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The convergence of sampling series based on multiresolution analysis
PERGAMON
Applied Mathematics Letters 14 (2001) 897-901
Applied Mathematics Letters www.elsevier.nl/locate/aml
T h e C o n v e r g e n c e of S a m p l i n g Series B a s e d on M u l t i r e s o l u t i o n A n a l y s i s W E N C H A N G S U N AND X I N G W E I Z H O U Nankai Institute of Mathematics Nankai University, Tianjin 300071, P.R. China©nankai. edu.
cn
(Received June 2000; accepted July 2000) Communicated by P. Butzer A b s t r a c t - - T h e sampling theory says that a function may be determined by its sampled values at some certain points provided the function satisfies some certain conditions. In this paper, we consider the convergence of the sampling series for bounded functions with the sampling function from some multiresolution analysis. We also present a convergence theorem on the sampling series with irregular sampling points and give the approximation error. (~) 2001 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - S a m p l i n g series, Sampling theorem, Wavelets, Multiresolution analysis.
1. I N T R O D U C T I O N The sampling theory says that if a function f(x) satisfies some certain conditions, then f(x) is uniquely determined by its sampled values at some discrete points (Xk : k C Z} and f ( x ) = E f (Xk) Sk(x), kEZ for some s a m p l i n g f u n c t i o n s {Sk(x)}. For example, if f • L 2 ( R ) w i t h supp f C [ - ~ , ~ t ] , t h e n f ( x ) = ~ k e z f(kTr/f~)(sin(gtx - k~r)/(f~x - kTr)). T h i s is the classical S h a n n o n s a m p l i n g t h e o r e m . Recently, the s a m p l i n g t h e o r e m has been e x t e n d e d to some wavelet subspaces (see [1-8]). Specifically, we have the following proposition. PROPOSITION 1.1. (See [4,7].) Let V0 be a dosed subspace of L 2 ( R ) a n d {~(- - n) : n • Z} be a Riesz basis for Vo. Suppose that ~ is continuous, SUPx ~ n ~ z 19~(x + n)l 2 < +oo, and that there
exist two constants C1, C2 > 0 such that C1 < IZ~(0, w)t < C2,
(1.1)
where Z~(x,w) := ~-]~ez ~(x + n)e -i'~"~ is the Zak transform of ~. Put S(w) = ~(w)/Z~(O,w). Then {S(- - n)} is a Riesz b ~ i s for V0 and for any f • Y0, f ( ~ ) = E k c z f ( k ) S ( ~ - k), where the convergence is both in L 2 ( R ) and uniform on R . The authors would like to thank the referees for their valuable suggestions which helped us to improve this paper. This work was supported by the National Natural Science Foundation of China (Grant No. 16971047), China Postdoctoral Science Foundation, and the Research Fund for the Doctoral Program of Higher Education. 0893-9659/01/$ - see front matter (~) 2001 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(01)00062-3
Typeset by ~4A/I,S-TEX
898
W. SUN AND X. ZHOU
For any k >_ 0, define n nEZ Fischer [9] proved that if f and ~ are smooth enough then ( R k f ) ( t ) converges to f ( t ) uniformly on R. In literature, the series in (1.2) is usually viewed as a sampling series. In this paper, we study the convergence of R k f for any bounded function f . We also study the case for which the sampling points are irregular and give the convergence rate for some continuous functions. NOTATIONS. In this paper, the Fourier transform of f E L2(R) is defined by ] ( w ) = f a f ( x ) e - ~ " dx. ACIo~ = {f(x) : f is locally absolutely continuous on R}.
2. M A I N R E S U L T S We begin with a simple lemma. LEMMA 2.1. Suppose that f E ACloc and f, f ' E L I ( R ) . Then for any bounded real sequence {e~}, Enezle~f(x - n)l c o n v e r g e s pointwisely to a continuous function. PROOF. Let C = supn [CnI. Since f l Y'~.n Ic~f( x -- n)] dx <_ f l C . ~ n If( x - n)l dx = C]IfiiL1 < +oo, y~,~ezic,~f(x - n)l is convergent ahnost everywhere on R. On the other hand, for any x < y < x + 1, we see from f ' E L I ( R ) that ~--~nez [c,~f'(t - n)[ is convergent in L l [ x , y ] and IE
ez Ic
f(y -
n)l -
leJ(x
-
n)ll -< f 'J
Icnf'(t
-
n)l
dr.
