outside [-[pi], [pi]]. Then f can be recovered by its uniformly spaced discrete values as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which converges absolutely and uniformly over R. This series is called the cardinal series or the Whittaker-Shannon-Kotel'nikov (WSK) sampling series. This formula tells us that once we know the values of a band-limited signal f at certain discrete points, we can recover f completely. In 1941 Hardy [4] recognized that this cardinal series is actually an orthogonal expansion.

WSK sampling series was generalized by Kramer [8] in 1957 as follows: Let k([xi], t) be a kernel on I x [OMEGA], where I is a bounded interval and [OMEGA] is a subset of R. Assume that k(*, t) [member of] [L.sup.2](I) for each t in [OMEGA] and there are points [{[t.sub.n]}.sub.n[member of]z] in such that [{k([xi],[t.sub.n])}.sub.n[member of]Z] is an orthonormal basis of [L.sup.2](I). Then any f(t) = [[Integral].sub.I]F([xi])K([xi]), t)d[xi] with F([xi]) [member of] [L.sup.2](I) can be expressed as a sampling series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which converges absolutely and uniformly over the subset D on which ||k(*, t)||[L.sup.2] (I) is bounded. While WSK sampling series treats sample values taken at uniformly spaced points, Kramer's series may take sample values at nonuniformly spaced points.

Recently, A. G. Garcia and A. Portal [3] extended the WSK and Kramer sampling formulas further to a more general setting using a suitable abstract Hilbert space valued kernel.

On the other hand, Papoulis [10] (see also [7]) introduced a multi-channel sampling formula for band-limited signals in which a signal is recovered from discrete sample values of several transformed versions of the signal.

In this work, following the setting introduced by Garcia and Portal [3], we first extend and modify Theorem 1 in [3] into a single channel sampling formula (see Theorem 3.2 below), which is more transparent. It is then easy to extend it to a multi-channel sampling formula in which each channel can be given rather arbitrary sampling rate. Comparing two-channel sampling formula, Theorem 3 in [3] and our multi-channel sampling formula, Theorem 3.3, reveals the advantage of modification made in Theorem 3.2. Finally, we also discuss the oversampling and recovery of missing samples in the single-channel sampling formula.

2 Preliminaries

For f(t) [member of] [L.sup.2](R), we let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

be the Fourier transform of f (t) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the inverse Fourier transform of f([xi]).

Definition 2.1. For any w > 0, the Paley-Wiener space, P[W.sub.[pi]w], is defined to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Note that P[W.sub.[pi]w] is isometrically isomorphic onto [L.sup.2][-[pi]w, [pi]w] under the Fourier transform.

We call a basis {[[phi].sub.n]} of a separable Hilbert space H to be an unconditional basis of H if for every f [member of] H the expansion f = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] still converges to f after any permutation of its terms. We also call a basis ([[phi].sub.n]} to be a Riesz basis of H if there is a linear isomorphism T from H onto H such that T([e.sub.n]) = [[phi].sub.n] where {[e.sub.n]} is an orthonormal basis for H. Then any Riesz basis of H is an unconditional basis of H but not conversely in general.

Definition 2.2. [12] A Hilbert space H consisting of complex-valued functions defined on a set D([not equal to] [??]) is called a reproducing kernel Hilbert space (RKHS in short) if there exists a function k(s, t) on D x D satisfying

(1) k(*,t) [member of] H for each t [member of] D;

(2) (f(s),k[(s,t)).sub.H] = f(t) for all f [member of] H and all t [member of] D.

Such a function k(s,t) is called a reproducing kernel of H.

We need some properties of RKHS's.

Proposition 2.3. [5] Let H be a Hilbert space as in Definition 2.2. Then we have:

(a) H is an RKHS if and only if the point evaluation map [l.sub.t](f) := f(t) is a bounded linear functional on H for each t [member of] D;

(b) an RKHS H has a unique reproducing kernel;

(c) the convergence of a sequence in an RKHS H implies its uniform convergence over any subset of D on which k(t, t) is bounded.

For example, the Paley-Wiener space P[W.sub.[pi]w] is an RKHS with the reproduc ing kernel [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3 Multi-channel sampling

Let H be a separable Hilbert space and k : [OMEGA] [right arrow] H be an H-valued function on a subset [OMEGA] of the real line R. Define a linear operator T on H by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We call k(t) the kernel of the linear operator T.

Lemma 3.1. ([3])

(a) T is one-to-one if and only if {k(t) | t [member of] [OMEGA]} is total in H.

Assume {k(t) | t [member of] [OMEGA]} is total in H so that T : H [right arrow] T(H) is a bijection. Then

(b) (T(x)* [T(y)).sub.T(H)] := [(x; Y).sub.H] defines an inner product on T(H) with which T(H) is a Hilbert space and T : H [right arrow] T(H) is unitary. Moreover, T(H) becomes an RKHS with the reproducing kernel k(s, t) := (k(t): [k(s)).sub.H].

Proof. (a) T is one-to-one if and only if [{k(t) | t [member of] [OMEGA]}.sup.[perpendicular to]] = {0} if and only if span {k(t) | t [member of] [OMEGA] } = H, that is, {k(t) | t [member of] [OMEGA]} is total in H.

