AbstractA necessary and sufficient condition for the linear independence of integer translates of Box splines with rational directions is presented in terms of intrinsic properties of the defining matrices. We also give a necessary and sufficient condition for the space of linear dependence relations to be finite dimensional. A method to compute the approximation order of these Box spline spaces is obtained. All these conditions can be tested by finite steps of computations based on elementary properties of the matrices. The method of proofs is from linear diophantine equations

Zhou, Ding-Xuan

Elsevier - Publisher Connector 

 .JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 203, 270]277 1996
ARTICLE NO. 0379
Box Splines with Rational Directions and Linear
Diophantine Equations
Ding-Xuan Zhou*
Department of Mathematical Sciences, Uni¨ ersity of Alberta, Edmonton,
Canada T6G 2G1
Submitted by William F. Ames
Received January 24, 1995
A necessary and sufficient condition for the linear independence of integer
translates of Box splines with rational directions is presented in terms of intrinsic
properties of the defining matrices. We also give a necessary and sufficient
condition for the space of linear dependence relations to be finite dimensional. A
method to compute the approximation order of these Box spline spaces is obtained.
All these conditions can be tested by finite steps of computations based on
elementary properties of the matrices. The method of proofs is from linear
diophantine equations. Q 1996 Academic Press, Inc.
1. INTRODUCTION AND MAIN RESULTS
Let J be an s = n matrix of nonzero columns called directions. The
Box spline M associated with J is the compactly supported distributionJ
defined by
 : ` sM , c [ c J x dx , c g C R . 1.1 .  .  .HJ
nw .0, 1
Its Fourier]Laplace transform is given by
1 y eyi j T v
sÃM v s , v g C . 1.2 .  .J Tij vjgJ
*The author is supported by the Alexander von Humboldt Foundation and NSERC Can-
ada under Grants OGP 121336 and A7687. E-mail address: dxzhou@approx.math.
ualberta.ca.
270
0022-247Xr96 $18.00
Copyright Q 1996 by Academic Press, Inc.
All rights of reproduction in any form reserved.
BOX SPLINES WITH RATIONAL DIRECTIONS 271
The first purpose of this paper is to characterize the linear indepen-
dence of integer translates of Box splines with rational directions. Define
the set of linear dependence relations among the integer translates of MJ
as
K [ f g CZ s : f a M ?y a s 0 . 1.3 .  .  .J J 5
sagZ
Then we say that the integer translates of M are linearly independent ifJ
K contains only the zero sequence. In case J ; Z s=n, necessary andJ
sufficient conditions for K to be null or finite dimensional were given inJ
w x1, 3, 5, 8 ; see also the monograph of de Boor, Hollig, and Riemenschnei-È
w xder 2 .
Initiated by his joint investigation with Jetter on cardinal interpolation
s w x w xon submodules of Z in 6 , Riemenschneider posed in his survey 10 the
problem of characterizing the linear independence of integer translates of
a Box spline with rational directions in terms of elementary properties of
the defining matrix J g Q s=n. Since then some partial results concerning
w xthis problem have been obtained in 9, 12 . The main results in the first
part of this paper, Theorems 1 and 2, give an answer to this problem.
 .We first note that any s = n rational matrix can be written as 1rP J
with P g N and J g Z s=n. So in what follows we always take such a form
for the defining matrix. For an l = m integer matrix A we also think of it
as the multiset of its column vectors and denote aA, d as its cardinalityA
 .   4.   4.and the greatest common divisor g.c.d. of all min l, m = min l, m
minors of A, respectively. For P, s g N, we denote
E [ g s g , . . . , g : 0 F g F P y 1 for 1 F j F s 1.4 .  . 4P , s 1 s j
