Fourier analysis of 2-point Hermite interpolatory subdivision schemes

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Fourier analysis of 2-point Hermite interpolatory subdivision schemes
Fourier analysis of 2-point Hermite
interpolatory subdivision schemes
Serge Dubuc, Daniel Lemireyand Jean-Louis Merrienz
12/14/2000
Abstract
Two subdivision schemes with Hermite data on Z are studied. These
schemes use 2 or 7 parameters respectively depending on whether Hermite data
involve only rst derivatives or include second derivatives. For a large region
in the parameter space, the schemes are convergent in the space of Schwartz
distributions. The Fourier transform of any interpolating function can be computed through products of matrices of order 2 or 3. The Fourier transform
is related to a specic system of functional equations whose analytic solution
is unique except for a multiplicative constant. The main arguments for these
results come from Paley-Wiener-Schwartz theorem on the characterization of
the Fourier transforms of distributions with compact support and a theorem of
Artzrouni about convergent products of matrices.
Math Subject Classication: 40A20, 42A38, 46F12, 65D05, 65D10
Keywords and Phrases: Hermite interpolation, curve tting, subdivision, Fourier
transform, distributions, convergence of in
nite products, products of matrices.
1 Introduction
Hermite interpolatory subdivision schemes have been introduced by Merrien 10, 11]
and Dyn-Levin 5, 6]. These authors studied the C k -convergence of these schemes.
What we would like to do here is to compute the Fourier transform of these interpolants and to provide an additional tool for studying these schemes and for generating functions which are not necessarily of class C 1 or C 2.
S. Dubuc, D
epartement de mathematiques et de statistique, Universite de Montreal, C.P. 6128
Succursale Centre-ville, Montreal (Quebec), H3C 3J7 Canada, email: [email protected]
y D. Lemire, Ondelette Inc., 2059, rue Wurtele, Montr
eal (Quebec), H2K 2P8 Canada, email:
[email protected]
z J.-L. Merrien, INSA de Rennes, 20 av. des Buttes de Co
esmes, CS 14315, 35043 RENNES
CEDEX, France, email: [email protected]
1
As we will see, it is possible to de
ne the notion of convergence of Hermite interpolatory subdivision schemes in the space of Schwartz distributions. The study
of the convergence of these schemes on the space D(R ) can be done through the
Fourier transform provided that suciently general results about convergence of in
nite products of matrices are available. We came to this approach after previous
works of Deslauriers-Dubuc 3] and of Deslauriers-Dubois-Dubuc 4]. These authors
considered interpolatory subdivision schemes which were not of Hermite type however. In this situation, the Fourier transform allowed the study of the convergence of
some schemes with the help of products of trigonometrical polynomials (and not of
matrices).
We would like to point out two previous works that use harmonic analysis and
which have points in common with our article. A rst analysis has been written by
Herve 7] about wavelets in two or more dimensions. He did not study convergence
conditions, but he was involved in products of matrices. Another analysis, in subdivision schemes this time, is due to Kobbelt 8]. He proved a general convergence
criterion for arbitrary interpolatory schemes via discrete Fourier transform (instead
of the Fourier integral transform).
We now summarize the contents of the paper. In Section 2, we study a rst
Hermite interpolation scheme on R of a function and its rst derivative with data on
Z. This scheme called HS 21 depends on two parameters and . We rst recall the
conditions which insure C 1 -convergence on R . Then we evaluate the Fourier transform
of the interpolating functions and we give the rst properties. The Paley-WienerSchwartz theorem on the characterization of the Fourier transform of a distribution
with compact support and a convergence theorem of in
nite products of matrices
proposed by Artzrouni allow us to conclude to the convergence of the scheme in the
space of distributions D(R ) for a large region in the parameter space: 2 (;3 1)
with no restriction on . Moreover, the Fourier transforms of two basic distributions
in the Hermite subdivision scheme is the unique (except for a multiplicative constant)
analytic solution of a system of two functional equations.
Section 3 is devoted to the study of a second scheme where we interpolate not
only a function and its rst derivative but also its second derivative. This scheme
is called HS 22 and depends on 7 parameters: 1 and i i i i = 2 3. We prove
analogous properties to the previous scheme depending on the parameters introduced
in the algorithm. In particular, we get a satisfactory convergence in distribution of
the scheme when 2 and 3 2 (;3=2 1=2) 3 2 (;3=2 1=2), 1 + 22 = 2 + 23 = 1
and no restriction on 2 3. Most proofs are similar to the previous ones. However,
a few matrix tools must be improved.
0
0
2 The Hermite subdivision scheme HS 21
We recall Merrien's construction 10]. We suppose that the function f and its rst
derivative p are known on Z. Precisely, we have two sequences fyk yk gk Z and we
0
2
2
suppose that f (k) = yk p(k) = yk . We build f and p on Z=2n by induction. At step
n, set h = 1=2n and Dn = fx = jh j 2 Zg. If a = jh and b = (j + 1)h are two
consecutive points of Dn, we compute f and p at x = (a + b)=2 which is the midpoint
of a b] by the formulae:
0
f (x) = f (a) +2 f (b) + hp(b) ; p(a)]
p(x) = (1 ; ) f (b) ;h f (a) + p(a) +2 p(b)
9
>
=
>
(1)
Hence f and p are de
ned on Dn+1. The construction depends on two parameters and . S
When we iterate the process, we de
ne f and p on the set of dyadic numbers
D = Dn which is dense in R . For some values of the parameters the functions
f and p may be uniformly continuous on D so that they may be extended to R sometimes we have in addition: p = f . In these cases the algorithm is said to be C 1convergent.
n) ; p(i2 n )
p
((
i
+
1)2
i
If we de
ne Un = 2n(f ((i + 1)2 n) ; f (i2 n)) ; (p((i + 1)2 n) + p(i2 n))=2 ,
i+1 = U i and U 2i = U i where
it is easy
Un2+1
1 n
1 n
n+1
to1 prove"(1that:
" = " 82+1 1+; ) " = 1. Then the scheme is C 1-convergent if and only
4
2
if the generalized spectral radius of f1 1g is < 1. An equivalent condition is
that there exists a matrix norm k k such that: k"k < 1 " = 1. This result can
be proved by techniques which are described in 11]. Details on generalized spectral
radii can be found in Daubechies-Lagarias 2], but one knows that their evaluation is
dicult or even impossible.