(b) It is trivial that (T(x), [T(y)).sub.T(H)] := [(x, Y).sub.H] defines an hmer product on T(H) with which T: H [right arrow] T(H) is unitary. Now for any f(*) = (x, k(*))H in T(H) and t [member of] [OMEGA]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If(t)| = | (x, [k(t)).sub.H]| [less than or equal to] ||x||H||k(t)||H = ||f||T(n)||k(t)||H so that [l.sub.t](f) = f(t) is a bounded linear functional on T(H). Hence, T(H) is an RKHS by Proposition 2.3. Since

f(t) = [(x, k(t)).sub.H] = (T(x)(s), T(k(t))(S))T(H) = (f(s), (k(t), k(S))H)T(H),

the reproducing kernel k(s, t) of T(H) is (k(t), [k(s)).sub.H].

First, we develop a single-channel sampling formula. Let [??] : [OMEGA] [right arrow] H be another H-valued function on [OMEGA] and [??] the linear operator on H defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Theorem 3.2. If K er T [subset or equal to] Ker[??] and there exists a sequence [{[t.sub.n]}.sub.n[member of]z] in [OMEGA] such that [{k([t.sub.n])}.sub.n[member of]Z] is a basis of H, then T is one-to-one so that T(H) becomes an RKHS under the inner product (T(x), [T(y)).sub.T(H)] := (x, y)H. Moreover, there is a basis [{[S.sub.n]}.sub.n[member of]Z] ofT(H) such that, for any [f.sub.x] [member of] T(H) the sampling expansion

[f.sub.x](t) = [summation (n)] [??]x([t.sub.n] [S.sub.n](t), t [member of] [OMEGA] (3.1)

holds. The convergence of the series is not only in T(H) but also uniform over any subset on which [[parallel k(t)].sub.H] is bounded.

Proof. Assume [??](x)(t) = <x,[??](t)) = 0 on [OMEGA]. Then (x,[??]([t.sub.n])> = 0 for any n [member of] Z so that x = 0 since [{[??]([t.sub.n])}.sub.n[member of]Z] is a basis of H. Hence, K er T = K er T = {0} and T(H) becomes an RKHS as in Lemma 3.1 (b).

Let [{[x.sub.n]}.sub.n[member of]Z] [{[??]([t.sub.n])}.sub.n[member of]Z] and {[[X.sup.*.sub.n]}n[member of]Z] be its dual. Then {T([x.sub.n])} and {T([x*.sub.n])} are bases of T(H), which are dual each other since T is unitary.

Expanding any [f.sub.x] = T(x) in T(H) via the basis [{[S.sub.n]}.sub.n[member of]Z] = {T([x.sub.n]*)} gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Uniform convergence of the series (3.1) follows from Proposition 2.3 (c).

The single channel sampling expansion (3.1) may not converge absolutely unless [{[x.sub.n]}.sun.n[member of]Z] is an unconditional basis and may not be stable. However, if [{[x.sub.n]}.sub.n[member of]Z ]is an unconditional basis and [sup.sub.n] [[parallel [[x.sup.*].sub.n] < [infinity], then (3.1) is a stable sampling expansion, which converges absolutely on [OMEGA] In fact, if then, [{[S.sub.n]}.sub.n[member of]Z] becomes an unconditional basis of T(H) and [sup.sub.n] [parallel [S.sub.n]] = [sup.sub.n] [parallel [x.sup.*.sub.n]*|| < [infinity]. Since {[1/[parallel [S.sub.n]] [S.sub.n]}.sub.n [member of] z] is a Riesz basis of T(H) by the K6the-Toeplitz Theorem [9], there is a constant B > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Furthermore, the sampling series expansion (3.1) remains valid when [{[??]([t.sub.n]).sub.n [member of] z] [is not a basis but a frame of H. When k(t) = k(t) on [OMEGA] so that T = [??], Theorem 3.2 is essentially Theorem 1 in [3]. However, Theorem 3.2 might have some advantage over Theorem 1 in [3]. While Theorem 1 in [3] requires first the expansion of the kernel k(t) in terms of a given basis of H and then the interpolatory condition for the expansion coefficients at some points in [OMEGA], Theorem 3.2 simply requires points in [OMEGA], whose values under k(*) form a basis of H.

Now, we can extend Theorem 3.2 naturally to a multi-channel setting. Let [{k.sub.i]}.sup.N.sub.i=1] be N H-valued functions on and [{T.sub.i]}.sup.N.sub.i=1] operators on by

[T.sub.i](x) = [f.sup.i.sub.x] := (x, [[k.sub.i](t)).sub.H], x [member of] H.

Theorem 3.3. (Asymmetric nonuniform multi-channel sampling formula) If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and there exist points [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and constants [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for some integer M [greater than or equal to] 1 such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is an unconditional basis of H, then there is a basis {[S.sub.j,n] | 1 [less than or equal to] j [less than or equal to] M and n [member of] Z} of T(H) such that for any [f.sub.x] = T(x) [member of] T(H),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.2)

which converges in T(H). Moreover, the series (3.2) converges absolutely and uniformly on any subset of [OMEGA] over which [[parallel k(t)].sub.H] is bounded.

Proof. First, we prove that T is one-to-one. Suppose T(x)(t) = (x,k(t)) = 0 for all t [member of] [OMEGA]. Then, (x, ki(t)) O, 1 [less than or equal to] i [less than or equal to] N oll fl since KerT G N In particular, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for all 1 [less than or equal to] j [less than or equal to] M and n [member of] Z so that x = 0 since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] M and n [member of] Z} is a basis of H. Therefore, T : H [right arrow] T(H) is a bijection and T(H) becomes an RKHS under the inner product (T(x),T(y))T(H) := (x, Y)H by Lemma 3.1.