and for 0 F k s s g P jy1 F P s y 1 with 0 F g F P y 1 definejs1 j j
g k s g , . . . , g g E . .  .1 s P , s
The following concept of correlation sets plays an important role in the
statements and proofs of our main results.
DEFINITION. For J g Z s=n, we define the following sets:
&
 41-1 J [ X ; J : rank X s aX ) 0, d ) 1 ; 1.5 .  .X
&
aX
T TF J [ X , b : X g 1-1 J , b g Z , d / d ; 1.6 .  .  .  . 5X w X , b x
sTÄ xF J [ X , b : X , b g F J , b g X y1, 0 ; 1.7 .  .  .  .  . 4
DING-XUAN ZHOU272
P sy1 sCor J , P [ X , b : X , b g F J for 0 F j F P y 1 ; .  . 4 .  . 5j j j jjs0
1.8 .
& sP y1 sÄCor J , P [ X , b : X , b g F J for 0 F j F P y 1 . .  . 4 .  . 5j j j jjs0
1.9 .
&
 .  .We call Cor J, P and Cor J, P the correlation set and the fundamental
 4correlation set of J, P , respectively.
 .4P sy1  .  4For Y s X , b g Cor J, P , called a correlation of J, P , wej j js0
define its correlation matrix X and correlation vector b asY Y
T
sX s X , X , . . . , X , 1.10 .  .Y 0 1 P y1
TT T TT T s
s sb s yP b , . . . , b y g 0 X , . . . , g P y 1 X . 1.11 .  .  . .  .Y 0 P y1 0 P y1
With these notations and definitions we can now state our main results
on linear independence and linear dependence relations as follows.
THEOREM 1. Let J g Z s=n and P g N. Then the integer translates of&
 .M are linearly independent if and only if for any Y g Cor J, P ,1r P .J
w xrank X - rank X , b . 1.12 .  .Y Y Y
THEOREM 2. Let J g Z s=n and P g N. Then dim K - ` if and1r P .J&
 .  . w x  .only if for any Y g Cor J, P , rank X s rank X , b implies rank XY Y Y Y
s s.
&Ä .  .Let us mention that F J , hence Cor J, P , is a finite set. Therefore
the conditions in Theorems 1 and 2 can be tested by finite steps of
computations based on elementary properties of the defining matrix.
The second purpose of this paper is to give a method to compute the
approximation order of spaces of Box splines with rational directions. An
w xinteresting characterization was presented by Ron and Sivakumar in 13 ,
which states that in our case of rational directions, the approximation
 .order of the Box spline space S M is the number1r P .J
s  4min aK J : a g Z _ 0 , 1.13 4 .  .a , P
where
T  4K J [ j g J : j a g PZ _ 0 . 1.14 4 .  .a , P
 .Note that 1.13 concerns taking the minimum over an infinite set. We
give a method here to calculate this approximation order by finite steps of
elementary computations based on the defining matrix.
BOX SPLINES WITH RATIONAL DIRECTIONS 273
w xTo this end, we use the following notations from 2, 14 :
 41-1 J [ X ; J : rank X s aX ) 0 ; 1.15 .  .
Jc [ j g J : j Tg g PZ , g g E . 1.16 4  .g P , s
Then our main result on approximation order can be stated as follows.
THEOREM 3. Let J g Z s=n with rank J s s and P g N. Then the
 .approximation order of the Box spline space S M is gi¨ en by1r P .J
min min a J _ Y : Y ; J , rank Y - s , min a Jc _ span X : 4 .   .g
 4ggE _ 0P , s
X g 1-1 J , X ; Jc , d T s d T T . 1.17 .  .4g X w X , 1r P . X g x 5
The proofs of our main results depend mainly upon linear diophantine
equations which were first introduced to the investigation of multivariate
w x w xsplines by Dahmen and Micchelli 4 . In 7 Jia used a solvability condition
for linear diophantine equations to solve the problem of linear indepen-
w xdence of discrete Box splines. It is the nice paper of Jia 7 that leads the
author to solve the problems in this paper. For our purpose we need the
following solvability condition for linear diophantine equations which can
w xbe found, e.g., in 7, Theorem 3.2 .
LEMMA. Let A be an l = m integer matrix with full row rank and b g Z l.
Then the following system of linear diophantine equations
Ay s b 1.18 .
has an integer solution for y if and only if d s d .A w A, b x
2. CHARACTERIZATIONS AND PROOFS FOR LINEAR