Remark 1: If = ;1=8 = ;1=2, then f is the Hermite interpolating cubic
polynomial between two consecutive integers. When = ;1=8 = ;1, then f is
the quadratic interpolating spline with knots on Z=2.
1
1
0
;
;
;
;
;
;
;
;
2.1 Elementary properties of
HS
21
We describe some elementary properties of Hermite subdivision schemes. We will use
those properties later. The rst one is the linearity of the scheme.
Lemma 1 Let fyk yk y~k y~kgk Z be 4 sequences. Assume that both pairs (f p) and
(f~ p~) are built by the subdivision scheme from: f (k) = yk p(k) = yk f~(k) = y~k p~(k) = y~k then the pair of functions (f + f~ p + p~) are obtained by the subdivision
scheme from the sequences: fyk + y~k yk + y~k gk Z.
Similarly, if c 2 R , the pair of functions (cf cp) is obtained by the scheme from
the sequences: fcyk cyk gk Z.
Then we have a second lemma about translation and scale change.
0
0
2
0
0
0
0
2
0
2
3
Lemma 2 Let fyk yk gk Z be two sequences from which we build the pair of functions
0
2
(f p) by the subdivision scheme with f (k) = yk p(k) = yk . If c 2 Z, then the
pair of functions (f (x + c) p(x + c)) is obtained by the scheme from the sequences:
fyk+c yk+cgk Z.
The pair of functions (f (x=2) p(x=2)=2) is built from the sequences:
ff (k=2) p(k=2)=2gk Z.
There are two basic solutions of our recursive system (1): the rst one is the pair
(f0 p0) which is solution of (1) with data f0(k) = k0 p0 (k) = 0 k 2 Z and the second
one is the pair (f1 p1) which is solution of (1) with data f1 (k) = 0 p1(k) = k0 k 2 Z.
These two pairs are important because with linear combinations of their tranlates we
can get all solutions (f p) of (1). For any dyadic number x:
0
0
2
2
f (x) =
9
>
f (k)f0 (x ; k) + p(k)f1(x ; k)] >
=
k=
X
>
f (k)p0 (x ; k) + p(k)p1(x ; k)] >
X
1
(2)
;1
p(x) =
1
k=
;1
Notice that both sums are nite as the supports of fi pi i = 0 1 lie in the set ;1 1].
Now, using relation (1) which is applied to the pair of functions
(f (x) = f0 (x=2) p(x) = p0 (x=2)=2) and then to the pair (f1(x=2) p1(x=2)=2), after
evaluations of the functions f0 p0 f1 p1 at the half-integers, we obtain a system of
functional equations for f0 p0 f1 p1 .
f0(x=2)
p0 (x=2)=2
f1(x=2)
p1 (x=2)=2
9
= 21 f0 (x ; 1) + f0(x) + 21 f0 (x + 1) ; 1 2 f1(x ; 1) + 1 2 f1 (x + 1) >
= 21 p0 (x ; 1) + p0(x) + 12 p0(x + 1) ; 1 2 p1 (x ; 1) + 1 2 p1(x + 1) =
= ;f0 (x ; 1) + f0(x + 1) + 4 f1 (x ; 1) + 12 f1 (x) + 4 f1 (x + 1) >
= ;p0 (x ; 1) + p0(x + 1) + 4 p1 (x ; 1) + 21 p1(x) + 4 p1 (x + 1): (3)
2.2 Fourier transform of
HS
;
;
;
;
21
In this section, we suppose that the system (3) of functional equations is valid not
only whenever x is a dyadic number, but also whenever x is an arbitrary real number. Moreover we assume that f0 p0 f1 p1 2 L1 (R ). Now, we compute the Fourier
transform f^ of a function f by
Z
^
f (
) =
f (x)e ixdx:
1
;
;1
Using this Fourier operator on each equation of the system (3), we get:
p^ (
=2) f^ (
=2) p^ (
) f^ (
) 0
0
0
p^ (
) = 2A(
=2) p^0(
=2) = A(
=2) ^
^
f1 (
=2)
f1 (
)
1
1
4
(4)
1 + 1 cos i 1 sin 2
A(
) = 2 2
where
;
i sin 14 + 4 cos We have two vector equations in (4). To study them, we now look at the product
of matrices:
Pn(
) = A(
=2)A(
=4) : : :A(
=2n):
(5)
Precisely, we look for conditions on the parameters to get convergence of the
sequence of matrices Pn(
). This convergence should happen for every real or complex
value of . The study of this sequence for complex values of is motivated by a
generalization of Paley-Wiener theorem as proposed by Schwartz 12].
Theorem 3 (Schwartz) Let F be a continuous function on the real axis which is
the Fourier transform of a tempered distribution T . The support of T is contained in
;C C ] if and only if F may be extended on the complex plane to an entire function
of exponential type C .