Hence,
En
zlcJ(x -
converges pointwisely to a continuous function. THEOREM 2.2. L e t qo(x) be continuous such that (i) ~ E ACloc and ~, ~o' E L I(R), or (ii) there exist s o m e C, ¢ > 0 such that [~(x)I G C . (1/(1 + [x[l+e)). S u p p o s e that Z ~ ( O , w ) satisfies (1.1) for some constants C1,C2 > O. L e t S(w) = ~ ( w ) / Z ~ ( O , w). / f q~(2n~r) = hA,O, then for a n y bounded function f and each point t E R of continuity of f , limk--,+oo(Rkf)(t) = f ( t ) . Furthermore, if f is uniformly continuous on R , then the convergence is uniform on R .
PROOF. First, we show that E k E z [ S ( x -- k)l is convergent uniformly on [0, 1]. For Condition (i), we see from Lemma 2.1 that ~ e z [~(n)l < +oo. For Condition (ii), it is easy to see that ~-~,~ez]~(n)l < +oo. Since Z~o(O,w) has no zero on R, by Wiener's lemma [10, pp. 367,368], there exists some {ak} E 11 such that 1Zq0(0, w) = ~ n e z c~ne-~'~" Hence,
S(x) =
-
Let S l ( x ) = ~-~.n~Zlt~,~(x - n)]. For Condition (i), S l ( x ) is continuous due to Lemma 2.1 and E S l ( x -- k) ~- E E IOLn~g~(X-- 1~-- n)[ = E Icon[ E I ~ ( x - k)]. kEZ kEZ nEZ nEZ kEZ Using Lemma 2.1 again, we see that ~ k ~ z S l ( x - k ) is continuous on R. For Condition (ii), since ~ n e z [ ~ ( x - n)[ is convergent uniformly on any compact set, it is easy to see that both S l ( x ) and ~ k E z S I ( x -- k) are continuous on R. By Dini's theorem [11, pp. 369,370], ~ k e z S l ( x - k) is convergent uniformly on [0, 1]. Since IS(x)[ < S I ( x ) , ~ k e z S ( z - k) is also convergent uniformly on [0, 1]. Next, we show that ~--~-kczS ( x - k) = 1, V x . For any k E Z, f01 Y~-nEZ~(X n)e -i2k~x dx = f2oo +oo ~(x)e -i2k~x dx = ~(2krr) = 5k,0. Hence, Y-~.nez ~ o ( x - n ) -- 1, Vx. In particular, Z~(0, 0) = ~-~.,~ez~(n) = 1 and so ~-'~kezak = 1/(Zqo(0, 0)) = 1. It follows that ~ k e z S ( x - k) = ~ k e Z Y~neZ anqo(x -- k - n) = Y~.~ez Y~.kez an~o(x - k -
-
n ) = 1. Now the conclusion follows by Theorem 1 in [12].
C o n v e r g e n c e of S a m p l i n g Series
899
Next we study the convergence of sampling series for which the sampling points are irregular. Let Ng 1 be the set of bounded functions f ( x ) such that ~ n e z l f ( x - n)l is convergent, uniformly on [0, 1] and that ~-jnezf(X - n) - 1. LEMMA 2.3. Suppose that h > 0 and that {a(h, n) : n c Z} is a real sequence satisfying lim sup la(h, n) - hn I = O.
(2.1)
h---~0 n
Then for any S • N~ 1 and f • L ~ , at each point t • R of continuity o f f , lim E
h-*O
f(a(h,n))S (h -n)
= f(t).
n6Z
PROOF. Let M = max{Nfll~, sup~ E~ezlS(x ~)1}. Then M < +oc. For any s > O, there exists some N > 0 such that -
x
ix/h--nl>_N
Since f is continuous at t, there exists some ~ > 0 such that for any x with ] x - t I < 5, If(x) - f(t)l < e. On the other hand, by (2.1), there exists some 0 < ~1 < 5 / 2 N such that for any 0 < h < r/, la( h,n) - hn I < 5/2,Vn. It follows that la(h,n) - t I < ]a( h,n) - hn I + Ihn - t I < ~/2 + N h < ~ if I t ~ h - n I < N. Hence,
n~czf(a(h,n"S(h-n) <-
E
- f ( t,
'f(a(h'n))-f(t)l"
S(h-n)
+
]t/h-nl
E
,f(a(h,n))-f(t),.