INDEPENDENCE
Let J g Z s=n and P g N. The following sets are essential for the
proofs of the main results on linear independence,
&
s sN [ v g C : M v q 2pa s 0 for any a g Z ; 2.1 .  . 5J , P 1r P .J
ssÄN [ v g C : v g N , Re v g 0, 2p , 2.2.  . 4J , P J , P
 .T s  .There for v s v , . . . , v g C , we denote Re v s Re v , . . . , Re v .1 s 1 s
DING-XUAN ZHOU274
 .When P s 1, the set N can be characterized by F J and theJ , 1
Lemma as follows.
THEOREM 4. Let J g Z s=n. Then
 s T 4N s 2pv g C : X v s yb . 2.3 .DJ , 1
 .  .X , b gF J
 .Proof. From the Fourier]Laplace transform 1.2 we see that for
v g C s, 2pv g N if and only if for any a g Z s there is some j g JJ , 1 a
T  .  4 T T  .such that j v q a g Z _ 0 , i.e., j v g Z and j v q a / 0. Ifa a a
2pv g N , we choose a greatest linearly independent subset X of theJ , 1 v
 s4  T 4 sset j : a g Z ; j g J : j v g Z , then for any a g Z there existsa
T  .some j g X such that j v q a / 0, i.e., the following system of linearv
diophantine equations
X T y s yX Tvv v
has no integer solutions for y. Let X s X and b s yX Tv g ZaXv, byv v
the Lemma we have d T ) d T G 1, from which it follows that XX w X , b x&
s T .  .  .  4g 1-1 J . Hence X, b g F J and 2pv g 2pv g C : X v s yb .
s  .  .On the other hand, if v g C and there exists X, b g F J such that
T s T T X v s yb, then by the Lemma for any a g Z , X a / b, hence X v q
. s T a / 0, i.e., for any a g Z we have some j g X ; J such that j v q$
.  4  .a g Z _ 0 which implies M 2pv q 2pa s 0. Therefore, 2pv g N .J J , 1
The proof of Theorem 4 is complete.
ÄFrom Theorem 4, the following characterization for N is easy.J , 1
THEOREM 5. Let J g Z s=n. Then
ss TÄN s 2pv g C : X v s yb, Re v g 0, 1. 4DJ , 1
Ä .  .X , b gF J
 s T 4; 2pv g C : X v s yb . 2.4 .D
Ä .  .X , b gF J
 . sWhen P g N is general, we know from 1.2 that for v g C , 2pv g
N if and only if for any a s g q Pb g Z s s E q PZ s, i.e., anyJ , P P , s
g g E and b g Z s, there exists some j g J such thatP , s
 . T  . T  . .  41rP j v q a s j v q g rP q b g Z _ 0 , which is equivalent to
 . .the statement that for any g g E , 2p v q g rP g N . Thus weP , s J , 1
conclude the following characterization for N .J , P
BOX SPLINES WITH RATIONAL DIRECTIONS 275
THEOREM 6. Let J g Z s=n and P g N. Then
v q g
sN s 2pv g C : 2p g N for any g g E . 2.5 .J , P J , 1 P , s 5P
Combining Theorems 4 and 6, for v g C s, 2pv g N if and only ifJ , P
 .4P sy1  .there exists a correlation Y s X , b g Cor J, P such thatj j js0
T   .. .X v q g j rP s yb , i.e., X v s b . Therefore, we have.j j Y Y
THEOREM 7. Let J g Z s=n and P g N. Then
N s 2pv g C s : X v s b . 2.6 4  .DJ , P Y Y
 .YgCor J , P
w . s  . w . sWe notice that for v g 0, 1 and g g E , v q g rP g 0, 1 . ThenP , s
in the same way, from Theorems 5, 6, and 7, the characterization of NJ , P
follows.
THEOREM 8. Let J g Z s=n and P g N. Then
Ä sN ; 2pv g C : X v s b ; N . 2.7 4  .DJ , P Y Y J , P&
 .YgCor J , P
After presenting the above characterization and observing that N sJ , P
Ä s w xN q 2p Z , the proof of Theorem 1 follows from 11 , while the proof ofJ , P
w xTheorem 2 is obtained directly from 3 .
It is worth mentioning that unlike the case with integer directions, linear
independence of integer translates of Box splines with rational directions
is not equivalent to local linear independence as shown by the following
 4   . 4simple example: s s 1, J s 3 , P s 2. Then M ?y a : a g Z are1r P .J
 .  .  .linearly independent, while on 0, 1r2 , M ? y M ?q 1 s 0.1r P .J 1r P .J
3. APPROXIMATION ORDER OF BOX SPLINE SPACES
In this section we prove the main result on approximation order.
Proof of Theorem 3. Let g g E and b g Z s, thenP , s
K J s j g Jc : j T g q Pb / 0 .  . 4gqPb , P g
1
c c T Ts J _ j g J : j b s y j g . 3.1 .g g 5P
DING-XUAN ZHOU276