We recall that an entire function F (z) is of exponential type C if
lim sup log jjFzj(z)j C:
z
j j!1
To study the convergence of the matrix products Pn(
), we will need a lemma to
bound the moduli of the elements of Pn(
). We will also need a convergence criterion
which has been found by Artzrouni 1].
Lemma 4 Let jj jj be a matrix norm on the space of complex matrices of order d,
and let An be a sequence in this space such that jjAPnnjj 1 + "n, with "n 0.
Set P0 = I Pn = Pn 1An n = 1 2 : : :.PThen
jjPnk e k=1 "k and the modulus of
n "
each component of Pn is bounded by Ce k=1 k where C is a constant which depends
only on the matrix norm jj jj.
C dd ,
;
Proof: Since we notice that 1 + "n e"n , we get the rst result by induction.
Then, for A = (aij ) 2 C d d , we choose a new vector norm N (A) = maxfjaij jg.
We know that there exists a constant CN such that for all A 2 C d d N (A) CN kAk.
Then we get the second upper bound. 2
Theorem 5 (Artzrouni) Let jjjj be a matrix norm on C d d . Let Mn be a sequence
in C d d such that for all n 2 N kMn k = 1 and for all m 2 N the sequence of
matrices Mm Mm+1 : : : Mn , n m converges. If Nn is a sequence in C d d such that
X
jjNn ; Mn jj < 1, then the sequence of matrices N1 N2 : : : Nn converges.
1
n=1
We are now ready for the main result.
5
Theorem 6 If ;5 < 3, then for all complex numbers the sequence of matrices
Pn(
) dened in (5) converges and the convergence is uniform on any bounded domain.
As functions of , the four components of the limit matrix P (
) are entire functions
of exponential type 1.
Proof: Let us start by proving that the moduli of all components of Pn(z) are
uniformly bounded whenever z 2 , a bounded domain in C .
If A = (aij ) is a matrix in C 2 2 , we recall that jjAjj = maxifjai1j + jai2 jg. Then,
we use Lemma 4 with the matrices An = A(z=2n) and the sequence
"n = max0 sup (jjA(z=2n)jj ; 1)]:
1
z
1
j j2
If the components
1 of0thematrix A(z) are aij (z), these functions are analytic at z = 0
with A(0) = 0 1+ therefore "n = O(1=2n) (remembering the hypothesis ;5 <
4
3). Lemma 4 shows that the moduli of all components of the matrices Pn(z) are
uniformly bounded whenever z 2 .
Now, we prove that for all z 2 C , the sequence of matrices Pn(z) converges. Let
Mn be the sequence of constant matrices M = Mn = A(0) and de
ne the sequence
Nn = A(z=2n). We know that jjNn ; Mn jj = O(1=2n). Moreover kM k = 1 and
with the hypothesis on , the sequence M k k 0 converges. All the hypotheses of
Theorem 5 are satis
ed, therefore the sequence Pn(z) converges.
As the sequence Pn(z) converges to a matrix P (z) and as the moduli of the components of the matrices Pn(z) are uniformly bounded whenever z 2 , the Lebesgue
dominated convergence theorem and the Cauchy formula give us the proof that all
the components of the matrix P (z) are analytic in z and that the convergence of the
sequence Pn(z) to P (z) is uniform on .
Finally, let us verify that each element of the matrix P (z) is an entire function of
exponential type. Firstly, there exists a real positive number C which depends on the
parameters such that for all z 2 C , jjA(z)jj Ce z . Secondly, we know that
there exists a real positive number M (depending on the parameters again) such that
for all z 2 C jzj 1 jjP (z)jj M .
Let z be a complex number such that 2n jzj 2n+1.
As P (z) = A(z=2)A(z=4) : : : A(z=2n )P (z=2n+1), we obtain the bound
Yn
Yn z=2k
n
+1
k
jjP (z )jj jjP (z=2 jj
jjA(z=2 )jj M Ce ] MC n e z :
1
1
j j
1
1
j j
1
1
1
k=1
k=1
Hence lim sup z log jjPjz(jz)jj 1 and the functions composing the matrix
P (z) are entire functions of exponential type 1. 2
The Paley-Wiener theorem implies the following corollary:
Corollary 7 Let us assume that ;5 < 3. Then each function composing the
limit matrix P (z) = lim Pn (z ) is the Fourier transform of a distribution whose support
lies in the interval ;1 1].
1
j j!1
6
2.3 Schwartz distributions associated with the scheme
We will link the computation of Fourier transforms of the previous subsection with the
limit matrix P (
). This link will come from four sequences of Schwartz distributions.
We set
X
Ti(n) = 21n fi(m=2n)m=2n i = 0 1
m
X
1
(n)
Ui = 2n pi(m=2n)m=2n i = 0 1
m
where h is the Dirac distribution at point h de
ned by h() = (h).
Notice that these sums are nite and that the distributions are compactly supported, the supports of fi pi i = 0 1 being in ;1 1].
We evaluate the Fourier transform of these four distributions:
T^i(n)(
) = Ti(n)(e ix) U^i(n) (
) = Ui(n) (e ix) i = 0 1:
;
;
Hence using the equalities (3), we verify that two simple inductions link both Fourier
transforms through the matrix A(
). Indeed,
X
T^0(n+1) (
) = 2n1+1 f0 (m=2n+1)e im=2n+1 :
m
;
In this last equation, we substitute to f0 (m=2n+1) by means of the rst equation of
system (3) with x = m=2n to obtain a rst recursion:
T^0(n+1) (
) = 21 + 21 cos( 2
)]T^0n(
=2) + i 1 ;2 sin( 2
)T^1n(
=2)
Similarly, we can evaluate T^1(n+1) (
) using the third equation of system (3) again
at x = m=2n and we get a second recursion:
T^1(n+1) (
) = i sin( 2
)T^0n(
=2) + 41 + 4 cos( 2
)]T^1n(
=2)
Both recursions may be linked in a single vector recursion through the matrix
A(
=2):
^(n+1) !