S(h-n)
lt/h-nl>N
< e. M + 2 M s = 3Me,
0 < h < r1.
This completes the proof. By Theorem 2.2 and Lemma 2.3, it is easy to prove the following theorem. THEOREM 2.4. Let the hypotheses be as in Theorem 2.2. Furthermore, suppose that {xn} is a real sequence such that iXn - n I < L for some constant L. Then for any bounded function f(t), at each point t 6 R of continuity o f f , limk--.+~ ~-~n f ( X n / 2 k ) s ( 2kt --n) = f(t). At the end of this paper, let us study the approximation error. The following two lemmas are easy to prove, and we leave them to the readers. LEMMA 2.5.
E
1
n e Z (1 + In -- xl) ~ --
a -- 1
(~)~
Vc~>I,
x•R.
'
LEMMA 2.6. Suppose that q~ e L2(R) and ~(w) = ~(w)~(w), where ~(w) is a bounded 27r-periodic function. Then ~ n e z I~P(x - n)l 2 -< IL~II2 ~-~.neZ [~(x -- n)l 2, a.e. THEOREM 2.7. Let the hypotheses be as in Theorem 2.4. Define B(.) as in Lemma 2.5. If I~(x)l < 6(1 + Ixl)-~,
f(t) - E nCZ
7 > 2, a n d I f ( x ) - f ( Y ) i -< C i I x - y t ~ , 0
n ) S (2kt - n ) f ( x-~-
< ~ < ]/2,
<_ 2 - ~ k C f (n~Mo,.y + M~,~),
then
Vt • R,
900
W.
where
AND
X. ZHOU
Mc,~ 2CB(7) ,/4(1-a)r2C2~ ' -
PROOF.
SUN
C----~ + v
1-2a
Cx/B(2",/ - 2)]
L c2('7-2)
+
C1
Put fl(w) = 1/Z~(O,w) = ~ k c z flk e-ik~" Then
--
(z~(0,~))
2
- c-7 ~
kEZ
Ik~(k)l _<
k~0
( 1 + Ikl)~ -1
<
2C
c~(.~ - 2)
and IIZ(~)lloo _< 1/C1. It follows that
C CB('-,/) _< 119(~o)11ook~z ~ (1 + Ix - kl)'~ -< --C,
kEZ
Since [-ixS(x)]^(w) = S'(w) = fl'(w)~(w) + 13(w)qY(w), by Lemma 2.6, we have \ 1/2
-< II/~'11oo
I~(x
-
4-ll~[Ioo(n~ezl(x--n)~(x--n)]2)l/2 +[l~llooCIn~ez(l+lx-n[)-2~'+21
n)l 2
1/2
_< IlZ'llooC
(1 + Iz
-
nl) -2"/
< 2c~,/N5-~ C , / B ( 2 . ~ - 2) + - c,~(~ - 2) c,
Therefore,
~' Ix - nl'~lS(x - ~)1 -< 211Slloo + nEZ
_< 211Slloo +
(
ln-xl_>l
Ix - ~l~-ll(x - n)S(x - ~)1
Iz - ~1 ~-2
< 2cB(~)
I,~-xl>l ,/4(1- c~) t.[2C2VfB~ +
-
v
C1
1-2~
/ c~(,~-2)
I(x - n)S(x Cx/B(2" f -
- n)l 2
)lj
2)] = 2kIa~.
C1
It follows that
-< E
c,, I~n -- 2"tl ° 2 -'~k IS (2kt- n)l
nEZ
-< E 2-~kc* (Ix,, - hi" + 12h- nl '~) Is (2hnEZ
-< 2-~cf L° Z Is (2~t- n)l nCZ
+ 2-'~cf F. I~t - ,~1" is (2~t- n)l nEZ
<_ 2-~kCf (L~Mo,.7 + M~,.y).
n)l
Convergence of Sampling Series
901
REMARK. The approximation order can also be derived from results in [12] or [13]. Here we give the explicit error bound. We point out that both scaling functions ~N(2 ~ N < 20) of Daubechies' and B-splines defined by
~(w) = (sin(w/2)\/m+l
\
/
'
re>l,
meet the requirements of these theorems (see [8]).