Let :[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for 1 [less than or equal to] j [less than or equal to] M and n [member of] Z and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] be the dual of {[x.sup.j.sub.n]}. Then,. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] becomes an unconditional basis of T(H) with the dual basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], which is also unconditional.

Expanding fx = T(x) in T(H) with respect to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Uniform convergence of the series (3.2) follows from Proposition 2.3 (c). Finally, the series (3.2) converges also absolutely since it is an unconditional basis expansion.

If either K er T = {0} or ki(t) = Ai(k(t)), 1 [less than or equal to] i [less than or equal to] N, where Ai's are auN tomorphisms of H, then the first assnmption [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of Theorem 3.3 is trivially satisfied. For example, it is so when H = [L.sup.2][-[pi], [pi], [OMEGA] = R and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the inverse Fourier transform. In particular, if N = M = 2, [k.sub.1](t) = k(t),[t.sub.1,n] = [t.sub.2,n] = [t.sub.n], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

1, 2 and n [member of] Z} is a Riesz basis of H, then Theorem 3.3 is essentially the same as Theorem 3 in [3].

When H = [L.sup.2] [-[pi]w, rw](w > 0), [OMEGA] = R and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for suitable bounded measurable functions [A.sub.i]([xi])(1 [less than or equal to] i [less than or equal to] N) on [-[pi]w, [pi]w], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence, T(H) becomes the Paley-Wiener space [PW.sub.[pi]w] and then Theorem 3.3 reduces to an asymmetric multi-channel sampling handled in [7].

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is a frame of H in Theorem 3.3, then the sampling series expansion (3.2) still holds.

As in the single channel case, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the multichannel sampling expansion (3.2) is also stable in the following sense.

Definition 3.4. (cf. Rawn [11] and Yao and Thomas [13]) We say that {[t.sub.i,n] | 1 [less than or equal to] i [less than or equal to] N and n [member of] Z} is a set of stable sampling for T(H) if there exists A > 0 which is independent of [f.sub.x] [member of] T(H) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let B > 0 be the upper Riesz bound for the Riesz basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of T(H). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

so that (3.2) is a stable sampling expansion with respect to {[t.sub.i,n]} when sup ||[[alpha].sup.j.sub.i,n]

We now discuss several examples in which we always take H = [L.sup.2] [-[pi], [pi]], [OMEGA] =

R, and k(t) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] so that T = [F.sup.-1] is the inverse Fourier transform and T(H) = [PW.sub.[pi]].

Example 3.5. (Sampling with Hilbert transform)

Take k(t) = i sgn([xi]) k(t) so that T(f)(t) = f(t) is the Hilbert transform of

f(t) in [PW.sub.[pi]. Choosing [{[t.sub.n]}.sub.n[member of]Z] = [{n}.sub.n[member of]Z], [{[x.sub.n]}.sub.n[member of]Z] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is an orthonormal basis of [L.sup.2][-[pi], [pi]] so that {[x*.sub.n]}.sub.n[member of]Z] = [{[x.sub.n]}.sub.n[member of]Z]. We then have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where sinct := sin [pi]t/[pi]t. Hence, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using the operational relation f = -f ([5, Appendix B]) and the fact that if f [member of] [PW.sub.[pi]], then so does f, we also have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Example 3.6. Here, we derive asymmetric derivative sampling formula on [PW.sub.[pi]], in which we take samples from f(t) and if(t) with ratio 2:1.

Take [k.sub.1](t) = k(t) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [k.sub.2](t) = -i[xi]k(t) = k'(t) so that [f.sup.1](t) = f(t) and [f.sup.2](t) = f'(t) for f(t) [member of] [PW.sub.[pi]]. Now, take the set of sampling points [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

if n is even,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

if n is odd, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Taking inverse Fourier transform on (3.3), we have a Riesz basis {Sl,n}n U {[S.sub.2,n]}.sub.n] of [PW.sub.[pi]] where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

With these setting we have the nonsymmetric derivative sampling formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Example 3.7. We now take kl(t) = k(t) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [k.sub.2](t) = [e.sup.i[xi]] k(t) so that [f.sup.1](t) = f(t) and [f.sup.2](t) = f(t - 1). We want to express f [member of] [PW.sup.[pi] via samples from f(t) and f(t - 1) with ratio 3: 2. Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.4)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then, we can obtain the sampling series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where {[S.sub.1,n]t(t)} U {[S.sub.2,n](t)} are the inverse Fourier transforms of functions in (3.4)

4 Oversampling and recovery of missing samples

We now develop an oversampling expansion, which extends the one in Kramer [2]. Again, let k and k : [OMEGA] [right arrow] H be H-valued functions. Assume that there exists {tn}~ez C [t such that {[x.sub.n] := k([t.sub.n])}.sub.[member of] Z is a basis of H with the dual basis {x~},ez. Define linear operators T and T on H by T(x) = <x, k(t))H := fx and T(x) = (x, k')H := .~x, respectively, and assume KerT C KerT. Then, both T and T are one-to-one, and so T(H) and T(H) become RKHS's.

Now, let G be a proper closed subspace of H and P : H [right arrow] G the orthogonal projection onto G. Then, fbr any x [member of] G we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Theorem 4.1. Under the above setting there is a sequence of sampling functions {[[T.sub.n]}.sub.n[member of]Z] in T(G) such that for any x [member of] G

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.2)

which converges in T(H) and uniformly on any subset of [OMEGA] over which ||k(t)||H is bounded. Moreover, if {[x.sub.n]} is a Riesz basis of H, then {[T.sub.n]}.sub.n[member of]Z] is a frame of T(G).