!
T0 (
) = A(
=2) T^0(n) (
=2)
(6)
T^1(n+1) (
)
T^1(n) (
=2)
Similarly, with the second and fourth equations of system (3), we obtain a second
vector recursion:
!
^ (n+1) !
U0 (
) = 2A(
=2) U^0(n) (
=2)
(7)
U^1(n) (
=2)
U^1(n+1) (
)
7
Since
T^0(0) (
)
T^1(0) (
)
! 1
=
0
and
The vector sequence
and
T^0(n)(
)
T^1(n)(
)
!
T^0(n) (
)
T^1(n) (
)
U^0(n) (
)
U^1(n) (
)
!
U^0(0) (
)
U^1(0) (
)
!
! 0
=
1 , we get
1
(8)
0
(9)
= Pn(
) 0
= 2nP
n(
)
1 :
converges to the rst column of the matrix P (
).
Theorem 8 If ;5 < 3, then each sequence of distributions Ti(n) converges to a
distribution Ti , i = 0 1. The vector (T^0 (
) T^1(
))T is the rst column of the matrix
P (
).
Proof: LetR a C -function with bounded support, we have the inversion formula
(x) =
1
2
^(
)e
1
;
ix d
.
So for i = 0 1,
X
Ti(n)() = 21n fi(m=2n)(m=2n)
mZ
1
= 2 T^i(n) (;
)^(
)d
:
R
As n tends to 1, we get limn Ti(n) () = 21 p^i+11 (;
)^(
)d
= Ti().
((p11(
) p21(
))T is the rst column of P (
).) Hence Ti(n) converges to Ti. From that,
it follows that the Fourier transform T^i(
) is limn T^i(n) (
) = pi+11(
). 2
We are ready to prove that the subdivision scheme is always convergent in the
space of distributions D(R) whenever ;5 < 3. In the following, we use Schwartz
notation for the translation operator h where h is a real number. If is a function
in C0 and if T is a distribution, then h (x) = (x ; h) and hT () = T ( h).
!1
!1
0
1
;
Theorem 9 Let us assume that ;5 < 3. If we build the pair (f p) by the
subdivision scheme (1) from the data fyk yk gk Z, then the sequence of distributions
X
Fn = 21n
f (m=2n)m=2n converges to the distribution
m=
0
2
1
;1
F=
X
1
k=
yk k T0 + yk k T1 ]:
0
;1
8
Proof: LetPN 2ben a function in C with support in ;N N ], then
1
Fn() = 21n
m= N 2n f (m=2
;
2nFn()
=
n )(m=2n).
N2
X
m= N 2n k= Nn
N +1
N2
X
1
;
yk f0 (m=2n) + yk f1(m=2n)](m=2n + k)
0
k= N 1 m= N 2n
N +1
2n
yk T0(n) + yk T1(n)](k ):
k= N 1
;
=
;
yk f0 (m=2n ; k) + yk f1(m=2n ; k)](m=2n)
0
X
;
=
X
N +1
n
We use relation (2) to get:
X
;
;
;
0
;
As n tends to in
nity, the limit of the sequence Fn() is:
X
1
k=
2
yk k T0 + yk k T1 ]():
0
;1
Theorem 10 If ;3 < < 1, then both sequences of distributions U0(n) U1(n) converge
respectively to the distributions T0 T1 which are the derivatives of the distributions
T0 T1 .
0
0
Proof: Using both relations (5) and (7), we have:
U^0(n) (
)
U^1(n) (
)
U^0(n) (
)
U^1(n) (
)
!
!
U^0(n) (
)
U^1(n) (
)
U^0(n) (
)
U^1(n) (
)
= i 1 ;2 !
!
= 2nP
n
= 2nPn
0
1
i 1 sin(
=2n) 2
1 (
)
;
1
4
;
T^0(n 1) (
)
T^1(n 1) (
)
;
!
;
n 1
X
= i 1 ;2 ( 1 +2 )k
k=0
;
1
;
(
)A(
=2n)
+ 4 cos(
=2n)
+ 1 +2 T^0(n
T^1(n
k) (
)
k) (
)
1
; ;
1
; ;
U^0(n 1) (
)
U^1(n 1) (
)
;
;
!
!
+ O(1=2n)
+ O(nmax( 21 j1 +2 j )]n)
If j1+ j < 2 which is the hypothesis, then the right member of the last vector equation
tends to the column vector whose components are i
T^0(
) i
T^1(
). They are the two
respective limits of the sequences U^0(n) (
) U^0(n)(
). Using the inverse Fourier transform
on each sequence, it is clear that U0(n) U1(n) converge respectively to the distributions
T0 T1 . 2
0
0
9
Theorem 11 Let us assume that ;3 < < 1. If we build the pair (f p) by the
subdivision scheme (1) from the data fyk yk gk Z, then the sequence of distributions
X
p(m=2n)m=2n converges to the distribution
Gn = 21n
m=
0
2
1
;1
G=
X
1
k=
yk k T0 + yk k T1 ]:
0
0
0
;1
Proof: The proof is similar to that of Theorem 9. 2
We conclude this subsection by a comment of the referee. It is amazing that there
is no restriction on the parameter in Theorems 8-11. We think that this is typical
of convergence in distribution and that for some values of , C 1-convergence will not
happen.