REFERENCES 1. A. Aldroubi and M. Unser, Families of wavelet transforms in connection with Shannon's sampling theory and the Gabor transform, In Wavelets--A Tutorial in Theory and Applications, (Edited by C.K. Chui), pp. 509-528, Academic Press, Boston, MA, (1992). 2. A.J.E.M. Janssen, The Zak transform and sampling theorem for wavelet subspaces, IEEE Trans. Signal Proc. 41, 3360-3364 (1993). 3. Y. Liu and G. Walter, Irregular sampling in wavelet subspaces, Fourier Anal. Appl. 2, 181-189 (1995). 4. W. Sun and X. Zhou, Frames and sampling theorem, Science in China 41, 606-612 (1998). 5. W. Sun and X. Zhou, Sampling theorem for multiwavelet subspaces, Chinese Science Bulletin 44, 1283-1286 (1999). 6. X.G. Xia and Z. Zhang, On sampling theorem, wavelets, and wavelet transforms, IEEE Trans. Signal Proc. 41, 3524-2535 (1993). 7. G. Walter, A sampling theorem for wavelet subspaces, IEEE Trans. Inform. Theory 38, 881-884 (1992). 8. X. Zhou and W. Sun, On the sampling theorem for wavelet subspaces, Y. Fourier Anal. Appl. 5, 347-354 (1999). 9. A. Fischer, Multiresolution analysis and multivariate approximation of smooth signals in CB (Rd), J. Fourier Anal. Appl. 2, 161-180 (1995). 10. W. Rudin, Real and Complex Analysis, McGraw-Hill, (1987). 11. S.C. Malik, Mathematical Analysis, John Wiley &: Sons, New York, (1984). 12. P.L. Butzer, S. Ries and R.L. Stens, Approximation of continuous and discontinuous functions by generalized sampling series, J. Approx. Theory 50, 25-39 (1987). 13. A. Fischer, On wavelets and prewavelets with vanishing moments in higher dimensions, Y. Approx. Theory 90, 46-74 (1997).
Applied Mathematics Letters 14 (2001) 897-901
Applied Mathematics Letters www.elsevier.nl/locate/aml
T h e C o n v e r g e n c e of S a m p l i n g Series B a s e d on M u l t i r e s o l u t i o n A n a l y s i s W E N C H A N G S U N AND X I N G W E I Z H O U Nankai Institute of Mathematics Nankai University, Tianjin 300071, P.R. China
cn
(Received June 2000; accepted July 2000) Communicated by P. Butzer A b s t r a c t - - T h e sampling theory says that a function may be determined by its sampled values at some certain points provided the function satisfies some certain conditions. In this paper, we consider the convergence of the sampling series for bounded functions with the sampling function from some multiresolution analysis. We also present a convergence theorem on the sampling series with irregular sampling points and give the approximation error. (~) 2001 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - S a m p l i n g series, Sampling theorem, Wavelets, Multiresolution analysis.
