SAMPLING THEORY IN SIGNAL AND IMAGE PROCESSING c
Vol. 2, No. 3, Sept. 2003, pp. 217-233
2003 SAMPLING PUBLISHING
ISSN: 1530-6429
Nonuniform Sampling of Complex-Valued Harmonic Functions
Gerhard Schmeisser Mathematical Institute, University of Erlangen{Nuremberg, Bismarckstrasse 1 D-91054 Erlangen, Germany
[email protected]
Dedicated to Professor Paul L. Butzer
th
on the occasion of his 75
birthday
Abstract The class
H;C
exponential type
of complex-valued entire harmonic functions of contains various interesting subclasses such as
bandlimited signals, Bernstein classes, Paley{Wiener classes, entire functions of exponential type, and real-valued entire harmonic functions of exponential type.
First, we mention some properties
H;C and its subclasses. Then we show that every w 2 H;C is uniquely determined by samples of w and
of the class function
of the Wirtinger derivative
@w=@z ,
the samples being taken on
two sequences of (not necessarily uniform) points on the real line. Finally, we establish formulae for the reconstruction of
w
from
these samples, treating the cases of `sampling at a lowest rate' and `oversampling' separately. to subclasses of
H;C ,
By a restriction of these formulae
several old and new results on nonuniform
sampling of functions can be deduced.
Key words and phrases
:
Complex-valued entire harmonic func-
tions of exponential type, uniqueness theorems, nonuniform sampling, reconstruction formulae
2000 AMS Mathematics Subject Classi cation
1
| 31A05, 94A20
Introduction
Throughout this paper, we shall use the following notation.
1 2
218
G. SCHMEISSER
De nition 1.1 For 0; we denote by E the set of all functions which are holomorphic in the whole complex plane and satisfy
lim sup r!+1
log maxjzj=r jf (z )j r
f
(1)
:
We call E the class of entire functions of exponential type . Furthermore, we denote by B the restrictions to R of the functions from E :
By the Paley{Wiener theorem (see [3, Theorem 6.8.1] or [11, Theorem 7.2]), the set B \ L2 (R) is the class of signals which are bandlimited to [ ; ]. More generally, the set Bp := B \ Lp(R), where p 2 [1; 1]; is called the Bernstein class [11, De nition 6.5]. For 1 < p 2, the Paley{ Wiener class PWp , introduced in [11, De nition 6.15], is a subclass of Bq ; where q := p=(p 1); see [11, x 6.3]. Next, we introduce two classes of harmonic functions. De nition 1.2 For 0; we denote by H;R the set of all functions f : C ! R which are harmonic on C , that is, @ 2 f (z )
+ @x2
@ 2 f (z ) @y 2
0
(z = x + iy;
x; y
2 R );
and satisfy (1). We call H;R the class of (planar) real-valued entire harmonic functions of exponential type : Furthermore, we denote by H;C the set of all functions w := u + iv, where u and v belong to H;R . We call H;C the class of (planar) complex-valued entire harmonic functions of exponential type :
It was pointed out by Boas [4] that H;R is constituted by the real parts of the functions from E . However, other than in E ; the functions in H;R are not uniquely determined by their values on R. For example, the function u : x +iy 7! xy belongs to H;R for every > 0. Obviously, u is identically zero on the real line, but it does not vanish identically on C : In his paper [4], Boas also showed that the functions from H;R are uniquely determined by certain sequences of samples taken on two appropriately speci ed parallel or intersecting lines. In connection with these uniqueness theorems, Boas asked for correponding sampling formulae. The paper of Boas [4] has inspired numerous investigations concerning uniqueness and sampling of functions from H;R ; see [1], [2], [5], [6],
SAMPLING OF HARMONIC FUNCTIONS
219
[7], [8], [14], [15], [16], [18], [19], [20], [22]. In most of these papers, sampling formulae for E are used as an important tool; unfortunately, there is no direct relation between sampling in E and sampling in H;R . However, this is not really surprising since the classes E and H;R are quite di erent. In fact, their intersection contains the real constants only (see Proposition 2.1 below). In [17] we showed that, in the case of uniform sampling, the reconstruction of entire harmonic functions of exponential type becomes more lucid if we consider the wider class H;C : We established a sampling formula for H;C which extends the classical sampling formulae for bandlimited signals. Moreover, that sampling formula for H;C implies a sampling formula for E and also one for H;R : The purpose of this paper is to establish results analogous to those in [17] but for nonuniform sampling. The former results on uniform sampling will be included as special cases.
