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Improved Haar and Walsh functions over triangular domains
Journal of the Franklin Institute 347 (2010) 1782–1794 www.elsevier.com/locate/jfranklin
Improved Haar and Walsh functions over triangular domains Ren-hong Wang, Wei Dan School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China Received 29 December 2008; received in revised form 29 April 2010; accepted 10 September 2010
Abstract Piecewise constant orthogonal functions over triangular domains play an important role in many applications. In the present paper, some Haar and Walsh functions over triangular domains are constructed. Compared with the previously proposed Haar functions in [5], the new Haar functions take only integer. For any continuous function, the uniform convergence of the new Haar–Fourier series is proved. Moreover, based on the relation between the new Haar and Walsh functions, the uniform convergence of the Walsh–Fourier series is studied. Additionally, we obtain the relation between the Walsh functions in the two different orders, Paley order and Hadamard order, which have not been discussed previously. & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction The orthogonal functions have been widely used in numerical analysis [3,4,12–14], signal analysis [1], image processing [7]. Compared with the classic orthogonal functions, such as the trigonometric, Chebyshev, Legendre, Hermite, etc. [15], the Haar and Walsh functions take essentially only two values [6,8,16]. Due to the above, the Walsh functions have been successfully applied to broad areas with the development of semiconductor technology and digital technology [2,17]. And the Haar functions become the typical representation of the basic wave functions in wavelet analysis. In many applications, such as system controlling [11], numerical analysis [12,13], it is necessary to have the piecewise constant orthogonal functions on O, where O is a compact subset of R2 . The most convenient method of Corresponding author.
E-mail address: [email protected] (W. Dan). 0016-0032/$32.00 & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2010.09.003
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constructing such functions is to take tensor products of one-dimensional orthogonal functions. Nevertheless, the orthogonal functions constructed in this manner may not be suited to handle problems over non-box domains, such as triangular or hexagonal domains. On the other hand, triangular meshes are preferred in applications due to their simplicity and efficient representation for any subset of R2 . The goal of this paper is to study the generalized Haar and Walsh functions over triangular domains which take essentially only two values. This paper is organized as follows. Section 2 gives a survey on the notations and preliminaries that are used in subsequent sections. In Section 3, the improved Haar functions are constructed. Compared with the proposed Haar functions in [5], the improved Haar functions take only integer. Thus, the improved Haar functions may be more useful in practice. We also prove the uniform convergence of the Haar–Fourier series for any continuous function. The Haar functions can be used as a very practical wavelet. In Section 4, the Walsh functions in Paley order and Hadamard order are given, respectively. With the help of the improved Haar functions in Section 3, the uniform convergence of the Walsh–Fourier series is also studied. Our result improves the result given in [5]. Additionally, the relation between the Walsh functions in the two different orders is obtained. We close with a short conclusion in Section 5.
2. Notations and preliminaries Let n (or nABC ) be a triangle on the plane. Suppose that the area of nABC equals 1. If D, E, and F are midpoints of AB, BC, and CA, respectively, by connecting DE, EF, and FD, we divide n into four similar sub-triangles nDEF ; nADF ; nBDE , and nCEF . Denote them by n1 ; n2 ; n3 , and n4 , respectively. The notation used here is not the same as given in [5,9,10]. Set n1;i ¼ ni
ði ¼ 1; 2; 3; 4Þ:
For each n1;i , we divide it into four similar small triangles, and order them in the same way as before. Denote them by n2;1 ; n2;2 ; . . . ; and n2;16 such that n1;i ¼ n2;4i3 [ n2;4i2 [ n2;4i1 [ n2;4i
ði ¼ 1; 2; 3; 4Þ:
We continue this process. For any n, we get a sequence nn;1 ; nn;2 ; . . . ; and nn;4n such that nn1;i ¼ nn;4i3 [ nn;4i2 [ nn;4i1 [ nn;4i
ði ¼ 1; 2; . . . ; 4n1 ; n ¼ 2; 3; . . .Þ:
If two points P and Q are in two similar triangles, respectively, and have the same area coordinate, then we denote them by PQ. The Haar functions X over n proposed in [5] are defined as follows: X 0 ðPÞ ¼ 1 ( X ð1Þ 1 ðPÞ
¼
for P 2 n; 1 1
for P 2 n2 [ n3 ; for P 2 n1 [ n4 ;
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8 pffiffiffi 2 > < p ffiffiffi 2 X ð2Þ ðPÞ ¼ 1 > : 0
for P 2 n2 ; for P 2 n3 ; for P 2 n1 [ n4 ;
8 pffiffiffi 2 > < p ffiffiffi ð3Þ X 1 ðPÞ ¼ 2 > : 0
for P 2 n4 ; for P 2 n1 ; for P 2 n2 [ n3 ;
^ ( n2 X nð34 jþiÞ ðPÞ
¼
2X ðiÞ n1 ðQÞ for P 2 njþ2 ; 0
for P 2 n\njþ2 ;
where Q 2 n; QP; j ¼ 0; 1; 2; 3; i ¼ 1; 2; . . . ; 3 4ðn2Þ ; n ¼ 2; 3; . . . : It should be pointed out that n5 means n1 in the last definition. Let (u,v,w) be the area coordinate of P with respect to n, and set u,v,w in binary notation u ¼ 0:u1 u2 u3 . . . ;
v ¼ 0:v1 v2 v3 . . . ;
w ¼ 0:w1 w2 w3 . . . ;
where ui ; vi ; wi 2 f0; 1g; i ¼ 1; 2; 3; . . . : The Rademacher functions R are defined by R0 ðmÞðu; v; wÞ ¼ 1; R1ðmÞ ðu; v; wÞ ¼ ð1Þvmþ1 þwmþ1 ; R2ðmÞ ðu; v; wÞ ¼ ð1Þumþ1 þwmþ1 ; R3ðmÞ ðu; v; wÞ ¼ ð1Þumþ1 þvmþ1
ðm ¼ 0; 1; 2; . . .Þ:
The Walsh functions W in [5] are then given by p Y W n ðu; v; wÞ ¼ Rni ðiÞ ðu; v; wÞ i¼0 p
for n ¼ np 4 þ np1 4p1 þ þ n1 4 þ n0 , where ni 2 f0; 1; 2; 3g; i ¼ 0; 1; . . . ; p. Pn1 RTheorem 1 (Feng and Qui [5]). If f 2 CðnÞ and fn ¼ i¼0 ai W i , where ai ¼ n f ðPÞ W i ðPÞ dP, then lim Jf f4n J1 ¼ 0:
n-1
3. The improved Haar functions I over triangular domains We define the improved Haar functions I by I 0 ðPÞ ¼ 1 for P 2 n; ( 1 for P 2 n1 [ n2 ; ð1Þ I 1 ðPÞ ¼ 1 for P 2 n3 [ n4 ;
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( I ð2Þ 1 ðPÞ
¼ (
I ð3Þ 1 ðPÞ ¼
1
for P 2 n1 [ n3 ;
1
for P 2 n2 [ n4 ;
1 1
for P 2 n1 [ n4 ; for P 2 n2 [ n3 ;
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^ ( I nð4ðj1ÞþiÞ ðPÞ
¼
ðjÞ ðQÞ for P 2 ni ; 2I n1
0
for P 2 n\ni ;
where Q 2 n; QP; i ¼ 1; 2; 3; 4; j ¼ 1; 2; . . . ; 3 4ðn2Þ ; n ¼ 2; 3; . . . : The value of above functions at any discontinuous point is defined by the corresponding averages. ð2Þ ð3Þ ð34n1 Þ Each of the functions I 0 ; I ð1Þ can be expressed linearly in terms of 1 ; I1 ; I1 ; . . . ; In n1
ð2Þ ð3Þ ð34 Þ the functions X 0 ; X ð1Þ , and conversely. Indeed, for n=1, we have 1 ; X1 ; X1 ; . . . ; Xn pffiffiffi pffiffiffi 2 ð2Þ 2 ð2Þ ð3Þ ð2Þ ð1Þ ðX ðX 1 þ X ð3Þ I 0 ¼ X 0 ; I ð1Þ ¼ X Þ; I ¼ I ð3Þ 1 1 1 1 1 Þ; 1 ¼ X 1 ; 2 2 and conversely, pffiffiffi pffiffiffi 2 ð1Þ ð2Þ 2 ð1Þ ð1Þ ð3Þ ð2Þ ð3Þ ðI 1 I 1 Þ; X 1 ¼ ðI 1 þ I ð2Þ X 0 ¼ I 0 ; X 1 ¼ I 1 ; X 1 ¼ 1 Þ: 2 2 The general fact appears by induction from the very definitions of the functions I and the X . The set X is known to be complete [5], it follows from the expression of the X in terms of the I that the set I is also complete.