1. I N T R O D U C T I O N The sampling theory says that if a function f(x) satisfies some certain conditions, then f(x) is uniquely determined by its sampled values at some discrete points (Xk : k C Z} and f ( x ) = E f (Xk) Sk(x), kEZ for some s a m p l i n g f u n c t i o n s {Sk(x)}. For example, if f • L 2 ( R ) w i t h supp f C [ - ~ , ~ t ] , t h e n f ( x ) = ~ k e z f(kTr/f~)(sin(gtx - k~r)/(f~x - kTr)). T h i s is the classical S h a n n o n s a m p l i n g t h e o r e m . Recently, the s a m p l i n g t h e o r e m has been e x t e n d e d to some wavelet subspaces (see [1-8]). Specifically, we have the following proposition. PROPOSITION 1.1. (See [4,7].) Let V0 be a dosed subspace of L 2 ( R ) a n d {~(- - n) : n • Z} be a Riesz basis for Vo. Suppose that ~ is continuous, SUPx ~ n ~ z 19~(x + n)l 2 < +oo, and that there
exist two constants C1, C2 > 0 such that C1 < IZ~(0, w)t < C2,
(1.1)
where Z~(x,w) := ~-]~ez ~(x + n)e -i'~"~ is the Zak transform of ~. Put S(w) = ~(w)/Z~(O,w). Then {S(- - n)} is a Riesz b ~ i s for V0 and for any f • Y0, f ( ~ ) = E k c z f ( k ) S ( ~ - k), where the convergence is both in L 2 ( R ) and uniform on R . The authors would like to thank the referees for their valuable suggestions which helped us to improve this paper. This work was supported by the National Natural Science Foundation of China (Grant No. 16971047), China Postdoctoral Science Foundation, and the Research Fund for the Doctoral Program of Higher Education. 0893-9659/01/$ - see front matter (~) 2001 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(01)00062-3
Typeset by ~4A/I,S-TEX
898
W. SUN AND X. ZHOU
For any k >_ 0, define n nEZ Fischer [9] proved that if f and ~ are smooth enough then ( R k f ) ( t ) converges to f ( t ) uniformly on R. In literature, the series in (1.2) is usually viewed as a sampling series. In this paper, we study the convergence of R k f for any bounded function f . We also study the case for which the sampling points are irregular and give the convergence rate for some continuous functions. NOTATIONS. In this paper, the Fourier transform of f E L2(R) is defined by ] ( w ) = f a f ( x ) e - ~ " dx. ACIo~ = {f(x) : f is locally absolutely continuous on R}.
2. M A I N R E S U L T S We begin with a simple lemma. LEMMA 2.1. Suppose that f E ACloc and f, f ' E L I ( R ) . Then for any bounded real sequence {e~}, Enezle~f(x - n)l c o n v e r g e s pointwisely to a continuous function. PROOF. Let C = supn [CnI. Since f l Y'~.n Ic~f( x -- n)] dx <_ f l C . ~ n If( x - n)l dx = C]IfiiL1 < +oo, y~,~ezic,~f(x - n)l is convergent ahnost everywhere on R. On the other hand, for any x < y < x + 1, we see from f ' E L I ( R ) that ~--~nez [c,~f'(t - n)[ is convergent in L l [ x , y ] and IE
ez Ic
f(y -
n)l -
leJ(x
-
n)ll -< f 'J
Icnf'(t
-
n)l
dr.
Hence,
En
zlcJ(x -
converges pointwisely to a continuous function. THEOREM 2.2. L e t qo(x) be continuous such that (i) ~ E ACloc and ~, ~o' E L I(R), or (ii) there exist s o m e C, ¢ > 0 such that [~(x)I G C . (1/(1 + [x[l+e)). S u p p o s e that Z ~ ( O , w ) satisfies (1.1) for some constants C1,C2 > O. L e t S(w) = ~ ( w ) / Z ~ ( O , w). / f q~(2n~r) = hA,O, then for a n y bounded function f and each point t E R of continuity of f , limk--,+oo(Rkf)(t) = f ( t ) . Furthermore, if f is uniformly continuous on R , then the convergence is uniform on R .
PROOF. First, we show that E k E z [ S ( x -- k)l is convergent uniformly on [0, 1]. For Condition (i), we see from Lemma 2.1 that ~ e z [~(n)l < +oo. For Condition (ii), it is easy to see that ~-~,~ez]~(n)l < +oo. Since Z~o(O,w) has no zero on R, by Wiener's lemma [10, pp. 367,368], there exists some {ak} E 11 such that 1Zq0(0, w) = ~ n e z c~ne-~'~" Hence,
S(x) =
-
Let S l ( x ) = ~-~.n~Zlt~,~(x - n)]. For Condition (i), S l ( x ) is continuous due to Lemma 2.1 and E S l ( x -- k) ~- E E IOLn~g~(X-- 1~-- n)[ = E Icon[ E I ~ ( x - k)]. kEZ kEZ nEZ nEZ kEZ Using Lemma 2.1 again, we see that ~ k ~ z S l ( x - k ) is continuous on R. For Condition (ii), since ~ n e z [ ~ ( x - n)[ is convergent uniformly on any compact set, it is easy to see that both S l ( x ) and ~ k E z S I ( x -- k) are continuous on R. By Dini's theorem [11, pp. 369,370], ~ k e z S l ( x - k) is convergent uniformly on [0, 1]. Since IS(x)[ < S I ( x ) , ~ k e z S ( z - k) is also convergent uniformly on [0, 1]. Next, we show that ~--~-kczS ( x - k) = 1, V x . For any k E Z, f01 Y~-nEZ~(X n)e -i2k~x dx = f2oo +oo ~(x)e -i2k~x dx = ~(2krr) = 5k,0. Hence, Y-~.nez ~ o ( x - n ) -- 1, Vx. In particular, Z~(0, 0) = ~-~.,~ez~(n) = 1 and so ~-'~kezak = 1/(Zqo(0, 0)) = 1. It follows that ~ k e z S ( x - k) = ~ k e Z Y~neZ anqo(x -- k - n) = Y~.~ez Y~.kez an~o(x - k -
-
n ) = 1. Now the conclusion follows by Theorem 1 in [12].