Pwof. Applying T on both sides of (4.1) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [T.sub.n](t) = T(P([x*.sub.n]))(t). Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the series (4.2) converges uniformly o,1 any subset over which ||k(t)||H is bounded. Finally, if {[x.sub.n]} is a Riesz basis of H, then {[x*.sub.n]} is also a R]esz basis of H so that {P([x*.sub,n]} is a frame of G since G is a closed subspace of H [1, Proposition 5.3.5]. Hence {[T.sub.n](t) = T(P([x*.sub.n]))(t)} is a frame of T(G).

Note that the sample set {[t.sub.n]}.sub.n[member of]Z] oversamples functions in T(G) in the sense that {[t.sub.n]}.sub.n[member of]Z] leads to a basis {Sn}~ez of T(H) (see Theorem 3.2), which properly contains T(G) but {Tn}nez in Theorem 4.1 may be overcomplete in T(G). Hence, we nmy call (4.2) an oversampling expansion of fx in T(G) for x E G.

Now assume that finitely many sample values {fz(tn) In E X = {nl, n2,'" , [n.sub.N]}} are missing. Applying T on both sides of (4.1) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.3)

which converges not only in T(H) but also pointwisely in [OMEGA] since T(H) is an RKHS. Setting t = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] in (4.3), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which can be rewritten in the matrix form as

(I - T) f = h

where f = ([f.sub.x]([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]), ..., [f.sub.x]([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.])).sup.T] is the column vector consisting of missing samples, h = ([h.sub.1], ... , [h.sub.N]).sup.T], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and T is the N x N matrix with entries

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Note that if I-T is invertible, tile missing samples f can be recovered uniquely. In particular, if <Tv, v> < [||v||.sup.2] for any v [member of] [C.sup.N] \ {0}, then I-T is invertible. We have:

Theorem 4.2. Under the same hypotheses as in Theorem 4.1, we assume further that [{[x.sub.n]}.sub.n] is a Riesz basis of H such that [x.sub.n] = U([e.sub.n]) where [{[e.sub.n]}.sub.n] is an orthonormal basis of H and U is an automorphism of H. If PU = UP and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCCII.] (4.4)

then any finitely many missing samples [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] in the oversampling expansion (4.2) can be uniquely recovered.

Proof. Note first that [x.sub.n] * = [(U *).sup.-1] ([e.sub.n] where {[x.sub.n] * is the dual of [{[x.sub.n].sub.n]. Hence, we have for any v = [([v.sub.1] ... ,[v.sub.N]).sup.T] [member of] [C.sup.N] \ {0},

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is an orthonormal basis of H. Hence, I-T is invertible.

If, moreover, [{[x.sub.n]}.sub.n] is an orthonormal basis of H in Theorem 4.2, then any finitely many missing samples [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] can be uniquely recovered when the condition (4.4) holds.

ACKNOWLEDGEMENTS

This work is partially supported by BK-21 project and KOSEF(R01-2006000-10424-0). Authors are grateful to the referee for his many valuable comments.

References

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[3] A. G. Garcia and A. Portal, Sampling in the functional Hilbert space induced by a Hilbert space valued kernel, Appl. Anal., 82(12), 1145-1158, 2003.

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[5] J. R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, Oxford Science Publications, 1996.

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[7] Jong M. Kim and Kil H. Kwon, Vector sampling expansion in Riesz basis setting and its aliasing error, preprint.

[8] H. P. Kramer, A generalised sampling theorem, J. Math. Phys., 63, 68-72, 1957.

[9] N. K. Nikol'ski[??], Treatise on the shift operator: Spectral function theory, Springer-Verlag, 1980.

[10] A. Papoulis, Generalized sampling expansion, IEEE Trans. Circuits & Systems, CAS-24, 652-654, 1975.

[11] M. D. Rawn, A stable nonuniform sampling expansion involving derivatives, IEEE Trans. Info. Theory, IT-35, 1223-1227, 1989.

[12] S. Saitoh, Integral Transforms, Reproducing kernels and Their Applications, Longman, Essex, England, 1997.

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Yoon Mi Hong

Division of applied mathematics, KAIST

Daejeon, 305-701, Korea

ymhong@amath.kaist.ac.kr

Jong Min Kim

Division of applied mathematics, KAIST

Daejeon, 305-701, Korea

franzkimQamath.kaist.ac.kr

Kil H. Kwon

Division of applied amthematics, KAIST

Daejeon, 305-701, Korea

khkwon@amath.kaist.ac.kr

Let H be a separable Hilbert space and k(t) an H-valued function on a subset [OMEGA] of tHe real line R such that {k(t) | T [member of] [OMEGA]} is total in H. Then

{[f.sub.x] := [<x, k(t)>.sub.H] | x [OMEGA] H}

becomes a reproducing kernel Hilbert space (RKHS) in a natural way. Here, we develop a sampling tbrmula for functions in this RKHS, which generalizes the well-known celebrated Whittaker-Shamton-Kotel'nikov sampling formula in tim Paley-Wiener space of band-limited signals. To be more precise, we develop a multi-channel sampling formula in which each channel is given a rather arbitrary sampling rate. We also discuss stability and oversampling.

Key words and phrases : Sampling, Reproducing Kernel Hilbert space, oversampling

1 Introduction

Let f(t) be a band-limited signal with band region [-[pi], [pi]], that is, a squre-integrable function on R of which the Fourier transform [??] vanishes outside [-[pi], [pi]]. Then f can be recovered by its uniformly spaced discrete values as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which converges absolutely and uniformly over R. This series is called the cardinal series or the Whittaker-Shannon-Kotel'nikov (WSK) sampling series. This formula tells us that once we know the values of a band-limited signal f at certain discrete points, we can recover f completely. In 1941 Hardy [4] recognized that this cardinal series is actually an orthogonal expansion.