2
Relations between the considered classes
In this section, we present relations between the classes E , B ; H;R , and H;C . Partly, they were already indicated in [17], but here we give complete proofs. Proposition 2.1 The following relations hold:
(i) H;R H;C
(ii) E H;C ;
Proof Since the zero function belongs to
(iii) E \ H;R =
R:
H;R ; De nition 1.1 shows
immediately that (i) holds. Now let f 2 E : Then u := 0 and, by (i), it also belongs to H;C , but u is not holomorphic on C : Hence E 6= H;C : This completes the proof of (ii). Finally, if f 2 E \ H;R ; then v := =f = 0, and so f = 0 and > 0, we nd that g
:
z
7 !f
z
belongs to E if f belongs to E . Hence, by scaling the argument of f appropriately, we can always establish that the sequences in N (L; Æ) give admissible sampling points for the `scaled' function g. In the case of oversampling, a uniqueness theorem for H;C is very simple. Theorem 3.1 Let (sn )n2Z and (tn )n2Z be two sequences from N (L; Æ), and let w 2 H;C , where < . If w(sn )
= 0 and
@w @z
tn
=0
(n 2 Z);
then w is identically zero on C :
Proof By Proposition 2.2, there exist functions f and g in E such that w(z )
= f (z ) + g(z ) and g(0) = 0. This implies that @w @z
(z ) =
@ @z
g (z )
= g0 (z );
224
G. SCHMEISSER
and so, by the assumption, g 0 (tn ) = 0 (n 2 Z): Furthermore, together with g, the derivative g0 also belongs to E (see [3, Theorem 2.4.1]). Hence a theorem of DuÆn and Schae er (see [9, Theorem 1] or [3, p. 191, Theorem 10.5.3]) applies and yields that g0 (z ) 0: Since also g(0) = 0, we have g(z ) 0. Thus w = f , and so w 2 E . Using now that w(sn ) = 0 for all n 2 Z, we nd, by employing the theorem of DuÆn and Schae er once again, that w is identically zero on C. 2 For a uniqueness theorem in the case of sampling at a lowest rate, additional hypotheses have to be imposed. We need something similar to w and @w=@z belonging to Lp (R) and, in addition, a restriction on the constant L specifying the class N (L; Æ) of admissible sampling points. However, there is some dependence between these two hypotheses. The restriction on L can be relaxed by replacing the Lp requirement by something stronger. This allows us some exibility. For easy reference, we introduce the following notation. De nition 3.2 Let p 2 [1; 1); and let k be a non-negative integer. Furthermore, let f be a complex-valued function whose domain contains the real line. Then we say that f belongs to Lpk (R) if the function (
g
belongs to
:
R x
! C 7 ! (1 + jxj)k f (x)
Lp (R ):
Clearly, p p Lk (R ) Lpm (R ) L0 (R ) = Lp (R ) for 0 < m < k : For describing our restrictions on the constant L, it is convenient to introduce k 2 N 0 ; p 2 [1; 1) : (5) (k; p) := min 1 +4 k ; 1 +2pkp The following uniqueness theorem is a generalization of a uniqueness theorem in [17, Theorem 2.1]. The latter is covered by the case k = m = 0 and p = q = 2.