Theorem 2. The functions I form a complete orthonormal function system. Proof. It is sufficient to show that the functions I are orthonormal. Indeed, if I ðkÞ n ðPÞ and ðsÞ I ðsÞ ðPÞ are two different functions of the set I , and n4v, then I ðPÞ has a constant value v v over the domain where I ðkÞ ðPÞ is different from zero. Therefore, we have n Z Z ðkÞ ðsÞ I n ðPÞI v ðPÞ dP ¼ const: I ðkÞ n ðPÞ dP ¼ 0: n
n
We now evaluate Z Z ðkÞ ðsÞ 1 1Þþk1 Þ I n ðPÞI n ðPÞ dP ¼ I ð4ðj ðPÞI vð4ðl1 1Þþs1 Þ ðPÞ dP n n n Z Z 1Þ 1Þ 1Þ 1Þ ¼ 4dk1 ;s1 I ðjn1 ðQÞI ðln1 ðQÞ dP ¼ dk1 ;s1 I ðjn1 ðPÞI ðln1 ðPÞ dP n k1 n Z Z ð4ðl2 1Þþs2 Þ 2 1Þþk2 Þ 2Þ 2Þ ðPÞI ðPÞ dP ¼ 4d d I ðjn1 ðQÞI ðln1 ðQÞ dP ¼ dk1 ;s1 I ð4ðj k ;s k ;s 1 1 2 2 n1 n1 n n k2 Z ¼ ¼ dk1 ;s1 dk2 ;s2 dkn1 ;sn1 I j1n1 ðPÞI l1n1 ðPÞ dP ¼ dk1 ;s1 dk2 ;s2 dkn1 ;sn1 djn1 ;ln1 : n
This finishes the proof of the theorem.
&
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We now investigate the uniform convergence of the series for each continuous function over n taken with respect to the I . Let f(P) be continuous over n, and define the finite sum [f(P)](k) n by Z Z ð1Þ ½f ðPÞðkÞ ¼ I ðPÞ f ðQÞI ðQÞ dQ þ I ðPÞ f ðQÞI ð1Þ 0 0 n 1 1 ðQÞ dQ þ n n Z ðPÞ f ðQÞI ðkÞ þ I ðkÞ n n ðQÞ dQ: n
Set ðkÞ KnðkÞ ðP; QÞ ¼ I 0 ðPÞI 0 ðQÞ þ þ I ðkÞ n ðPÞI n ðQÞ;
then ½f ðPÞðkÞ n ¼
Z n
KnðkÞ ðP; QÞf ðQÞ dQ:
Direct calculations show that 2 1 1 1 61 1 1 6 I 0 ðPÞI 0 ðQÞ2s0 ¼ 6 41 1 1 1 1 1 2
1
3 1 17 7 7; 15 1 3
1
1
1
6 1 6 ð1Þ I ð1Þ ðPÞI ðQÞ2s ¼ 6 1 1 1 4 1
1 1
1 1
1 7 7 7; 1 5
1
1
1
1 6 1 6 ð2Þ I ð2Þ 1 ðPÞI 1 ðQÞ2s2 ¼ 6 4 1
1 1
1 1
1
1
1
1
1 1
1 1
1 1
1 1
2
1 2
1 6 1 6 ð3Þ I ð3Þ 1 ðPÞI 1 ðQÞ2s3 ¼ 6 4 1 1
1 3 1 1 7 7 7; 1 5 1 3 1 1 7 7 7; 1 5 1
where the notation GðP; QÞ2A means that the value of G(P,Q) is aij when P 2 nn;i ; Q 2 nn;j . We now obtain 2 2 2 2 K1ð1Þ ðP; QÞ2s0 þ s1 ¼ diag block ; ; 2 2 2 2
R.-h. Wang, W. Dan / Journal of the Franklin Institute 347 (2010) 1782–1794
2
3
6 1 6 K1ð2Þ ðP; QÞ2s0 þ s1 þ s2 ¼ 6 4 1 1
1
1
1
3
3 1
1 3
1 1
7 7 7; 5
1
1
3
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K1ð3Þ ðP; QÞ2s0 þ s1 þ s2 þ s3 ¼ diag block½4; 4; 4; 4: More precisely, if we write k ¼ 4n1 i þ 4n2 k1 þ þ 4kn2 þ kn1 þ 1ð1rkr3 4n 1; i 2 f0; 1; 2g; k1 ; . . . ; kn1 2 f0; 1; 2; 3gÞ, and set k0 ¼ 4n2 kn1 þ þ 4k2 þ k1 þ 1, then we have n1 ðkÞ I ðkÞ siþ1 ; 04 ; . . . ; 04 Þ; n ðPÞI n ðQÞ ¼ diag blockð04 ; . . . ; 04 ; 4
where the term 4n1 siþ1 is the k0 th block. Therefore, in the general case ! iþ1 i X X n1 n1 ðkÞ Kn ðP; QÞ ¼ diag block 4 sm ; . . . ; 4 sm ; . . . ; m¼0
m¼0
P where the j 0 ¼ 4n2 jn1 þ þ 4j2 þ j1 þ 1th block is 4n1 iþ1 m¼0 sm provided that j ¼ 4n1 i þ 4n2 j1 þ þ 4jn2 þ jn1 þ 1 satisfies jrk, and the remained blocks are P 4n1 im¼0 sm . In particular, Knð34
n1
Þ
¼ 4n I4n :
Without loss of generality, we assume for the moment that a 2 nn;4l3 nn1;l ð1rlr4n1 Þ is not on the boundaries of nn;1 ; . . ., and nn;4n , then the function (k) ðkÞ K(k) equals n (a,Q) of Q equals zero except for a domain Dn nn1;l whose area Sn n n1 n1 1=4 ; 1=2 4 , or 1=4 .