WSK sampling series was generalized by Kramer [8] in 1957 as follows: Let k([xi], t) be a kernel on I x [OMEGA], where I is a bounded interval and [OMEGA] is a subset of R. Assume that k(*, t) [member of] [L.sup.2](I) for each t in [OMEGA] and there are points [{[t.sub.n]}.sub.n[member of]z] in such that [{k([xi],[t.sub.n])}.sub.n[member of]Z] is an orthonormal basis of [L.sup.2](I). Then any f(t) = [[Integral].sub.I]F([xi])K([xi]), t)d[xi] with F([xi]) [member of] [L.sup.2](I) can be expressed as a sampling series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which converges absolutely and uniformly over the subset D on which ||k(*, t)||[L.sup.2] (I) is bounded. While WSK sampling series treats sample values taken at uniformly spaced points, Kramer's series may take sample values at nonuniformly spaced points.

Recently, A. G. Garcia and A. Portal [3] extended the WSK and Kramer sampling formulas further to a more general setting using a suitable abstract Hilbert space valued kernel.

On the other hand, Papoulis [10] (see also [7]) introduced a multi-channel sampling formula for band-limited signals in which a signal is recovered from discrete sample values of several transformed versions of the signal.

In this work, following the setting introduced by Garcia and Portal [3], we first extend and modify Theorem 1 in [3] into a single channel sampling formula (see Theorem 3.2 below), which is more transparent. It is then easy to extend it to a multi-channel sampling formula in which each channel can be given rather arbitrary sampling rate. Comparing two-channel sampling formula, Theorem 3 in [3] and our multi-channel sampling formula, Theorem 3.3, reveals the advantage of modification made in Theorem 3.2. Finally, we also discuss the oversampling and recovery of missing samples in the single-channel sampling formula.

2 Preliminaries

For f(t) [member of] [L.sup.2](R), we let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

be the Fourier transform of f (t) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the inverse Fourier transform of f([xi]).

Definition 2.1. For any w > 0, the Paley-Wiener space, P[W.sub.[pi]w], is defined to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Note that P[W.sub.[pi]w] is isometrically isomorphic onto [L.sup.2][-[pi]w, [pi]w] under the Fourier transform.

We call a basis {[[phi].sub.n]} of a separable Hilbert space H to be an unconditional basis of H if for every f [member of] H the expansion f = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] still converges to f after any permutation of its terms. We also call a basis ([[phi].sub.n]} to be a Riesz basis of H if there is a linear isomorphism T from H onto H such that T([e.sub.n]) = [[phi].sub.n] where {[e.sub.n]} is an orthonormal basis for H. Then any Riesz basis of H is an unconditional basis of H but not conversely in general.

Definition 2.2. [12] A Hilbert space H consisting of complex-valued functions defined on a set D([not equal to] [??]) is called a reproducing kernel Hilbert space (RKHS in short) if there exists a function k(s, t) on D x D satisfying

(1) k(*,t) [member of] H for each t [member of] D;

(2) (f(s),k[(s,t)).sub.H] = f(t) for all f [member of] H and all t [member of] D.

Such a function k(s,t) is called a reproducing kernel of H.

We need some properties of RKHS's.

Proposition 2.3. [5] Let H be a Hilbert space as in Definition 2.2. Then we have:

(a) H is an RKHS if and only if the point evaluation map [l.sub.t](f) := f(t) is a bounded linear functional on H for each t [member of] D;

(b) an RKHS H has a unique reproducing kernel;

(c) the convergence of a sequence in an RKHS H implies its uniform convergence over any subset of D on which k(t, t) is bounded.

For example, the Paley-Wiener space P[W.sub.[pi]w] is an RKHS with the reproduc ing kernel [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3 Multi-channel sampling

Let H be a separable Hilbert space and k : [OMEGA] [right arrow] H be an H-valued function on a subset [OMEGA] of the real line R. Define a linear operator T on H by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We call k(t) the kernel of the linear operator T.

Lemma 3.1. ([3])

(a) T is one-to-one if and only if {k(t) | t [member of] [OMEGA]} is total in H.

Assume {k(t) | t [member of] [OMEGA]} is total in H so that T : H [right arrow] T(H) is a bijection. Then

(b) (T(x)* [T(y)).sub.T(H)] := [(x; Y).sub.H] defines an inner product on T(H) with which T(H) is a Hilbert space and T : H [right arrow] T(H) is unitary. Moreover, T(H) becomes an RKHS with the reproducing kernel k(s, t) := (k(t): [k(s)).sub.H].

Proof. (a) T is one-to-one if and only if [{k(t) | t [member of] [OMEGA]}.sup.[perpendicular to]] = {0} if and only if span {k(t) | t [member of] [OMEGA] } = H, that is, {k(t) | t [member of] [OMEGA]} is total in H.