2.4 A characterization of Fourier transforms ^0 ^1
T T
In this subsection, we characterize the pair of functions T^0(
) T^1(
) without using
in
nite iteration of the subdivision scheme for two systems of initial data.
Theorem 12 If ;5 < < 3, then (T^0(
) T^1(
) is the unique pair of analytic functions (0 (
) 1 (
)) which is a solution of the vector equation
(2
) (
) 0
0
(10)
1(2
) = A(
) 1(
) :
and which satises 0(0) = 1.
Proof: Under the hypothesis ;5 < 3, we know that the sequence of matrices
Pn(
) converges. The limit P satis
es the equation P (2
) = P (
)A(
) as it can be
seen from Equation (5). By Theorem 8, (T^0 (
) T^1(
))T is the rst column of the
matrix P (
), it follows that 0(
) = T^0(
) 1(
) = T^1 (
) provide a solution to the
system of equations (10). Moreover we have (8n 2 N )T^0(n) (0) = 1, hence 0(0) = 1.
Let (0 1) be a pair of analytic functions solution of (10) and such that 0(0) = 1.
By setting = 0 in equation (10), we have that 1(0) = 0 (because 6= 3). We expand the functions 0 (
) 1(
) and the components aij (
) of the matrix A(
) in power
series of P
:
P
P
0(
) = n=0 (0n) n, 1(
) = n=0 (1n)
n, aij (
) = n=0 a(ijn) n.
Then we substitute these expansions in system (10). We develop the products and
we reorganize the result in terms of powers of . We use the hypothesis
0(0) = 1 1(0) = 0 to compute the Maclaurin series of the functions 0 1 recursively, for n = 1 2 3 : : ::
1
1
(n)
i;1
2 X
n
X
=(
j =1 k=1
1
a(ijk) (jn 1k))=(2n ; aii (0)) i = 1 2
;
;
10
(11)
Notice that a11 (0) = 1 a22(0) = (1 + )=4. Therefore 0 and 1 are uniquely determined and are respectively T^0 and T^1 . 2
Remark 2: In this last theorem, the analyticity hypothesis on the functions 0 1 is
critical. If one uses the continuity hypothesis combined with the values of 0 (0) 1(0)
one does not have a unique solution of system (10). Kuczma proved in a general way
in his book (9], p. 245, Theorem 12.1), that the set of solutions of a system of type
(10) depends on an arbitrary function.
3 The Hermite subdivision scheme HS 22
We will see how we can generalize the previous results to the scheme HS 22 which is
based on bisections that use not only the values of a function and of its rst derivative
but also the values of its second derivative. We recall Merrien's construction 11]. If
the values of a function f , and of its rst and second derivative p s are known on Z,
we build the functions f p and s by a recursion on n as in the previous section. If
a = j=2n and b = (j + 1)=2n are two successive points of Dn, we compute f p and s
at the midpoint x = (a + b)=2 by the formulae:
9
f (x) = 1=2f (b) + f (a)] + 2 hp(b) ; p(a)] + 3 h2 s(b) + s(a)] >
>
=
f (b) ; f (a)
p(x) = 1
+
2 p(a) + p(b)] + 3 hs(b) ; s(a)]
(12)
h
>
>
p(b) ; p(a)
+ s(b) + s(a)]
s(x) = 2
h
3
The algorithm is said to be C 2-convergent if, for any data fyk yk yk gk Z, the three
functions f p s can be extended from D to R with f 2 C 2(R ), p = f and s = f .
The second author found a necessary condition to get C 2-convergence (Proposition 3
in 11]):
82 + 163 = ;1 1 + 22 = 1 2 + 23 = 1:
We will nd again the last two conditions later in the convergence theorems in the
space of distributions. If these conditions are satis
ed, then the algorithm depends
only on 4 parameters: 2 1 3 and 2.
As for HS 21, with this necessary hypothesis, a necessary and sucient condition
to get C 2-convergence is ^() < 1 where = f1 1g and
0
00
2
0
00
1
0
1
2
" = @ "( 14 + 23 ; 61 )
; 81 + 3 ; 121
;
"2
0
1 ; 22
"21
"( 21 + 42 + 122 ) 2 ; 1
1
A " 1:
An equivalent condition is that there exists a matrix norm kk such that: k"k < 1 " = 1.
The proof of this result is given in 11].
Remark 3: For some values of the parameters, the function f on each interval
k k + 1] k 2 Z is a usual polynomial or a piecewise polynomial. For example:
11
1. 2 = ;5=32 3 = 1=64 1 = 15=8 2 = ;7=16 3 = 1=32 2 = 3=2
3 = ;1=4, then f is the Hermite interpolating quintic polynomial on each
interval k k + 1] k 2 Z,
2. 2 = ;23=144 3 = 5=288 1 = 9=4 2 = ;5=8 3 = 1=16 2 = 3=2
3 = ;1=4, then f is the cubic spline with knots at Z=3,
3. 2 = ;5=32 3 = 1=64 1 = 2 2 = ;1=2 3 = 1=24 2 = 3=2 3 = ;1=4,
then f is the quartic spline with knots at Z=2.
3.1 Elementary properties of
HS
22
Of course, we have lemmas equivalent to Lemma 1 and 2 on linearity, translation and
scale change for the scheme HS 22 they are not written again.