n ðkÞ (1) If DðkÞ n ¼ nn;4l3 , and Sn ¼ 1=4 , then Z Z 1 n ½f ðaÞðkÞ ¼ 4 f ðQÞ dQ ¼ f ðQÞ dQ: n sðkÞ nn;4l3 DðkÞ n n n1 ðkÞ (2) If DðkÞ , then n ¼ nn;4l3 [ nn;4l2 , and Sn ¼ 1=2 4 Z Z 1 ½f ðaÞðkÞ 2 4n1 f ðQÞ dQ ¼ ðkÞ f ðQÞ dQ: n ¼ ðkÞ Sn DðkÞ Dn n n1 ðkÞ , then (3) If DðkÞ n ¼ nn1;l , and Sn ¼ 1=4 Z Z 1 ðkÞ n1 4 f ðQÞ dQ ¼ ðkÞ f ðQÞ dQ ½f ðaÞn ¼ Sn DðkÞ nn1;l n
or n1 ½f ðaÞðkÞ n ¼4
"Z
Z
Z
3f ðQÞ dQ þ nn;4l3
Z
f ðQÞ dQ þ nn;4l2
f ðQÞ dQ nn;4l1
# f ðQÞ dQ
nn;4l
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¼
1 SnðkÞ
Z
f ðQÞ dQ þ 2 4n1
"Z
DðkÞ n
Z f ðQÞ dQ
nn;4l3
# f ðQÞ dQ :
nn;4l
In any cases, by the continuity of f, we can conclude lim ½f ðaÞðkÞ n ¼ f ðaÞ:
n-1
This equality also holds when a is on the boundary, the argument is similar. Thus, we establish: Theorem 3. If f 2 CðnÞ, then lim Jf ðPÞ½f ðPÞðkÞ n J1 ¼ 0:
n-1
4. The improved Walsh functions over triangular domains 4.1. The Walsh functions W in Paley order In order to improve Theorem 1, we define the Rademacher functions f by f2n ðu; v; wÞ ¼ ð1Þvnþ1 þwnþ1 ; f2nþ1 ðu; v; wÞ ¼ ð1Þunþ1 þwnþ1
ðn ¼ 0; 1; 2; . . .Þ:
With the help of the f, the Walsh functions W are defined by W0 ¼ 1 and W n ¼ fn1 fn2 . . . fnr for n ¼ 2n1 þ 2n2 þ þ 2nr , where the integers ni are uniquely determined by niþ1 oni . The value of above functions at any discontinuous point is defined by the corresponding averages. Clearly, our definition is equivalent to the definition given in [9] with the relations R1ðpÞ ¼ f2p ;
R2ðpÞ ¼ f2pþ1 ;
R3ðpÞ ¼ R1ðpÞ R2ðpÞ ¼ f2p f2pþ1
ðp ¼ 0; 1; 2; . . .Þ:
Lemma 4. f2nþ2 ðPÞ ¼ f2n ðQÞ; f2nþ3 ðPÞ ¼ f2nþ1 ðQÞ for P 2 ni , where QP; i ¼ 1; 2; 3; 4. Proof. Assume for the moment that P 2 n1 . Let (u,v,w) be the area coordinate of P with respect to n, and in binary notation we have u ¼ 0:0u2 u3 . . . ;
v ¼ 0:0v2 v3 . . . ;
w ¼ 0:0w2 w3 . . .
since P 2 n1 . It is clear that the area coordinate of Q with respect to n, that is, the area coordinate of P with respect to n1 , is (12u,12v,12w), and then in binary notation 12u ¼ 0:ð1u2 Þð1u3 Þ . . . ;
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12v ¼ 0:ð1v2 Þð1v3 Þ . . . ; 12w ¼ 0:ð1w2 Þð1w3 Þ . . . : Hence f2nþ2 ðPÞ ¼ ð1Þvnþ2 þwnþ2 ¼ ð1Þð1vnþ2 Þþð1wnþ2 Þ ¼ f2n ðQÞ; f2nþ3 ðPÞ ¼ ð1Þunþ2 þwnþ2 ¼ ð1Þð1unþ2 Þþð1wnþ2 Þ ¼ f2nþ1 ðQÞ: The argument is similar to the above process when i=2,3 or 4. This completes the proof. & Lemma 5. f2n ¼
1 ð1Þ ð2Þ ð4n Þ n ðI nþ1 þ I nþ1 þ þ I nþ1 Þ; 2
f2nþ1 ¼
1 ð4n þ1Þ ð4n þ2Þ ð24n Þ n ðI nþ1 þ I nþ1 þ þ I nþ1 Þ 2
ðn ¼ 0; 1; 2; . . .Þ:
Proof. A direct calculation shows that f0 ¼ I ð1Þ 1 ;
f1 ¼ I ð2Þ 1 :
Suppose that the lemma holds for nrN. For n=Nþ1 and P 2 ni ði ¼ 1; 2; 3; 4Þ, by Lemma 4, the definition of the I and induction hypothesis, we have 1 ð2Þ ð4N Þ f2ðNþ1Þ ðPÞ ¼ f2N ðQÞ ¼ N ðI ð1Þ Nþ1 ðQÞ þ I Nþ1 ðQÞ þ þ I Nþ1 ðQÞÞ 2 1 ðiÞ ð4Nþ1 4þiÞ ¼ Nþ1 ðI Nþ2 ðPÞ þ I ð4þiÞ ðPÞÞ Nþ2 ðPÞ þ þ I Nþ2 2 1 ð2Þ ð4Nþ1 Þ ¼ Nþ1 ðI ð1Þ Nþ2 ðPÞ þ I Nþ2 ðPÞ þ þ I Nþ2 ðPÞÞ 2 and 1 ð4N þ1Þ ð4N þ2Þ ð24N Þ f2ðNþ1Þþ1 ðPÞ ¼ f2Nþ1 ðQÞ ¼ N ðI Nþ1 ðQÞ þ I Nþ1 ðQÞ þ þ I Nþ1 ðQÞÞ 2 Nþ1 Nþ1 1 ð4Nþ1 þiÞ þ4þiÞ 4þiÞ ¼ Nþ1 ðI Nþ2 ðPÞ þ I ð4 ðPÞ þ þ I ð24 ðPÞÞ Nþ2 Nþ2 2 Nþ1 1 ð4Nþ1 þ1Þ ð4Nþ1 þ2Þ Þ ¼ Nþ1 ðI Nþ2 ðPÞ þ I Nþ2 ðPÞ þ þ I ð24 Nþ2 ðPÞÞ: 2 Therefore, the lemma holds for n=Nþ1, and this finishes the induction. & We have the improved uniform convergence of the Walsh–Fourier series as follows: R P Theorem 6. If f 2 CðnÞ and fn ¼ n1 i¼0 ai W i , where ai ¼ n f ðPÞ W i ðPÞ dP, then lim Jf f2n J1 ¼ 0:
n-1
Proof. From Theorems 1 and 3, it obviously suffices to show that the functions ð4n Þ W 0 ; W 1 ; . . . ; W 22nþ1 1 can be expressed linearly in terms of the functions I 0 ; I ð1Þ 1 ; . . . ; I nþ1 .