 . c T T TIf X g 1-1 J , X ; J , and, d s d when g / 0; rank Xg X w X , 1r P . X g x
- s when g s 0. Then, by the Lemma, there exists some b g Z s when
s  4g / 0; b g Z _ 0 when g s 0, such that
1
T TX b s y X g ,
P
 . c  4which implies K J ; J _ span X. Hence for g g E _ 0 ,gqPb , P g P , s
min aK J : b g Z s F min a Jc _ span X : X g 1-1 J , .  . 4   .gqPb , P g
X ; Jc , d T s d T T ,4g X w X , 1r P . X g x
while for g s 0,
s  4min aK J : b g Z _ 0 F min a J _ Y : Y ; J , rank Y - s . 4 .  . 4Pb , P
  . s  44Thus, min aK J : a g Z _ 0 is not more than the number given bya , P
 .1.17 .
On the other hand, for g g E and b g Z s with b / 0 when g s 0, weP , s
 c T  .take a greatest linearly independent subset X of j g J : j b q grPg
4  . cs 0 . Evidently, X g 1-1 J , X ; J .g
If g s 0, then rank X - s since b / 0 while X Tb s 0. Hence
J _ span X ; K J .Pb , P
and
s  4min a J _ Y : Y ; J , rank Y - s F min aK J : b g Z _ 0 . 4 .  . 4Pb , P
If g / 0, then b is an integer solution to the following system of linear
diophantine equations
1
T TX y s y X g ,
P
which implies by the Lemma d T s d T T . Moreover,X w X , 1r P . X xg
Jc _ span X ; K J . .g gqPb , P
Hence
min a Jc _ span X : X g 1-1 J , X ; Jc , d T .  .g g X
s d T T F min aK J : b g Z s . . 44w X , 1r P . X g x gqPb , P
Combining the above two cases, we know that the number given by
 .   . s  441.17 is not more than min aK J : a g Z _ 0 .a , P
The proof of Theorem 3 is complete.
BOX SPLINES WITH RATIONAL DIRECTIONS 277
REFERENCES
 .1. C. de Boor and K. Hollig, B-splines from parallelepipeds, J. Analyse Math. 42 1982]1983 ,È
99]115.
2. C. de Boor, K. Hollig, and S. D. Riemenschneider, ``Box Splines,'' Springer-Verlag,È
BerlinrNew York, 1993.
3. W. Dahmen, R. Q. Jia, and C. A. Micchelli, On linear dependence relations for integer
 .translates of compactly supported distributions, Math. Nachr. 151 1991 , 303]310.
4. W. Dahmen and C. A. Micchelli, The number of solutions to linear diophantine
 .equations and multivariate splines, Trans. Amer. Math. Soc. 308 1988 , 509]532.
5. W. Dahmen and C. A. Micchelli, Translates of multivariate splines, Linear Algebra Appl.
 .52 r 53 1983 , 217]234.
6. K. Jetter and S. D. Riemenschneider, Cardinal interpolation, submodules, and the
 .4-direction mesh, Constr. Approx. 3 1987 , 169]188.
7. R. Q. Jia, Multivariate discrete splines and linear diophantine equations, Trans. Amer.
 .Math. Soc. 340 1993 , 179]198.
 .8. R. Q. Jia, Linear independence of translates of a box spline, J. Approx. Theory 40 1984 ,
158]160.
9. R. Q. Jia and N. Sivakumar, On the linear independence of integer translates of a box
 .spline with rational directions. Linear Algebra Appl. 135 1990 , 19]31.
10. D. S. Riemenschneider, Multivariate cardinal interpolation, in ``Approximation Theory,
 .VI'' C. K. Chui, L. L. Schumaker, and J. D. Ward, Eds. , pp. 561]580, Academic Press,
New York, 1989.
11. A. Ron, A necessary and sufficient condition for the linear independence of the integer
 .translates of a compactly supported distribution, Constr. Approx. 5 1989 , 297]308.
12. A. Ron, Remarks on the linear independence of the integer translates of exponential box
 .splines, J. Approx. Theory 71 1992 , 61]66.
13. A. Ron and N. Sivakumar, The approximation order of Box spline spaces, Proc. Amer.
 .Math. Soc. 117 1993 , 473]482.
14. Ding-Xuan Zhou, Some characterizations for Box spline wavelets and linear diophantine
equations, manuscript.


Box Splines with Rational Directions and Linear Diophantine Equations 

file:///data/remote/core/dit/data/elsevier/pdf/4a0/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDAyMjI0N3g5NjkwMzc5MA==.pdf

Box Splines with Rational Directions and Linear Diophantine Equations

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Elsevier - Publisher Connector