SAMPLING OF HARMONIC FUNCTIONS
225
Theorem 3.2 For non-negative integers k; m and real numbers p; q 2 [1; 1), let @w p w 2 H;C \ Lk (R ) and 2 Lqm(R): @z Furthermore, for L1 2 0; (k; p) and L2 2 0; (m; q) ; where is given by (5), let (sn )n2Z 2 N (L1 ; Æ) and (tn )n2Z 2 N (L2 ; Æ): If @w w(sn ) = 0 and tn = 0 (n 2 Z); @z then w is identically zero on C :
For the proof, we proceed as in the proof of Theorem 3.1 except that, instead of the theorem of DuÆn and Schae er, we employ a result by Voss [21, Corollary 2.2.3] (also see Chapter3/phd thesis voss.zip on the CD-ROM in [12]), which we specialize for our needs and state it as follows. Theorem A For a non-negative integer k and a real number p 2 [1; 1), let f 2 E \ Lpk (R): Furthermore, for L 2 0; (k; p) ; where is given by (5), let s = (sn )n2Z 2 N (L; Æ): Then f (z )
=
1 X
n=
f (sn )Gn (s; z )
(z 2 C );
1 where Gn is given by (4). The series converges uniformly on every compact subset of C :
Proof of Theorem 3.2 Some indications will suÆce. Again, we use
Proposition 2.2 and write w as w(z ) = f (z ) + g(z ); where f; g 2 E and = 0. The assumptions on @w=@z imply that g 0 2 E \ Lqm (R ) and g 0 (tn ) = 0 (n 2 Z): Now Theorem A applies to g0 and yields that g0 = 0. From this, we conclude that w = f , and thus p w 2 E \ Lk (R ) and w(sn ) = 0 (n 2 Z): Finally, using Theorem A once again, we nd that w is identically zero on C : 2 g (0)
226
4
G. SCHMEISSER
Sampling theorems
For sampling complex-valued entire harmonic functions of exponential type at a lowest rate, we can extend the sampling formula of Theorem A. In addition to (4), we need a second sequence of fundamental functions, which we de ne by Hn (t; x + iy )
:= i
Z y y
(x; y 2 R; n 2 Z);
Gn (t; x + i ) d
(6)
where t is a sequence from the class N (L; Æ) for some L. Since Gn (t; z ) is real for real z , we observe that Hn (t; z )
=
Hn (t; z )
=
Theorem 4.1 For non-negative integers [1; 1), let p w 2 H;C \ Lk (R ) and Furthermore, for L1 given by (5), let
2 (0; (k; p))
s := (sn )n2Z 2 N (L1 ; Æ )
Then w (z )
=
1 X n=
1
w(sn )Gn (s; z ) +
(n 2 Z):
Hn (t; z ) k; m
and real numbers
2 Lqm(R): 2 (0; (m; q));
@z
2
where is
t := (tn )n2Z 2 N (L2 ; Æ ):
1 @w X (tn )Hn (t; z ) @z
n=
p; q
@w
and L2
and
(7)
1
(z 2 C ); (8)
where Gn and Hn are de ned by (4) and (6), respectively. The series converge uniformly on every compact subset of C :
Proof By Proposition 2.2, we can represent w as w(z ) = f (z ) + g(z ); where f; g 2 E : This can be rewritten as w(z )
= F (z ) + W (z );
(9)
where F (z )
:= f (z ) + g(z ) and
W (z )
:= g(z )
g (z ):
227
SAMPLING OF HARMONIC FUNCTIONS
The functions F and G have the following properties:
2 E W 2 H;C F
Furthermore, since obtain
F
w(x) (x 2 R); (10) and W (x) 0 (x 2 R ): (11) and the function z 7! g(z ) are holomorphic, we and
@w @z
=
F (x)
@W @z
= g0 (z ):
(12)
Now the hypotheses of Theorem 4.1 together with (10), (12), and the fact that g0 2 E show that Theorem A applies to F and to g0 . Thus, we obtain 1 X F (z ) = w(sn )Gn (s; z ) (13) n=
and
g 0 (z ) =
1
1 @w X (tn )Gn (t; z ); @z
n=
1
where both series converge uniformly on all compact subsets of C . Therefore, when we integrate the second series, we are allowed to interchange summation and integration. Taking also (7) into account, we nd that g (z )
g (z )
= i =
Z y y
g 0 (x + i ) d =
1 @w X (tn ) @z
n=
1
i
Z y y
Gn (t; x + i ) d
1 @w X (tn ) Hn (t; z ): @z
n=
1
Conjugation of both sides of these equations yields W (z )
=
1 @w X (tn ) Hn (t; z ): @z
n=
1
Finally, combining (9), (13), and (14), we arrive at (8).