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The definition of the W shows that W 1 ; . . . ; W 22nþ1 1 can be expressed by the product of f0 ; f1 ; . . . ; f2n . From Lemma 5, we obtain that W 1 ; . . . ; W 22nþ1 1 can be expressed by the sum of the products ðki1 Þ
c I i1
ðli Þ
ðki2 Þ
I i1 1 I i2
ðli Þ
ðkij Þ
I i2 2 I ij
ðli Þ
I ij j ;
ð1Þ
where c is a constant and 1rki r4i ; 4i þ 1rli r2 4i ; i ¼ 1; 2; . . . ; n þ 1; 1ri1 o has a constant value over the domain where I ðkÞ oij rn þ 1. Since I ðsÞ v u is different from zero provided that vou and the fact that for 1rs1 r4v1 ; 4v1 þ 1rs2 r2 4v1 8 < 1 I s2 þ4v1 if s s ¼ 4v1 ; 2 1 v ðs2 Þ 1Þ I ðs I ¼ 2v1 v v : 0 else; n
ð4 Þ then the product (1) must be a function of I 0 ; I ð1Þ 1 ; . . . ; I nþ1 multiplies a constant, and hence the theorem is proven. &
n
ð4 Þ Note: The functions X 0 ; X ð1Þ 1 ; . . . ; X nþ1 do not possess this property, that is, the functions
W 0 ; W 1 ; . . . ; W 22nþ1 1 cannot be expressed linearly in terms of the functions X 0 ; X ð1Þ 1 ;...; n
ð4 Þ X nþ1 .
4.2. The Walsh functions H in Hadamard order The Walsh functions H in Hadamard order are generated when the standard Kronecker product of the elementary Hadamard matrix H4 is performed with itself: 0 1 1 1 1 1 B 1 1 1 1 C B C H4 ¼ B C @ 1 1 1 1 A 1
1 1
1
and HN ¼ HN=4 H4 ; where N ¼ 4n ; n ¼ 1; 2; 3; . . . ; # denotes the Kronecker product. The Hadamard matrix ð1Þ ð4n 1Þ H4n corresponds to the Walsh functions Hð0Þ : n ; Hn ; . . . ; Hn
n ¼ 1;
Hð0Þ 1
2 þ
þ
þ
þ
Hð1Þ 1 Hð2Þ 1 Hð3Þ 1
2 þ
þ
2 þ
þ
2 þ
þ
where we omit 1 in these elements of the Hadamard matrix.
R.-h. Wang, W. Dan / Journal of the Franklin Institute 347 (2010) 1782–1794
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2 3
1
4 (2)
(1)
(0)
H1
H1
H1
(3)
H1
Fig. 1.
Figs. 1 and 2 show the Walsh functions associated with the Hadamard matrix. Black and white areas represent 1 and 1, respectively. The triangle above in Fig. 1 shows a certain order.
n ¼ 2;
Hð0Þ 2
2
þ þ
þ
þ
þ
þ
þ
þ
þ þ
þ
þ
þ
þ
þ
þ
Hð1Þ 2 Hð2Þ 2 Hð3Þ 2 Hð4Þ 2 Hð5Þ 2 Hð6Þ 2 Hð7Þ 2 Hð8Þ 2 Hð9Þ 2 ð10Þ H2 H2ð11Þ H2ð12Þ H2ð13Þ H2ð14Þ H2ð15Þ
2
þ
þ
þ
þ
þ
þ
þ
þ
2
þ þ
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þ þ
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2
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2
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þ
þ
Our next theorem dealing with the relation between the W and the H has an exact analogue in the one-dimensional case [17]. Theorem 7. Suppose 0rkr4n 1, and write k ¼ k2n1 22n1 þ k2n2 22n2 þ þ k1 2þ k0 , where ki 2 f0; 1g; i ¼ 0; 1; . . . ; 2n1. Set k0 ¼ k0 22n1 þ k1 22n2 þ þ k2n2 2þ k2n1 , then HðkÞ n ¼ W k0 .
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(1)
H2
(2)
H2
(4)
H2
(5)
H2
(6)
H2
(8)
H2
(9)
H2
(10)
H2
(12)
H2
(13)
H2
(14)
H2
H2
(3)
H2
(7)
H2
(11)
H2
(15)
H2
Fig. 2.