C o n v e r g e n c e of S a m p l i n g Series
899
Next we study the convergence of sampling series for which the sampling points are irregular. Let Ng 1 be the set of bounded functions f ( x ) such that ~ n e z l f ( x - n)l is convergent, uniformly on [0, 1] and that ~-jnezf(X - n) - 1. LEMMA 2.3. Suppose that h > 0 and that {a(h, n) : n c Z} is a real sequence satisfying lim sup la(h, n) - hn I = O.
(2.1)
h---~0 n
Then for any S • N~ 1 and f • L ~ , at each point t • R of continuity o f f , lim E
h-*O
f(a(h,n))S (h -n)
= f(t).
n6Z
PROOF. Let M = max{Nfll~, sup~ E~ezlS(x ~)1}. Then M < +oc. For any s > O, there exists some N > 0 such that -
x
ix/h--nl>_N
Since f is continuous at t, there exists some ~ > 0 such that for any x with ] x - t I < 5, If(x) - f(t)l < e. On the other hand, by (2.1), there exists some 0 < ~1 < 5 / 2 N such that for any 0 < h < r/, la( h,n) - hn I < 5/2,Vn. It follows that la(h,n) - t I < ]a( h,n) - hn I + Ihn - t I < ~/2 + N h < ~ if I t ~ h - n I < N. Hence,
n~czf(a(h,n"S(h-n) <-
E
- f ( t,
'f(a(h'n))-f(t)l"
S(h-n)
+
]t/h-nl
E
,f(a(h,n))-f(t),.
S(h-n)
lt/h-nl>N
< e. M + 2 M s = 3Me,
0 < h < r1.
This completes the proof. By Theorem 2.2 and Lemma 2.3, it is easy to prove the following theorem. THEOREM 2.4. Let the hypotheses be as in Theorem 2.2. Furthermore, suppose that {xn} is a real sequence such that iXn - n I < L for some constant L. Then for any bounded function f(t), at each point t 6 R of continuity o f f , limk--.+~ ~-~n f ( X n / 2 k ) s ( 2kt --n) = f(t). At the end of this paper, let us study the approximation error. The following two lemmas are easy to prove, and we leave them to the readers. LEMMA 2.5.
E
1
n e Z (1 + In -- xl) ~ --
a -- 1
(~)~
Vc~>I,
x•R.
'
LEMMA 2.6. Suppose that q~ e L2(R) and ~(w) = ~(w)~(w), where ~(w) is a bounded 27r-periodic function. Then ~ n e z I~P(x - n)l 2 -< IL~II2 ~-~.neZ [~(x -- n)l 2, a.e. THEOREM 2.7. Let the hypotheses be as in Theorem 2.4. Define B(.) as in Lemma 2.5. If I~(x)l < 6(1 + Ixl)-~,
f(t) - E nCZ
7 > 2, a n d I f ( x ) - f ( Y ) i -< C i I x - y t ~ , 0
n ) S (2kt - n ) f ( x-~-
< ~ < ]/2,
<_ 2 - ~ k C f (n~Mo,.y + M~,~),
then
Vt • R,
900
W.
where
AND
X. ZHOU
Mc,~ 2CB(7) ,/4(1-a)r2C2~ ' -
PROOF.
