Fourier analysis of 2-point Hermite interpolatory subdivision schemes
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(2) 10, 11] and Dyn-Levin
(3) 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: dubucs@dms.umontreal.ca y D. Lemire, Ondelette Inc., 2059, rue Wurtele, Montr eal (Quebec), H2K 2P8 Canada, email: lemire@ondelette.com z J.-L. Merrien, INSA de Rennes, 20 av. des Buttes de Co esmes, CS 14315, 35043 RENNES CEDEX, France, email: Jean-Louis.Merrien@insa-rennes.fr. 1.
(4) 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
(5) 3] and of Deslauriers-Dubois-Dubuc
(6) 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) 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) 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
(9) 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.
(10) 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
(11) a b] by the formulae: 0. f (x) = f (a) +2 f (b) + h
(12) p(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
(13) 11]. Details on generalized spectral radii can be found in Daubechies-Lagarias
(14) 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.
(15) 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 >
(16) f (k)f0 (x ; k) + p(k)f1(x ; k)] > = k= X >
(17) 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
(18) ;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).
(19) 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
(20) 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
(21) ;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
(22) 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.
(23) 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 = max
(24) 0 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
(25) 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
(26) ;1 1]. 1. j j!1. 6.
(27) 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
(28) ;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) ( ) =
(29) 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) +
(30) 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.
(31) 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
(32) 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=.
(33) yk k T0 + yk k T1 ]: 0. ;1. 8.
(34) Proof: LetP N 2ben a function in C with support in
(35) ;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. ;.
(36) yk f0 (m=2n) + yk f1(m=2n)] (m=2n + k) 0. k= N 1 m= N 2n N +1 2n
(37) yk T0(n) + yk T1(n)]( k ): k= N 1 ;. =. ;.
(38) 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.
(39) 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(n
(40) max( 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.
(41) 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=.
(42) 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).
(43) 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 (
(44) 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
(45) 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=2
(46) f (b) + f (a)] + 2 h
(47) p(b) ; p(a)] + 3 h2
(48) s(b) + s(a)] > > = f (b) ; f (a) p(x) = 1 + 2
(49) p(a) + p(b)] + 3 h
(50) s(b) ; s(a)] (12) h > > p(b) ; p(a) +
(51) 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
(52) 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
(53) 11]. Remark 3: For some values of the parameters, the function f on each interval
(54) k k + 1] k 2 Z is a usual polynomial or a piecewise polynomial. For example: 11.
(55) 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
(56) 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
(57) f (k)f0(x ; k) + p(k)f1 (x ; k) + s(k)f2 (x ; k)] > > > k= > > = X
(58) f (k)p0(x ; k) + p(k)p1(x ; k) + s(k)p2(x ; k)] > k= > > X >
(59) 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
(60) ;f1 (x ; 1) + f1 (x + 1)] > f1 (x=2) = 2
(61) ;f0 (x ; 1) + f0 (x + 1)] + 22 f1 (x ; 1) + 12 f1(x) + 22 f1 (x + 1) > = 2 + 4
(62) ;f2 (x ; 1) + f2 (x + 1)] > > f2 (x=2) = 3
(63) f0 (x ; 1) + f0(x + 1)] + 23
(64) ;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.
(65) 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
(66) A(0)]n = @ 1 1 n n 3 3 3 0 (8 + 4 ) 1 + 3 1
(67) ( 8 + 4 ) ; 1] 8 4 ;. 13.
(68) 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
(69) ;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) B@ T^1(n+1)( ) CA = A( =2) B@ T^1(n)( =2) CA B@ U^1(n+1)( ) CA = 2A( =2) B@ 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) B@ V^1(n+1)( ) CA = 4A( =2) B@ 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.
(70) 0 ^ (n) 1 0 0 1 0 V^ (n)( ) 1 001 U0 ( ) 0 B@ U^1(n) ( ) CA = 2nPn( ) @ 1 A B@ 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=.
(71) 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.
(72) 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 =
(73) 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 +
(74) 1=2 + 2 cos( =2n)]un 1 + i 23 2n sin( =2n) 4n1 1 vn 1 vn = i 22 2n sin( =2n)un 1 +
(75) 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.
(76) 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.
(77) 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.
(78) 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.
(79) 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
(80) 1] M. Artzrouni, On the convergence of in nite products of matrices. Linear Algebra Appl. 74 (1986) 11-21.
(81) 2] I. Daubechies and J. C. Lagarias, Set of matrices all in nite products of which converge. Linear Algebra Appl. 161 (1992) 227-263.
(82) 3] G. Deslauriers, S. Dubuc, Transformees de Fourier de courbes irreguli!eres. Ann. Sc. Math. Qu ebec 11 (1987) 25-44. 18.
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(84) 4] G. Deslauriers, J. Dubois, S. Dubuc, Multidimensional Iterative Interpolation. Canad. J. Math. 43 (1991) 297-312.
(85) 5] N. Dyn, D. Levin, Analysis of Hermite-type Subdivision Schemes. In Approximation Theory VIII. Vol 2: Wavelets and Multilevel Approximation, Ed: C. K. Chui and L.L. Schumaker. World Scienti c, Singapore, 1995, pp. 117-124.
(86) 6] N. Dyn, D. Levin, Analysis of Hermite-interpolatory subdivision schemes. In Spline Functions and the Theory of Wavelets, S. Dubuc and G. Deslauriers, Ed. Amer. Math. Soc., Providence R. I., 1999, pp. 105-113.
(87) 7] L. Herve, Multi-Resolution Analysis of Multiplicity d: Applications to Dyadic Interpolation. Applied Comput. Harmon. Anal. 1 (1994) 299-315.
(88) 8] L. Kobbelt, Using the discete Fourier transform to analyse the convergence of subdivision schemes. Applied Comput. Harmon. Anal. 5 (1998) 68-91.
(89) 9] M. Kuczma, Functional Equations in a Single Variable. Panstwowe Wydawnictwo Naukowe, Warsaw, 1968.
(90) 10] J.-L. Merrien, A family of Hermite interpolants by bissection algorithms. Numer. Algorithms 2 (1992) 187-200.
(91) 11] J.-L. Merrien, Interpolants d'Hermite C 2 obtenus par subdivision. M2AN Math. Model. Numer. Anal. 33 (1999) 55-65.
(92) 12] L. Schwartz, Th eorie des distributions, Tomes 1 and 2. Hermann, Paris, 1956.. 19.
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