Proof. If j ¼ j2n1 22n1 þ j2n2 22n2 þ þ j1 2 þ j0 , then the element hkj of the Hadamard matrix H4n , that is, the value of HðkÞ n over nn;jþ1 , can be expressed as P2n1 hkj ¼ ð1Þ s¼0 ks js : Q On the other hand, we have W k0 ¼ ki ¼1;i2f0;1;...;2n1g f2n1i from the definition of the W. In order to prove the theorem, we only need to show that P2n2lþ1 Y f2n1i ðPÞ ¼ ð1Þ s¼2n2l ks js ð2Þ ki ¼1;i2f2n2l;2n2lþ1g
for l ¼ 1; 2; . . . ; n and P 2 nn;jþ1 . Let (u,v,w) be the area coordinate of P with respect to n, and in binary notation u ¼ 0:u1 u2 u3 . . . ;
v ¼ 0:v1 v2 v3 . . . ;
We notice the fact that 8 ð0; 0; 0Þ > > > > < ð1; 0; 0Þ ðu1 ; v1 ; w1 Þ ¼ ð0; 1; 0Þ > > > > : ð0; 0; 1Þ
w ¼ 0:w1 w2 w3 . . . :
if j2n1 ¼ 0 and j2n2 ¼ 0; if j2n1 ¼ 0 and j2n2 ¼ 1; if j2n1 ¼ 1 and j2n2 ¼ 0; if j2n1 ¼ 1 and j2n2 ¼ 1:
If k2n1=1 and k2n2=1, then f0 ðPÞ f1 ðPÞ ¼ ð1Þv1 þw1 ð1Þu1 þw1 ¼ ð1Þu1 þv1 ¼ ð1Þj2n2 þj2n1 :
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If k2n1=1 and k2n2=0, then f0 ðPÞ ¼ ð1Þv1 þw1 ¼ ð1Þj2n1 : If k2n1=0 and k2n2=1, then f1 ðPÞ ¼ ð1Þu1 þw1 ¼ ð1Þj2n2 : If k2n1=0 and k2n2=0, then the equality is trivial. Therefore, (2) holds for l=1. Similarly, we can prove (2) for l ¼ 2; 3; . . . ; n. The theorem is true. & 5. Conclusion This paper deals with piecewise constant orthogonal functions over nested triangular domains. Firstly, we have constructed the improved Haar functions and proved the uniform convergence of the corresponding Haar–Fourier series. Secondly, the Walsh functions in Paley order and Hadamard order have been designed, respectively. Based on the relation between the improved Haar and Walsh functions, we have shown the uniform convergence of the Walsh–Fourier series. Finally, the relation between the Walsh functions in the two different orders has been proposed. Acknowledgements The authors would like to thank the reviews for their helpful comments and suggestions which have improved the presentation of the paper. This work was supported by the National Natural Science Foundation of China (Nos. U0935004, 11071031, 11001037, 10801024 and the Fundamental Research Funds for the Central Universities (Nos. DUT10ZD112, DUT10JS02. References [1] N. Ahmed, K.R. Rao, Orthogonal Transforms for Digital Signal Processing, Springer-Verlag, Berlin, 1975. [2] K.G. Beauchamp, Walsh Functions and their Applications, Academic Press, London, New York, San Francisco, 1975. [3] C.F. Chen, C.H. Hsiao, A Walsh series direct method for solving variational problems, J. Franklin Inst. 300 (1975) 265–280. [4] C.F. Chen, Y.T. Tsay, T.T. Wu, Walsh operational matrices for fractional calculus and their application to distributed systems, J. Franklin Inst. 303 (1977) 267–284. [5] Y.Y. Feng, D.X. Qi, On the Haar and Walsh systems on a triangle, J. Comput. Math. 1 (1983) 223–232. [6] A. Haar, Zur Theorie der orthogonalen Funktionen systeme, Math. Ann. 69 (1910) 331–371. [7] H.F. Harmuth, Transmission of Information by Orthogonal Functions, second ed., Springer-Verlag, Berlin, Heidelberg, New York, 1972. [8] R.E.A.C. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932) 241–279. [9] D.X. Qi, Walsh function on the triangular area, Chinese Sci. Bull. 33 (1988) 715–716. [10] D.X. Qi, New definition for a kind of self-similar structure and Walsh function, Chinese Ann. Math. Ser. A 12 (1991) 103–105. [11] G.P. Rao, Piecewise Constant Orthogonal Functions and their Application to System and Control, SpringerVerlag, Berlin, 1983. [12] B.G. Sloss, W.F. Blyth, A priori error estimates for Corrington’s Walsh function method, J. Franklin Inst. 331 (1994) 273–283.
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[13] B.G. Sloss, W.F. Blyth, A Walsh function method for a non-linear Volterra integral equation, J. Franklin Inst. 340 (2003) 25–41. [14] G. Szego, Orthogonal Polynomials, American Mathematical Society, New York, 1937. [15] R.H. Wang, Numerical Approximation, Higher Education Press, Beijing, 1999. [16] J.L. Walsh, A closed set of normal orthogonal functions, Am. J. Math. 45 (1923) 5–24. [17] W.X. Zheng, W.Y. Su, F.X. Ren, Walsh Function Theory and its Application, Shang Hai Science and Technology Press, Shang Hai, 1983.
Improved Haar and Walsh functions over triangular domains Ren-hong Wang, Wei Dan School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China Received 29 December 2008; received in revised form 29 April 2010; accepted 10 September 2010
Abstract Piecewise constant orthogonal functions over triangular domains play an important role in many applications. In the present paper, some Haar and Walsh functions over triangular domains are constructed. Compared with the previously proposed Haar functions in [5], the new Haar functions take only integer. For any continuous function, the uniform convergence of the new Haar–Fourier series is proved. Moreover, based on the relation between the new Haar and Walsh functions, the uniform convergence of the Walsh–Fourier series is studied. Additionally, we obtain the relation between the Walsh functions in the two different orders, Paley order and Hadamard order, which have not been discussed previously. & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction The orthogonal functions have been widely used in numerical analysis [3,4,12–14], signal analysis [1], image processing [7]. Compared with the classic orthogonal functions, such as the trigonometric, Chebyshev, Legendre, Hermite, etc. [15], the Haar and Walsh functions take essentially only two values [6,8,16]. Due to the above, the Walsh functions have been successfully applied to broad areas with the development of semiconductor technology and digital technology [2,17]. And the Haar functions become the typical representation of the basic wave functions in wavelet analysis. In many applications, such as system controlling [11], numerical analysis [12,13], it is necessary to have the piecewise constant orthogonal functions on O, where O is a compact subset of R2 . The most convenient method of Corresponding author.
E-mail address: [email protected] (W. Dan). 0016-0032/$32.00 & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2010.09.003
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constructing such functions is to take tensor products of one-dimensional orthogonal functions. Nevertheless, the orthogonal functions constructed in this manner may not be suited to handle problems over non-box domains, such as triangular or hexagonal domains. On the other hand, triangular meshes are preferred in applications due to their simplicity and efficient representation for any subset of R2 . The goal of this paper is to study the generalized Haar and Walsh functions over triangular domains which take essentially only two values. This paper is organized as follows. Section 2 gives a survey on the notations and preliminaries that are used in subsequent sections. In Section 3, the improved Haar functions are constructed. Compared with the proposed Haar functions in [5], the improved Haar functions take only integer. Thus, the improved Haar functions may be more useful in practice. We also prove the uniform convergence of the Haar–Fourier series for any continuous function. The Haar functions can be used as a very practical wavelet. In Section 4, the Walsh functions in Paley order and Hadamard order are given, respectively. With the help of the improved Haar functions in Section 3, the uniform convergence of the Walsh–Fourier series is also studied. Our result improves the result given in [5]. Additionally, the relation between the Walsh functions in the two different orders is obtained. We close with a short conclusion in Section 5.