SUN
C----~ + v
1-2a
Cx/B(2",/ - 2)]
L c2('7-2)
+
C1
Put fl(w) = 1/Z~(O,w) = ~ k c z flk e-ik~" Then
--
(z~(0,~))
2
- c-7 ~
kEZ
Ik~(k)l _<
k~0
( 1 + Ikl)~ -1
<
2C
c~(.~ - 2)
and IIZ(~)lloo _< 1/C1. It follows that
C CB('-,/) _< 119(~o)11ook~z ~ (1 + Ix - kl)'~ -< --C,
kEZ
Since [-ixS(x)]^(w) = S'(w) = fl'(w)~(w) + 13(w)qY(w), by Lemma 2.6, we have \ 1/2
-< II/~'11oo
I~(x
-
4-ll~[Ioo(n~ezl(x--n)~(x--n)]2)l/2 +[l~llooCIn~ez(l+lx-n[)-2~'+21
n)l 2
1/2
_< IlZ'llooC
(1 + Iz
-
nl) -2"/
< 2c~,/N5-~ C , / B ( 2 . ~ - 2) + - c,~(~ - 2) c,
Therefore,
~' Ix - nl'~lS(x - ~)1 -< 211Slloo + nEZ
_< 211Slloo +
(
ln-xl_>l
Ix - ~l~-ll(x - n)S(x - ~)1
Iz - ~1 ~-2
< 2cB(~)
I,~-xl>l ,/4(1- c~) t.[2C2VfB~ +
-
v
C1
1-2~
/ c~(,~-2)
I(x - n)S(x Cx/B(2" f -
- n)l 2
)lj
2)] = 2kIa~.
C1
It follows that
-< E
c,, I~n -- 2"tl ° 2 -'~k IS (2kt- n)l
nEZ
-< E 2-~kc* (Ix,, - hi" + 12h- nl '~) Is (2hnEZ
-< 2-~cf L° Z Is (2~t- n)l nCZ
+ 2-'~cf F. I~t - ,~1" is (2~t- n)l nEZ
<_ 2-~kCf (L~Mo,.7 + M~,.y).
n)l
Convergence of Sampling Series
901
REMARK. The approximation order can also be derived from results in [12] or [13]. Here we give the explicit error bound. We point out that both scaling functions ~N(2 ~ N < 20) of Daubechies' and B-splines defined by
~(w) = (sin(w/2)\/m+l
\
/
'
re>l,
meet the requirements of these theorems (see [8]).
REFERENCES 1. A. Aldroubi and M. Unser, Families of wavelet transforms in connection with Shannon's sampling theory and the Gabor transform, In Wavelets--A Tutorial in Theory and Applications, (Edited by C.K. Chui), pp. 509-528, Academic Press, Boston, MA, (1992). 2. A.J.E.M. Janssen, The Zak transform and sampling theorem for wavelet subspaces, IEEE Trans. Signal Proc. 41, 3360-3364 (1993). 3. Y. Liu and G. Walter, Irregular sampling in wavelet subspaces, Fourier Anal. Appl. 2, 181-189 (1995). 4. W. Sun and X. Zhou, Frames and sampling theorem, Science in China 41, 606-612 (1998). 5. W. Sun and X. Zhou, Sampling theorem for multiwavelet subspaces, Chinese Science Bulletin 44, 1283-1286 (1999). 6. X.G. Xia and Z. Zhang, On sampling theorem, wavelets, and wavelet transforms, IEEE Trans. Signal Proc. 41, 3524-2535 (1993). 7. G. Walter, A sampling theorem for wavelet subspaces, IEEE Trans. Inform. Theory 38, 881-884 (1992). 8. X. Zhou and W. Sun, On the sampling theorem for wavelet subspaces, Y. Fourier Anal. Appl. 5, 347-354 (1999). 9. A. Fischer, Multiresolution analysis and multivariate approximation of smooth signals in CB (Rd), J. Fourier Anal. Appl. 2, 161-180 (1995). 10. W. Rudin, Real and Complex Analysis, McGraw-Hill, (1987). 11. S.C. Malik, Mathematical Analysis, John Wiley &: Sons, New York, (1984). 12. P.L. Butzer, S. Ries and R.L. Stens, Approximation of continuous and discontinuous functions by generalized sampling series, J. Approx. Theory 50, 25-39 (1987). 13. A. Fischer, On wavelets and prewavelets with vanishing moments in higher dimensions, Y. Approx. Theory 90, 46-74 (1997).