As for HS 21, we introduce three basic solutions for the recursive system (12):
the triples fi pi si i = 0 1 2 which are solutions of (12) with data,
f0(k) = k0 p0(k) = 0 s0(k) = 0
8k 2 Z f1 (k) = 0 p1 (k) = k0 s1 (k) = 0
f2(k) = 0 p2(k) = 0 s2(k) = k0:
Then, for any construction produced by (12), we have for any dyadic number x:
f (x) =
9
f (k)f0(x ; k) + p(k)f1 (x ; k) + s(k)f2 (x ; k)] >
>
>
k=
>
>
=
X
f (k)p0(x ; k) + p(k)p1(x ; k) + s(k)p2(x ; k)]
>
k=
>
>
X
>
f (k)s0(x ; k) + p(k)s1(x ; k) + s(k)s2(x ; k)] >
X
1
;1
p(x) =
1
(13)
;1
s(x) =
1
k=
;1
From these formulae we can deduce a system of functional equations:
9
f0 (x=2) = f0(x ; 1)=2 + f0(x) + f0(x + 1)=2 + 21 ;f1 (x ; 1) + f1 (x + 1)] >
f1 (x=2) = 2;f0 (x ; 1) + f0 (x + 1)] + 22 f1 (x ; 1) + 12 f1(x) + 22 f1 (x + 1) >
=
2
+ 4 ;f2 (x ; 1) + f2 (x + 1)]
>
>
f2 (x=2) = 3f0 (x ; 1) + f0(x + 1)] + 23 ;f1 (x ; 1) + f1(x + 1)]
1
3
3
+ 4 f2 (x ; 1) + 4 f2(x) + 4 f2(x + 1)]
(14)
and similar equations in terms of pi and si (cf (3)).
12
3.2 Fourier transform of
HS
22
When the Fourier operator is applied to each equation of the system (14) and to
similar equations for pi si, we obtain
0 f^ (
) 1
0 f^ (
=2) 1 0 p^ (
) 1
0 p^ (
=2) 1
0
0
0
@ f^1(
) A = A(
=2) @ f^1(
=2) A @ p^1(
) A = 2A(
=2) @ p^01(
=2) A p^2(
)
p^2 (
=2)
f^2 (
)
f^2 (
=2)
0 s^ (
) 1
0 s^ (
=2) 1
0
@ s^1(
) A = 4A(
=2) @ s^01(
=2) A where
s^2(
)
(15)
s^2 (
=2)
1
0 1 + 1 cos i 1 sin 0
2
2
2
A(
) = @ i2 sin 14 + 22 cos i 42 sin A
1
3 cos i 23 sin 8 + 43 cos Therefore, again, we will have to study the sequence of matrix products:
Pn(
) = A(
=2)A(
=4) : : :A(
=2n):
(16)
Using Lemma 4 and Theorem 5, we prove a similar theorem to Theorem 6 and its
corollary. But before that, we must change the matrix norm.
Let = (1 2 : : : d ) be an element of R d with positive i. We de
ne a new
norm k k on C d by kX k = maxi=1d(i jxij) P
X 2 C d. It is easy to prove that the
associated matrix norm satis
es kAk = maxi j i =j jaij j whenever A = (aij ) lies
in C d d . Obviously if 1 = : : : = d = 1, we obtain the usual norm k k .
Theorem 13 If ;5=2 < 2 3=2 and if ;9=2 < 3 < 7=2, then for any complex
number , the sequence of matrices Pn(
) dened in (16) converges and the conver
1
gence is uniform on any boounded domain. The nine components of the limit matrix
P (
) are entire functions of exponential type 1.
Proof: We proceed as in Theorem 6 except that on C 3 , we use the norm kX k
with = (1 1 3) where 3 is chosen small enough to get 3 j3j + j1=8 + 3=4j 1.
Then it is easy to prove0that kA(z=2n)k 1 1
+ O(1=2n) in the disk jzj R.
1
0
0
1
2
@
Notice that A(0) = 0 4 + 2
0 A so that kA(0)k = 1 with the above
3 0 18 + 43
condition on 3 .
By induction, we prove that
0
1
1
0
0
A n 1
0
( 41 + 22 )n
0
A(0)]n = @ 1
1
n
n
3
3
3
0
(8 + 4 )
1 + 3 1 ( 8 + 4 ) ; 1]
8 4
;
13
0
therefore, using the hypothesis, this sequence converges to the matrix @
0 01
0 0 A.
3
; 1 + 3 1 0 0
8 4
We can complete the proof as in the previous section because the hypotheses of Theorem 5 are satis
ed. 2
Corollary 14 If ;5=2 < 2 3=2 and if ;9=2 < 3 < 7=2, then each component
of the limit matrix P (z ) = lim Pn(z ) is the Fourier transform of a distribution with
support in the interval ;1 1].