2. Notations and preliminaries Let n (or nABC ) be a triangle on the plane. Suppose that the area of nABC equals 1. If D, E, and F are midpoints of AB, BC, and CA, respectively, by connecting DE, EF, and FD, we divide n into four similar sub-triangles nDEF ; nADF ; nBDE , and nCEF . Denote them by n1 ; n2 ; n3 , and n4 , respectively. The notation used here is not the same as given in [5,9,10]. Set n1;i ¼ ni
ði ¼ 1; 2; 3; 4Þ:
For each n1;i , we divide it into four similar small triangles, and order them in the same way as before. Denote them by n2;1 ; n2;2 ; . . . ; and n2;16 such that n1;i ¼ n2;4i3 [ n2;4i2 [ n2;4i1 [ n2;4i
ði ¼ 1; 2; 3; 4Þ:
We continue this process. For any n, we get a sequence nn;1 ; nn;2 ; . . . ; and nn;4n such that nn1;i ¼ nn;4i3 [ nn;4i2 [ nn;4i1 [ nn;4i
ði ¼ 1; 2; . . . ; 4n1 ; n ¼ 2; 3; . . .Þ:
If two points P and Q are in two similar triangles, respectively, and have the same area coordinate, then we denote them by PQ. The Haar functions X over n proposed in [5] are defined as follows: X 0 ðPÞ ¼ 1 ( X ð1Þ 1 ðPÞ
¼
for P 2 n; 1 1
for P 2 n2 [ n3 ; for P 2 n1 [ n4 ;
R.-h. Wang, W. Dan / Journal of the Franklin Institute 347 (2010) 1782–1794
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8 pffiffiffi 2 > < p ffiffiffi 2 X ð2Þ ðPÞ ¼ 1 > : 0
for P 2 n2 ; for P 2 n3 ; for P 2 n1 [ n4 ;
8 pffiffiffi 2 > < p ffiffiffi ð3Þ X 1 ðPÞ ¼ 2 > : 0
for P 2 n4 ; for P 2 n1 ; for P 2 n2 [ n3 ;
^ ( n2 X nð34 jþiÞ ðPÞ
¼
2X ðiÞ n1 ðQÞ for P 2 njþ2 ; 0
for P 2 n\njþ2 ;
where Q 2 n; QP; j ¼ 0; 1; 2; 3; i ¼ 1; 2; . . . ; 3 4ðn2Þ ; n ¼ 2; 3; . . . : It should be pointed out that n5 means n1 in the last definition. Let (u,v,w) be the area coordinate of P with respect to n, and set u,v,w in binary notation u ¼ 0:u1 u2 u3 . . . ;
v ¼ 0:v1 v2 v3 . . . ;
w ¼ 0:w1 w2 w3 . . . ;
where ui ; vi ; wi 2 f0; 1g; i ¼ 1; 2; 3; . . . : The Rademacher functions R are defined by R0 ðmÞðu; v; wÞ ¼ 1; R1ðmÞ ðu; v; wÞ ¼ ð1Þvmþ1 þwmþ1 ; R2ðmÞ ðu; v; wÞ ¼ ð1Þumþ1 þwmþ1 ; R3ðmÞ ðu; v; wÞ ¼ ð1Þumþ1 þvmþ1
ðm ¼ 0; 1; 2; . . .Þ:
The Walsh functions W in [5] are then given by p Y W n ðu; v; wÞ ¼ Rni ðiÞ ðu; v; wÞ i¼0 p
for n ¼ np 4 þ np1 4p1 þ þ n1 4 þ n0 , where ni 2 f0; 1; 2; 3g; i ¼ 0; 1; . . . ; p. Pn1 RTheorem 1 (Feng and Qui [5]). If f 2 CðnÞ and fn ¼ i¼0 ai W i , where ai ¼ n f ðPÞ W i ðPÞ dP, then lim Jf f4n J1 ¼ 0:
n-1
3. The improved Haar functions I over triangular domains We define the improved Haar functions I by I 0 ðPÞ ¼ 1 for P 2 n; ( 1 for P 2 n1 [ n2 ; ð1Þ I 1 ðPÞ ¼ 1 for P 2 n3 [ n4 ;
R.-h. Wang, W. Dan / Journal of the Franklin Institute 347 (2010) 1782–1794
( I ð2Þ 1 ðPÞ
¼ (
I ð3Þ 1 ðPÞ ¼
1
for P 2 n1 [ n3 ;
1
for P 2 n2 [ n4 ;
1 1
for P 2 n1 [ n4 ; for P 2 n2 [ n3 ;
1785
^ ( I nð4ðj1ÞþiÞ ðPÞ
¼
ðjÞ ðQÞ for P 2 ni ; 2I n1
0
for P 2 n\ni ;
where Q 2 n; QP; i ¼ 1; 2; 3; 4; j ¼ 1; 2; . . . ; 3 4ðn2Þ ; n ¼ 2; 3; . . . : The value of above functions at any discontinuous point is defined by the corresponding averages. ð2Þ ð3Þ ð34n1 Þ Each of the functions I 0 ; I ð1Þ can be expressed linearly in terms of 1 ; I1 ; I1 ; . . . ; In n1
ð2Þ ð3Þ ð34 Þ the functions X 0 ; X ð1Þ , and conversely. Indeed, for n=1, we have 1 ; X1 ; X1 ; . . . ; Xn pffiffiffi pffiffiffi 2 ð2Þ 2 ð2Þ ð3Þ ð2Þ ð1Þ ðX ðX 1 þ X ð3Þ I 0 ¼ X 0 ; I ð1Þ ¼ X Þ; I ¼ I ð3Þ 1 1 1 1 1 Þ; 1 ¼ X 1 ; 2 2 and conversely, pffiffiffi pffiffiffi 2 ð1Þ ð2Þ 2 ð1Þ ð1Þ ð3Þ ð2Þ ð3Þ ðI 1 I 1 Þ; X 1 ¼ ðI 1 þ I ð2Þ X 0 ¼ I 0 ; X 1 ¼ I 1 ; X 1 ¼ 1 Þ: 2 2 The general fact appears by induction from the very definitions of the functions I and the X . The set X is known to be complete [5], it follows from the expression of the X in terms of the I that the set I is also complete.
Theorem 2. The functions I form a complete orthonormal function system. Proof. It is sufficient to show that the functions I are orthonormal. Indeed, if I ðkÞ n ðPÞ and ðsÞ I ðsÞ ðPÞ are two different functions of the set I , and n4v, then I ðPÞ has a constant value v v over the domain where I ðkÞ ðPÞ is different from zero. Therefore, we have n Z Z ðkÞ ðsÞ I n ðPÞI v ðPÞ dP ¼ const: I ðkÞ n ðPÞ dP ¼ 0: n
n
We now evaluate Z Z ðkÞ ðsÞ 1 1Þþk1 Þ I n ðPÞI n ðPÞ dP ¼ I ð4ðj ðPÞI vð4ðl1 1Þþs1 Þ ðPÞ dP n n n Z Z 1Þ 1Þ 1Þ 1Þ ¼ 4dk1 ;s1 I ðjn1 ðQÞI ðln1 ðQÞ dP ¼ dk1 ;s1 I ðjn1 ðPÞI ðln1 ðPÞ dP n k1 n Z Z ð4ðl2 1Þþs2 Þ 2 1Þþk2 Þ 2Þ 2Þ ðPÞI ðPÞ dP ¼ 4d d I ðjn1 ðQÞI ðln1 ðQÞ dP ¼ dk1 ;s1 I ð4ðj k ;s k ;s 1 1 2 2 n1 n1 n n k2 Z ¼ ¼ dk1 ;s1 dk2 ;s2 dkn1 ;sn1 I j1n1 ðPÞI l1n1 ðPÞ dP ¼ dk1 ;s1 dk2 ;s2 dkn1 ;sn1 djn1 ;ln1 : n
This finishes the proof of the theorem.