(b) It is trivial that (T(x), [T(y)).sub.T(H)] := [(x, Y).sub.H] defines an hmer product on T(H) with which T: H [right arrow] T(H) is unitary. Now for any f(*) = (x, k(*))H in T(H) and t [member of] [OMEGA]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If(t)| = | (x, [k(t)).sub.H]| [less than or equal to] ||x||H||k(t)||H = ||f||T(n)||k(t)||H so that [l.sub.t](f) = f(t) is a bounded linear functional on T(H). Hence, T(H) is an RKHS by Proposition 2.3. Since

f(t) = [(x, k(t)).sub.H] = (T(x)(s), T(k(t))(S))T(H) = (f(s), (k(t), k(S))H)T(H),

the reproducing kernel k(s, t) of T(H) is (k(t), [k(s)).sub.H].

First, we develop a single-channel sampling formula. Let [??] : [OMEGA] [right arrow] H be another H-valued function on [OMEGA] and [??] the linear operator on H defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Theorem 3.2. If K er T [subset or equal to] Ker[??] and there exists a sequence [{[t.sub.n]}.sub.n[member of]z] in [OMEGA] such that [{k([t.sub.n])}.sub.n[member of]Z] is a basis of H, then T is one-to-one so that T(H) becomes an RKHS under the inner product (T(x), [T(y)).sub.T(H)] := (x, y)H. Moreover, there is a basis [{[S.sub.n]}.sub.n[member of]Z] ofT(H) such that, for any [f.sub.x] [member of] T(H) the sampling expansion

[f.sub.x](t) = [summation (n)] [??]x([t.sub.n] [S.sub.n](t), t [member of] [OMEGA] (3.1)

holds. The convergence of the series is not only in T(H) but also uniform over any subset on which [[parallel k(t)].sub.H] is bounded.

Proof. Assume [??](x)(t) = <x,[??](t)) = 0 on [OMEGA]. Then (x,[??]([t.sub.n])> = 0 for any n [member of] Z so that x = 0 since [{[??]([t.sub.n])}.sub.n[member of]Z] is a basis of H. Hence, K er T = K er T = {0} and T(H) becomes an RKHS as in Lemma 3.1 (b).

Let [{[x.sub.n]}.sub.n[member of]Z] [{[??]([t.sub.n])}.sub.n[member of]Z] and {[[X.sup.*.sub.n]}n[member of]Z] be its dual. Then {T([x.sub.n])} and {T([x*.sub.n])} are bases of T(H), which are dual each other since T is unitary.

Expanding any [f.sub.x] = T(x) in T(H) via the basis [{[S.sub.n]}.sub.n[member of]Z] = {T([x.sub.n]*)} gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Uniform convergence of the series (3.1) follows from Proposition 2.3 (c).

The single channel sampling expansion (3.1) may not converge absolutely unless [{[x.sub.n]}.sun.n[member of]Z] is an unconditional basis and may not be stable. However, if [{[x.sub.n]}.sub.n[member of]Z ]is an unconditional basis and [sup.sub.n] [[parallel [[x.sup.*].sub.n] < [infinity], then (3.1) is a stable sampling expansion, which converges absolutely on [OMEGA] In fact, if then, [{[S.sub.n]}.sub.n[member of]Z] becomes an unconditional basis of T(H) and [sup.sub.n] [parallel [S.sub.n]] = [sup.sub.n] [parallel [x.sup.*.sub.n]*|| < [infinity]. Since {[1/[parallel [S.sub.n]] [S.sub.n]}.sub.n [member of] z] is a Riesz basis of T(H) by the K6the-Toeplitz Theorem [9], there is a constant B > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Furthermore, the sampling series expansion (3.1) remains valid when [{[??]([t.sub.n]).sub.n [member of] z] [is not a basis but a frame of H. When k(t) = k(t) on [OMEGA] so that T = [??], Theorem 3.2 is essentially Theorem 1 in [3]. However, Theorem 3.2 might have some advantage over Theorem 1 in [3]. While Theorem 1 in [3] requires first the expansion of the kernel k(t) in terms of a given basis of H and then the interpolatory condition for the expansion coefficients at some points in [OMEGA], Theorem 3.2 simply requires points in [OMEGA], whose values under k(*) form a basis of H.

Now, we can extend Theorem 3.2 naturally to a multi-channel setting. Let [{k.sub.i]}.sup.N.sub.i=1] be N H-valued functions on and [{T.sub.i]}.sup.N.sub.i=1] operators on by

[T.sub.i](x) = [f.sup.i.sub.x] := (x, [[k.sub.i](t)).sub.H], x [member of] H.

Theorem 3.3. (Asymmetric nonuniform multi-channel sampling formula) If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and there exist points [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and constants [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for some integer M [greater than or equal to] 1 such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is an unconditional basis of H, then there is a basis {[S.sub.j,n] | 1 [less than or equal to] j [less than or equal to] M and n [member of] Z} of T(H) such that for any [f.sub.x] = T(x) [member of] T(H),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.2)

which converges in T(H). Moreover, the series (3.2) converges absolutely and uniformly on any subset of [OMEGA] over which [[parallel k(t)].sub.H] is bounded.

Proof. First, we prove that T is one-to-one. Suppose T(x)(t) = (x,k(t)) = 0 for all t [member of] [OMEGA]. Then, (x, ki(t)) O, 1 [less than or equal to] i [less than or equal to] N oll fl since KerT G N In particular, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for all 1 [less than or equal to] j [less than or equal to] M and n [member of] Z so that x = 0 since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] M and n [member of] Z} is a basis of H. Therefore, T : H [right arrow] T(H) is a bijection and T(H) becomes an RKHS under the inner product (T(x),T(y))T(H) := (x, Y)H by Lemma 3.1.

Let :[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for 1 [less than or equal to] j [less than or equal to] M and n [member of] Z and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] be the dual of {[x.sup.j.sub.n]}. Then,. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] becomes an unconditional basis of T(H) with the dual basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], which is also unconditional.