(14)
2
The hypotheses of Theorem A and Theorem 4.1 imply that the moduli of the sampled functions have a majorant that decays on the real line as x ! 1: In the case of oversampling, this is no longer a requirement.
228
G. SCHMEISSER
We can even admit that the moduli of the sampled functions grow as x ! 1: In fact, they may grow faster than any polynomial. The idea is to multiply Gn (s; z ) by (z sn), where is an entire function of small exponential type whose modulus has a very rapidly decaying majorant to compensate for the growth of the samples. The situation of oversampling leaves some room for incorporating such a multiplier. Based on this technique, Voss [21, Corollary 2.3.4] (also see Chapter3/phd thesis voss.zip on the CD-ROM in [12]) obtained a result, which may be stated as follows. Theorem B For any positive L and Æ 2 (0; 1]; let s := (sn )n2Z N (L; Æ): Furthermore, let f 2 E ; where < , and suppose that
j sn j f (sn ) = O exp (log jsnj) > 0. Then, for " 2 (; ), there
where depending also on , such that 1 X f (z )
=
n=
1
f (sn )(z
as
n
! 1 ;
exists a function
sn )Gn (s; z )
2
2 E",
(z 2 C ) ;
where Gn is given by (4). The series converges absolutely and uniformly on every compact subset of C :
Remark 4.1 The statement concerning absolute and uniform conver-
gence is not explicitly contained in Voss's corollary, but it can be veri ed in a standard way (cf. [10] and [13, Theorem 1]). Proceeding as in the proof of Theorem 4.1, but using Theorem B instead of Theorem A, we easily obtain the following result on oversampling of functions from H;C : Theorem 4.2 For any positive L and Æ 2 (0; 1]; let s := (sn )n2Z 2 N (L; Æ )
and
t := (tn )n2Z 2 N (L; Æ ):
Furthermore, let w 2 H;C ; where < , and suppose that j sn j w(sn ) = O exp (log jsnj)
as
n
! 1
229
SAMPLING OF HARMONIC FUNCTIONS
and
(tn ) = O exp (logjtjntj j) @z n where > 0. Then @w
w(z )
=
1 X n=
1
where
e n (s ; z ) + w(sn )G
e n (s ; z ) G
and e n (t; x + iy ) H
:= i
Z y y
(x
as
n
! 1 ;
1 @w X (tn )Hen (t; z ) @z
n=
1
:= (z
(z 2 C ); (15)
sn )Gn (s; z )
tn + i )Gn (t; x + i ) d
(x; y 2 R)
with and Gn as in Theorem B. The series converge absolutely and uniformly on every compact subset of C :
5
Concluding remarks
Remark 5.1 The sampling formulae of Theorems 4.1 and 4.2 allow us
to deduce sampling formulae for various subclasses of H;C . (i) On the real line, the fundamental functions Hn (t; ) and Hen (t; ) vanish identically. As a result, in restrictions to the real line, the second series in the formulae (8) and (15) disappear. In particular, this is the case for sampling in B : (ii) If w 2 E ; then @w=@z vanishes identically on C , and thus, again, the second series in the formulae (8) and (15) disappear. (iii) If w 2 H;R , then w(z )