&
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We now investigate the uniform convergence of the series for each continuous function over n taken with respect to the I . Let f(P) be continuous over n, and define the finite sum [f(P)](k) n by Z Z ð1Þ ½f ðPÞðkÞ ¼ I ðPÞ f ðQÞI ðQÞ dQ þ I ðPÞ f ðQÞI ð1Þ 0 0 n 1 1 ðQÞ dQ þ n n Z ðPÞ f ðQÞI ðkÞ þ I ðkÞ n n ðQÞ dQ: n
Set ðkÞ KnðkÞ ðP; QÞ ¼ I 0 ðPÞI 0 ðQÞ þ þ I ðkÞ n ðPÞI n ðQÞ;
then ½f ðPÞðkÞ n ¼
Z n
KnðkÞ ðP; QÞf ðQÞ dQ:
Direct calculations show that 2 1 1 1 61 1 1 6 I 0 ðPÞI 0 ðQÞ2s0 ¼ 6 41 1 1 1 1 1 2
1
3 1 17 7 7; 15 1 3
1
1
1
6 1 6 ð1Þ I ð1Þ ðPÞI ðQÞ2s ¼ 6 1 1 1 4 1
1 1
1 1
1 7 7 7; 1 5
1
1
1
1 6 1 6 ð2Þ I ð2Þ 1 ðPÞI 1 ðQÞ2s2 ¼ 6 4 1
1 1
1 1
1
1
1
1
1 1
1 1
1 1
1 1
2
1 2
1 6 1 6 ð3Þ I ð3Þ 1 ðPÞI 1 ðQÞ2s3 ¼ 6 4 1 1
1 3 1 1 7 7 7; 1 5 1 3 1 1 7 7 7; 1 5 1
where the notation GðP; QÞ2A means that the value of G(P,Q) is aij when P 2 nn;i ; Q 2 nn;j . We now obtain 2 2 2 2 K1ð1Þ ðP; QÞ2s0 þ s1 ¼ diag block ; ; 2 2 2 2
R.-h. Wang, W. Dan / Journal of the Franklin Institute 347 (2010) 1782–1794
2
3
6 1 6 K1ð2Þ ðP; QÞ2s0 þ s1 þ s2 ¼ 6 4 1 1
1
1
1
3
3 1
1 3
1 1
7 7 7; 5
1
1
3
1787
K1ð3Þ ðP; QÞ2s0 þ s1 þ s2 þ s3 ¼ diag block½4; 4; 4; 4: More precisely, if we write k ¼ 4n1 i þ 4n2 k1 þ þ 4kn2 þ kn1 þ 1ð1rkr3 4n 1; i 2 f0; 1; 2g; k1 ; . . . ; kn1 2 f0; 1; 2; 3gÞ, and set k0 ¼ 4n2 kn1 þ þ 4k2 þ k1 þ 1, then we have n1 ðkÞ I ðkÞ siþ1 ; 04 ; . . . ; 04 Þ; n ðPÞI n ðQÞ ¼ diag blockð04 ; . . . ; 04 ; 4
where the term 4n1 siþ1 is the k0 th block. Therefore, in the general case ! iþ1 i X X n1 n1 ðkÞ Kn ðP; QÞ ¼ diag block 4 sm ; . . . ; 4 sm ; . . . ; m¼0
m¼0
P where the j 0 ¼ 4n2 jn1 þ þ 4j2 þ j1 þ 1th block is 4n1 iþ1 m¼0 sm provided that j ¼ 4n1 i þ 4n2 j1 þ þ 4jn2 þ jn1 þ 1 satisfies jrk, and the remained blocks are P 4n1 im¼0 sm . In particular, Knð34
n1
Þ
¼ 4n I4n :
Without loss of generality, we assume for the moment that a 2 nn;4l3 nn1;l ð1rlr4n1 Þ is not on the boundaries of nn;1 ; . . ., and nn;4n , then the function (k) ðkÞ K(k) equals n (a,Q) of Q equals zero except for a domain Dn nn1;l whose area Sn n n1 n1 1=4 ; 1=2 4 , or 1=4 .
n ðkÞ (1) If DðkÞ n ¼ nn;4l3 , and Sn ¼ 1=4 , then Z Z 1 n ½f ðaÞðkÞ ¼ 4 f ðQÞ dQ ¼ f ðQÞ dQ: n sðkÞ nn;4l3 DðkÞ n n n1 ðkÞ (2) If DðkÞ , then n ¼ nn;4l3 [ nn;4l2 , and Sn ¼ 1=2 4 Z Z 1 ½f ðaÞðkÞ 2 4n1 f ðQÞ dQ ¼ ðkÞ f ðQÞ dQ: n ¼ ðkÞ Sn DðkÞ Dn n n1 ðkÞ , then (3) If DðkÞ n ¼ nn1;l , and Sn ¼ 1=4 Z Z 1 ðkÞ n1 4 f ðQÞ dQ ¼ ðkÞ f ðQÞ dQ ½f ðaÞn ¼ Sn DðkÞ nn1;l n
or n1 ½f ðaÞðkÞ n ¼4
"Z
Z
Z
3f ðQÞ dQ þ nn;4l3
Z
f ðQÞ dQ þ nn;4l2
f ðQÞ dQ nn;4l1
# f ðQÞ dQ
nn;4l
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¼
1 SnðkÞ
Z
f ðQÞ dQ þ 2 4n1
"Z
DðkÞ n
Z f ðQÞ dQ
nn;4l3
# f ðQÞ dQ :
nn;4l
In any cases, by the continuity of f, we can conclude lim ½f ðaÞðkÞ n ¼ f ðaÞ:
n-1
This equality also holds when a is on the boundary, the argument is similar. Thus, we establish: Theorem 3. If f 2 CðnÞ, then lim Jf ðPÞ½f ðPÞðkÞ n J1 ¼ 0:
n-1
4. The improved Walsh functions over triangular domains 4.1. The Walsh functions W in Paley order In order to improve Theorem 1, we define the Rademacher functions f by f2n ðu; v; wÞ ¼ ð1Þvnþ1 þwnþ1 ; f2nþ1 ðu; v; wÞ ¼ ð1Þunþ1 þwnþ1
ðn ¼ 0; 1; 2; . . .Þ:
With the help of the f, the Walsh functions W are defined by W0 ¼ 1 and W n ¼ fn1 fn2 . . . fnr for n ¼ 2n1 þ 2n2 þ þ 2nr , where the integers ni are uniquely determined by niþ1 oni . The value of above functions at any discontinuous point is defined by the corresponding averages. Clearly, our definition is equivalent to the definition given in [9] with the relations R1ðpÞ ¼ f2p ;
R2ðpÞ ¼ f2pþ1 ;
R3ðpÞ ¼ R1ðpÞ R2ðpÞ ¼ f2p f2pþ1
ðp ¼ 0; 1; 2; . . .Þ:
Lemma 4. f2nþ2 ðPÞ ¼ f2n ðQÞ; f2nþ3 ðPÞ ¼ f2nþ1 ðQÞ for P 2 ni , where QP; i ¼ 1; 2; 3; 4. Proof. Assume for the moment that P 2 n1 . Let (u,v,w) be the area coordinate of P with respect to n, and in binary notation we have u ¼ 0:0u2 u3 . . . ;
v ¼ 0:0v2 v3 . . . ;
w ¼ 0:0w2 w3 . . .