1
0
;
3.3 Schwartz distributions associated to
HS
We introduce 9 sequences of distributions in Schwartz sense:
Ti(n) =
Ui(n) =
Vi(n) =
1
2n
1
2n
1
2n
22
P f (m=2n) i = 0 1 2
Pm pi(m=2n)m=2 i = 0 1 2
Pm si(m=2n)m=2 i = 0 1 2:
n
n
m i
m=2n
When we compute the Fourier transform of these distributions, we obtain:
T^i(n) (
) = Ti(n) (e ix) U^i(n) (
) = Ui(n) (e ix) V^i(n) (
) = Vi(n) (e ix):
;
;
;
Then using the equations (14), we may verify that three recursions link these Fourier
transform through the matrix A(
):
0 ^(n+1) 1
0 ^(n)
1 0 ^ (n+1) 1
0 ^ (n)
1
T0 (
)
T0 (
=2)
U0 (
)
U0 (
=2)
[email protected] T^1(n+1)(
) CA = A(
=2) [email protected] T^1(n)(
=2) CA [email protected] U^1(n+1)(
) CA = 2A(
=2) [email protected] U^1(n)(
=2) C
A
(n+1)
(n)
(n+1)
(n)
T^2
T^2 (
=2)
(
)
U^2
(
)
0 ^ (n+1) 1
0 ^ (n)
1
V0 (
)
V0 (
=2)
[email protected] V^1(n+1)(
) CA = 4A(
=2) [email protected] V^1(n)(
=2) CA :
(n+1)
(n)
V^2
U^2 (
=2)
(17)
V^2 (
=2)
(
)
0 ^(0) 1 0 1 0 ^ (0) 1 0 1 0 ^ (0) 1 0 1
T0 ( )
1 B U0 ( ) C
0 B V0 ( ) C
0
B
C
(0)
(0)
(0)
@
A
@
A
@
^
^
^
As @ T1 () A = 0 @ U1 () A = 1 @ V1 () A = 0 A , we deT^2
(0)
duce that
and
( )
0
^2
U
(0)
0
( )
V^2
0 ^(n) 1
011
T0 (
)
B
@ T^1((nn))(
) CA = Pn(
) @ 0 A
T^2 (
)
0
14
(0)
( )
1
0 ^ (n) 1
0 0 1 0 V^ (n)(
) 1
001
U0 (
)
0
[email protected] U^1(n) (
) CA = 2nPn(
) @ 1 A [email protected] V^1(n)(
) CA = 4nPn(
) @ 0 A :
(n)
(n)
(18)
^U2 (
)
^V2 (
)
0
1
Like Theorems 8 and 9 for HS 21, we have the two following theorems whose
proofs are similar.
Theorem 15 If ;5=2 < 2 3=2 and if ;9=2 < 3 < 7=2, then the three sequences
of distributions T0(n) T1(n) T2(n) converge respectively to the distributions T0 T1 T2 . The
vector (T^0 (
) T^1 (
) T^2(
))T is the rst column of the matrix P (
).
Theorem 16 We suppose that ;5=2 < 2 3=2 et ;9=2 < 3 < 7=2. If we build the
triple (f p s) by the subdivision scheme (12) from the data fyk yk yk gk Z, then the
1 X f (m=2n) n converges to the distribution
sequence of distributions Fn = n
m=2
2 m=
0
00
2
1
;1
F=
X
1
k=
yk k T0 + yk k T1 + yk k T2]:
0
00
;1
To study the convergence of the sequences U^i(n) and V^i(n) i = 0 1 2, we need a
new lemma which gives the convergence of an in
nite product of matrices.
01 0
Lemma 17 Let M = @ a b
0
0
0 c d
n
lim
+
!
1
1
A2C 3
with jbj < 1 jdj < 1 then
3
0
Mn = @
1
a
ac
b)(1 d)
1;b
(1;
;
0 0
0 0
0 0
1
A
Proof: By induction, we obtain that for all n 2 and b 6= d
0
Mn = @
1n
0
0
1 b
n
a1 b
b
0
ac ( 1 bn ; 1 dn ) c bn dn dn
b d 1 b
1 d
b d
;
;
;
;
and for b = d,
0
Mn = B
@
It is now easy to conclude. 2
;
;
;
1n
a 11 bb
;
ac
(n;1)bn ;nbn;1 +1
(1;b)2
;
15
1
A
;
;
0
bn
cnbn
1
;
0
0
bn
1
CA
Theorem 18 If ;3=2 < 2 < 1=2 and ;3=2 < 3 < 1=2 then both sequences Ui(n)
and Vi(n) i = 0 1 2 converge respectively to Ui and Vi.
If we add 1 + 22 = 1, then Ui = Ti i = 0 1 2
Moreover, if we add 2 + 23 = 1, then Vi = Ti i = 0 1 2.
0
00
0 ^(n) 1
0 ^ (n) 1
0 ^ (n) 1
T0
U0
V0
B
B
B
(n) C
(n) C
^
^
Proof: Let us set tn = @ T1 A un = @ U1 A vn = @ V^1(n) C
A. With (17), we
(n)
(n)
(n)
T^2
get
tn =
U^2
V^2
011
0 1=2 + cos(
=2n)=2 1
Pn 1A(
=2n) @ 0 A = Pn 1 @ i2 sin(
=2n) A
n
;
;
0
3 cos(
=2 )
1
n
= 1=2 + cos(
=2 )=2]tn 1 + 2n;1 i2 sin(
=2n)un 1 + 4n1;1 3 cos(
=2n)vn 1:
;
;
;
Similarly, we obtain:
un = i 21 2n sin(
=2n)tn 1 + 1=2 + 2 cos(
=2n)]un 1 + i 23 2n sin(
=2n) 4n1 1 vn 1
vn = i 22 2n sin(
=2n)un 1 + 1=2 + 3 cos(
=2n)]vn 1:
0t 1
0t 1
n
n 1
These equations can be written in a vector form: @ un A = Nn @ un 1 A where
vn
vn 1
;
;
;
;
;
;
;
;
;
0 1=2 + cos(
=2n)=2 1 i sin(
=2n) 1 i cos(
=2n) 1
2
3
2 ;1
4 ;1
Nn = @ i 21 2n sin(
=2n) 1=2 + 2 cos(
=2n) i3 2 1;1 sin(
=2n) A
2 n
n
n
n
0
0 1
0
1
Setting M = @ i 2 1=2 + 2
2
n
i 2 2 sin(
=2 )
1
n
1=2 + 3 cos(
=2 )
0
0 A and using the previous lemma and the
0
i 2 1=2 + 3
hypothesis on 2 and 3, we get that the sequence M n converges.