Expanding fx = T(x) in T(H) with respect to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Uniform convergence of the series (3.2) follows from Proposition 2.3 (c). Finally, the series (3.2) converges also absolutely since it is an unconditional basis expansion.

If either K er T = {0} or ki(t) = Ai(k(t)), 1 [less than or equal to] i [less than or equal to] N, where Ai's are auN tomorphisms of H, then the first assnmption [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of Theorem 3.3 is trivially satisfied. For example, it is so when H = [L.sup.2][-[pi], [pi], [OMEGA] = R and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the inverse Fourier transform. In particular, if N = M = 2, [k.sub.1](t) = k(t),[t.sub.1,n] = [t.sub.2,n] = [t.sub.n], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

1, 2 and n [member of] Z} is a Riesz basis of H, then Theorem 3.3 is essentially the same as Theorem 3 in [3].

When H = [L.sup.2] [-[pi]w, rw](w > 0), [OMEGA] = R and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for suitable bounded measurable functions [A.sub.i]([xi])(1 [less than or equal to] i [less than or equal to] N) on [-[pi]w, [pi]w], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence, T(H) becomes the Paley-Wiener space [PW.sub.[pi]w] and then Theorem 3.3 reduces to an asymmetric multi-channel sampling handled in [7].

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is a frame of H in Theorem 3.3, then the sampling series expansion (3.2) still holds.

As in the single channel case, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the multichannel sampling expansion (3.2) is also stable in the following sense.

Definition 3.4. (cf. Rawn [11] and Yao and Thomas [13]) We say that {[t.sub.i,n] | 1 [less than or equal to] i [less than or equal to] N and n [member of] Z} is a set of stable sampling for T(H) if there exists A > 0 which is independent of [f.sub.x] [member of] T(H) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let B > 0 be the upper Riesz bound for the Riesz basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of T(H). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

so that (3.2) is a stable sampling expansion with respect to {[t.sub.i,n]} when sup ||[[alpha].sup.j.sub.i,n]

We now discuss several examples in which we always take H = [L.sup.2] [-[pi], [pi]], [OMEGA] =

R, and k(t) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] so that T = [F.sup.-1] is the inverse Fourier transform and T(H) = [PW.sub.[pi]].

Example 3.5. (Sampling with Hilbert transform)

Take k(t) = i sgn([xi]) k(t) so that T(f)(t) = f(t) is the Hilbert transform of

f(t) in [PW.sub.[pi]. Choosing [{[t.sub.n]}.sub.n[member of]Z] = [{n}.sub.n[member of]Z], [{[x.sub.n]}.sub.n[member of]Z] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is an orthonormal basis of [L.sup.2][-[pi], [pi]] so that {[x*.sub.n]}.sub.n[member of]Z] = [{[x.sub.n]}.sub.n[member of]Z]. We then have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where sinct := sin [pi]t/[pi]t. Hence, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using the operational relation f = -f ([5, Appendix B]) and the fact that if f [member of] [PW.sub.[pi]], then so does f, we also have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Example 3.6. Here, we derive asymmetric derivative sampling formula on [PW.sub.[pi]], in which we take samples from f(t) and if(t) with ratio 2:1.

Take [k.sub.1](t) = k(t) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [k.sub.2](t) = -i[xi]k(t) = k'(t) so that [f.sup.1](t) = f(t) and [f.sup.2](t) = f'(t) for f(t) [member of] [PW.sub.[pi]]. Now, take the set of sampling points [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

if n is even,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

if n is odd, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Taking inverse Fourier transform on (3.3), we have a Riesz basis {Sl,n}n U {[S.sub.2,n]}.sub.n] of [PW.sub.[pi]] where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

With these setting we have the nonsymmetric derivative sampling formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Example 3.7. We now take kl(t) = k(t) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [k.sub.2](t) = [e.sup.i[xi]] k(t) so that [f.sup.1](t) = f(t) and [f.sup.2](t) = f(t - 1). We want to express f [member of] [PW.sup.[pi] via samples from f(t) and f(t - 1) with ratio 3: 2. Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.4)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then, we can obtain the sampling series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where {[S.sub.1,n]t(t)} U {[S.sub.2,n](t)} are the inverse Fourier transforms of functions in (3.4)

4 Oversampling and recovery of missing samples

We now develop an oversampling expansion, which extends the one in Kramer [2]. Again, let k and k : [OMEGA] [right arrow] H be H-valued functions. Assume that there exists {tn}~ez C [t such that {[x.sub.n] := k([t.sub.n])}.sub.[member of] Z is a basis of H with the dual basis {x~},ez. Define linear operators T and T on H by T(x) = <x, k(t))H := fx and T(x) = (x, k')H := .~x, respectively, and assume KerT C KerT. Then, both T and T are one-to-one, and so T(H) and T(H) become RKHS's.

Now, let G be a proper closed subspace of H and P : H [right arrow] G the orthogonal projection onto G. Then, fbr any x [member of] G we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Theorem 4.1. Under the above setting there is a sequence of sampling functions {[[T.sub.n]}.sub.n[member of]Z] in T(G) such that for any x [member of] G

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.2)

which converges in T(H) and uniformly on any subset of [OMEGA] over which ||k(t)||H is bounded. Moreover, if {[x.sub.n]} is a Riesz basis of H, then {[T.sub.n]}.sub.n[member of]Z] is a frame of T(G).

Pwof. Applying T on both sides of (4.1) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [T.sub.n](t) = T(P([x*.sub.n]))(t). Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the series (4.2) converges uniformly o,1 any subset over which ||k(t)||H is bounded. Finally, if {[x.sub.n]} is a Riesz basis of H, then {[x*.sub.n]} is also a R]esz basis of H so that {P([x*.sub,n]} is a frame of G since G is a closed subspace of H [1, Proposition 5.3.5]. Hence {[T.sub.n](t) = T(P([x*.sub.n]))(t)} is a frame of T(G).

Note that the sample set {[t.sub.n]}.sub.n[member of]Z] oversamples functions in T(G) in the sense that {[t.sub.n]}.sub.n[member of]Z] leads to a basis {Sn}~ez of T(H) (see Theorem 3.2), which properly contains T(G) but {Tn}nez in Theorem 4.1 may be overcomplete in T(G). Hence, we nmy call (4.2) an oversampling expansion of fx in T(G) for x E G.

Now assume that finitely many sample values {fz(tn) In E X = {nl, n2,'" , [n.sub.N]}} are missing. Applying T on both sides of (4.1) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.3)

which converges not only in T(H) but also pointwisely in [OMEGA] since T(H) is an RKHS. Setting t = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] in (4.3), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which can be rewritten in the matrix form as

(I - T) f = h

where f = ([f.sub.x]([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]), ..., [f.sub.x]([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.])).sup.T] is the column vector consisting of missing samples, h = ([h.sub.1], ... , [h.sub.N]).sup.T], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and T is the N x N matrix with entries

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Note that if I-T is invertible, tile missing samples f can be recovered uniquely. In particular, if <Tv, v> < [||v||.sup.2] for any v [member of] [C.sup.N] \ {0}, then I-T is invertible. We have:

Theorem 4.2. Under the same hypotheses as in Theorem 4.1, we assume further that [{[x.sub.n]}.sub.n] is a Riesz basis of H such that [x.sub.n] = U([e.sub.n]) where [{[e.sub.n]}.sub.n] is an orthonormal basis of H and U is an automorphism of H. If PU = UP and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCCII.] (4.4)

then any finitely many missing samples [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] in the oversampling expansion (4.2) can be uniquely recovered.

Proof. Note first that [x.sub.n] * = [(U *).sup.-1] ([e.sub.n] where {[x.sub.n] * is the dual of [{[x.sub.n].sub.n]. Hence, we have for any v = [([v.sub.1] ... ,[v.sub.N]).sup.T] [member of] [C.sup.N] \ {0},

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is an orthonormal basis of H. Hence, I-T is invertible.

If, moreover, [{[x.sub.n]}.sub.n] is an orthonormal basis of H in Theorem 4.2, then any finitely many missing samples [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] can be uniquely recovered when the condition (4.4) holds.

ACKNOWLEDGEMENTS

This work is partially supported by BK-21 project and KOSEF(R01-2006000-10424-0). Authors are grateful to the referee for his many valuable comments.

References

[1] O. Christensen, An introduction to Frames and Riesz bases, Birkhauser, Boston, 2003.

[2] P. J. S. G. Ferreira, Incomplete sampling series and the recovery of missing samples from oversampled band-limited signals, IEEE Trans. Signal Processing, SP-40(1), 225-227, 1992.

[3] A. G. Garcia and A. Portal, Sampling in the functional Hilbert space induced by a Hilbert space valued kernel, Appl. Anal., 82(12), 1145-1158, 2003.

[4] G. H. Hardy, Notes on special systems of orthogonal functions, IV: The orthogonal functions of Whittaker's cardinal series, Proc. Camb. Phil. soc., 37, 331-348, 1941.

[5] J. R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, Oxford Science Publications, 1996.

[6] J. R. Higgins and R. L. Stens, Sampling Theory in Fourier and Signal Analysis: Advanced Topics, Oxford Science Publications, 1999.

[7] Jong M. Kim and Kil H. Kwon, Vector sampling expansion in Riesz basis setting and its aliasing error, preprint.

[8] H. P. Kramer, A generalised sampling theorem, J. Math. Phys., 63, 68-72, 1957.

[9] N. K. Nikol'ski[??], Treatise on the shift operator: Spectral function theory, Springer-Verlag, 1980.

[10] A. Papoulis, Generalized sampling expansion, IEEE Trans. Circuits & Systems, CAS-24, 652-654, 1975.

[11] M. D. Rawn, A stable nonuniform sampling expansion involving derivatives, IEEE Trans. Info. Theory, IT-35, 1223-1227, 1989.

[12] S. Saitoh, Integral Transforms, Reproducing kernels and Their Applications, Longman, Essex, England, 1997.

[13] K. Yao and J. B. Thomas, On some stability and interpolatory properties of nonuniform sampling expansions, IEE Trans. Circuit Theory, CT-14, 404-408, 1967.

Yoon Mi Hong

Division of applied mathematics, KAIST

Daejeon, 305-701, Korea

ymhong@amath.kaist.ac.kr

Jong Min Kim

Division of applied mathematics, KAIST

Daejeon, 305-701, Korea

franzkimQamath.kaist.ac.kr

Kil H. Kwon

Division of applied amthematics, KAIST

Daejeon, 305-701, Korea

khkwon@amath.kaist.ac.kr

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Author: | Hong, Yoon Mi; Kim, Jong Min; Kwon, Kil H. |
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Publication: | Sampling Theory in Signal and Image Processing |

Geographic Code: | 9SOUT |

Date: | Jan 1, 2007 |

Words: | 4332 |

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