We choose a matrix norm k k = (1 2 3) with 2 and 3 such that:
2 j 21 j + j1=2 + 2j < 1 then 3 j 22 j + j1=2 + 3j < 1.
2
Now kM k = 1 and kM ; Nnk = O(1=2n). Using again Theorem 5 we conclude that the sequence NnNn 1 : : : N1 converges, so that the sequence (tn un vn)T
converges to (t u v)T .
When we reach the limit, then u = i 21 t +(1=2+ 2)u and v = i 22 u +(1=2+ 3)v.
This can be written u = i 1 212 t and v = i 1 223 u. By the inverse Fourier transform,
we have the convergence result for Ui(n) and Vi(n) i = 0 1 2.
;
;
;
16
With the additional hypothesis 1 + 22 = 1, we have u = i
t,
therefore Ui = Ti i = 0 1 2.
Finally, if moreover 2 + 23 = 1, then v = i
u so that Vi = Ti i = 0 1 2. 2
Now we have a last theorem on the convergence of derivatives of distributions as
we had for HS 21.
0
00
Theorem 19 We suppose that ;3=2 < 2 < 1=2 ;3=2 < 3 < 1=2 and 1 + 22 =
1 2 + 23 = 1. If we build the triple (f p s) by the subdivision scheme (12) from the
1 X p(m=2n) n
data fyk yk yk gk Z, then the sequence of distributions Gn = n
m=2
2 m=
1
0
00
2
converges to the distribution G =
X
1
k=
;1
yk k T0 + yk k T1 + yk k T2 ],
0
0
0
00
0
1 X s(m=2n) n converges to the disand the sequence of distributions Hn = n
m=2
2 m=
;1
tribution H =
X
1
k=
1
;1
yk k T0 + yk k T1 + yk k T2 ].
00
0
00
00
00
;1
3.4 A characterization of Fourier transforms ^0 ^1 ^2
T T T
We characterize the triple of functions i(
) = T^i (
) i = 0 1 2 without computing all
the subdivision scheme for three triples of initial data.
Theorem 20 If ;5=2 < 2 < 3=2 and ;9=2 < 3 < 7=2, then (T^0(
) T^1(
) T^2(
)) is
the unique triple of analytic functions (0(
) 1 (
2(
))) which is a solution of the
vector equation
0 (2
) 1
0 (
) 1
0
@ 1(2
) A = A(
) @ 01(
) A (19)
2(2
)
2(
)
and which satises 0(0) = 1.
Proof: One must rst notice that the matrix A(0) is a lower triangular matrix
whose diagonal components are successively 1 1=4 + =2 1=8 + 3=4. The proof that
(0 = T^0 1 = T^1 2T^2 )) which is a solution of the vector equation (19) is entirely
similar to that one of Theorem 12. By setting = 0 in Equation (19),we also remark
that (T^0 (0) T^1(0)T^1(0))T is a column eigenvector for the eigenvalue 1 of the matrix
A(0).
Now, let (0 1 2) be a triple of analytic functions solution of (19) and such that
0(0) = 1. As in Theorem 12, we expand the functions 0 1 2 and the components
of A(
) in
power series of :
P
P
i(
) = n=0 (in) n i = 1 2 3 A(
) = n=0 nAn.
1
1
17
When we substitute 0 to in equation (19), we obtain that (0(0) 1(0) 2(0))T
is a column eigenvector for the eigenvalue 1 of the matrix A(0). Since A(0) is triangular with only one 1 on its main diagonal, this eigenvector is unique when its rst
component is 1. Hence i(0) = T^i(0) i = 0 1 2. By replacing i with their powers
series in the system (19), by developing the products and reorganizing the results in
terms of powers of , we obtain the series of equations depending on n = 1 2 3 : : ::
0 (n) 1
0 (n k) 1
0
n
X B 0(n k) C
(n) C
2n B
=
Ak @ 1 A @ 1(n) A
(n k)
;
;
(20)
k=0
2
2
Since for all n > 0, the matrix 2nI ; A(0) has an inverse, all the components in
the expansions of (in) i = 0 1 2 are uniquely determined. Hence i = T^i i = 0 1 2.
2
;
4 Conclusion
We de
ned the notion of convergence in distribution of Hermite subdivision schemes
and we studied two classes of subdivision schemes for their convergence. For each
class, we were able to nd a large region of the parameter space for which distributional convergence happens. Is it possible to enlarge the region for which convergence
in distribution will be veri
ed? More generally, is it possible to nd simple necessary
and sucient conditions for distributional convergence for any Hermite subdivision
scheme?
After de
ning Hermite subdivision schemes on the space of distributions, one
may think of many other questions. One of these is the following. In the scheme
HS 11, can one characterize the largest regionR of the parameters such that for
the corresponding Fourier transforms T^0 T^1,
(1 + 2)jT^i(
)j2d
< 1 i = 0 1?
This characterization would be useful to specifying all interpolating functions f of
the scheme such that f and f are in L2 (R ).
1
;1
0
References
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nite products of matrices. Linear Algebra
Appl. 74 (1986) 11-21.
2] I. Daubechies and J. C. Lagarias, Set of matrices all in
nite products of which
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Sc. Math. Quebec 11 (1987) 25-44.
18
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19