since P 2 n1 . It is clear that the area coordinate of Q with respect to n, that is, the area coordinate of P with respect to n1 , is (12u,12v,12w), and then in binary notation 12u ¼ 0:ð1u2 Þð1u3 Þ . . . ;
R.-h. Wang, W. Dan / Journal of the Franklin Institute 347 (2010) 1782–1794
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12v ¼ 0:ð1v2 Þð1v3 Þ . . . ; 12w ¼ 0:ð1w2 Þð1w3 Þ . . . : Hence f2nþ2 ðPÞ ¼ ð1Þvnþ2 þwnþ2 ¼ ð1Þð1vnþ2 Þþð1wnþ2 Þ ¼ f2n ðQÞ; f2nþ3 ðPÞ ¼ ð1Þunþ2 þwnþ2 ¼ ð1Þð1unþ2 Þþð1wnþ2 Þ ¼ f2nþ1 ðQÞ: The argument is similar to the above process when i=2,3 or 4. This completes the proof. & Lemma 5. f2n ¼
1 ð1Þ ð2Þ ð4n Þ n ðI nþ1 þ I nþ1 þ þ I nþ1 Þ; 2
f2nþ1 ¼
1 ð4n þ1Þ ð4n þ2Þ ð24n Þ n ðI nþ1 þ I nþ1 þ þ I nþ1 Þ 2
ðn ¼ 0; 1; 2; . . .Þ:
Proof. A direct calculation shows that f0 ¼ I ð1Þ 1 ;
f1 ¼ I ð2Þ 1 :
Suppose that the lemma holds for nrN. For n=Nþ1 and P 2 ni ði ¼ 1; 2; 3; 4Þ, by Lemma 4, the definition of the I and induction hypothesis, we have 1 ð2Þ ð4N Þ f2ðNþ1Þ ðPÞ ¼ f2N ðQÞ ¼ N ðI ð1Þ Nþ1 ðQÞ þ I Nþ1 ðQÞ þ þ I Nþ1 ðQÞÞ 2 1 ðiÞ ð4Nþ1 4þiÞ ¼ Nþ1 ðI Nþ2 ðPÞ þ I ð4þiÞ ðPÞÞ Nþ2 ðPÞ þ þ I Nþ2 2 1 ð2Þ ð4Nþ1 Þ ¼ Nþ1 ðI ð1Þ Nþ2 ðPÞ þ I Nþ2 ðPÞ þ þ I Nþ2 ðPÞÞ 2 and 1 ð4N þ1Þ ð4N þ2Þ ð24N Þ f2ðNþ1Þþ1 ðPÞ ¼ f2Nþ1 ðQÞ ¼ N ðI Nþ1 ðQÞ þ I Nþ1 ðQÞ þ þ I Nþ1 ðQÞÞ 2 Nþ1 Nþ1 1 ð4Nþ1 þiÞ þ4þiÞ 4þiÞ ¼ Nþ1 ðI Nþ2 ðPÞ þ I ð4 ðPÞ þ þ I ð24 ðPÞÞ Nþ2 Nþ2 2 Nþ1 1 ð4Nþ1 þ1Þ ð4Nþ1 þ2Þ Þ ¼ Nþ1 ðI Nþ2 ðPÞ þ I Nþ2 ðPÞ þ þ I ð24 Nþ2 ðPÞÞ: 2 Therefore, the lemma holds for n=Nþ1, and this finishes the induction. & We have the improved uniform convergence of the Walsh–Fourier series as follows: R P Theorem 6. If f 2 CðnÞ and fn ¼ n1 i¼0 ai W i , where ai ¼ n f ðPÞ W i ðPÞ dP, then lim Jf f2n J1 ¼ 0:
n-1
Proof. From Theorems 1 and 3, it obviously suffices to show that the functions ð4n Þ W 0 ; W 1 ; . . . ; W 22nþ1 1 can be expressed linearly in terms of the functions I 0 ; I ð1Þ 1 ; . . . ; I nþ1 .
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The definition of the W shows that W 1 ; . . . ; W 22nþ1 1 can be expressed by the product of f0 ; f1 ; . . . ; f2n . From Lemma 5, we obtain that W 1 ; . . . ; W 22nþ1 1 can be expressed by the sum of the products ðki1 Þ
c I i1
ðli Þ
ðki2 Þ
I i1 1 I i2
ðli Þ
ðkij Þ
I i2 2 I ij
ðli Þ
I ij j ;
ð1Þ
where c is a constant and 1rki r4i ; 4i þ 1rli r2 4i ; i ¼ 1; 2; . . . ; n þ 1; 1ri1 o has a constant value over the domain where I ðkÞ oij rn þ 1. Since I ðsÞ v u is different from zero provided that vou and the fact that for 1rs1 r4v1 ; 4v1 þ 1rs2 r2 4v1 8 < 1 I s2 þ4v1 if s s ¼ 4v1 ; 2 1 v ðs2 Þ 1Þ I ðs I ¼ 2v1 v v : 0 else; n
ð4 Þ then the product (1) must be a function of I 0 ; I ð1Þ 1 ; . . . ; I nþ1 multiplies a constant, and hence the theorem is proven. &
n
ð4 Þ Note: The functions X 0 ; X ð1Þ 1 ; . . . ; X nþ1 do not possess this property, that is, the functions
W 0 ; W 1 ; . . . ; W 22nþ1 1 cannot be expressed linearly in terms of the functions X 0 ; X ð1Þ 1 ;...; n
ð4 Þ X nþ1 .
4.2. The Walsh functions H in Hadamard order The Walsh functions H in Hadamard order are generated when the standard Kronecker product of the elementary Hadamard matrix H4 is performed with itself: 0 1 1 1 1 1 B 1 1 1 1 C B C H4 ¼ B C @ 1 1 1 1 A 1
1 1
1
and HN ¼ HN=4 H4 ; where N ¼ 4n ; n ¼ 1; 2; 3; . . . ; # denotes the Kronecker product. The Hadamard matrix ð1Þ ð4n 1Þ H4n corresponds to the Walsh functions Hð0Þ : n ; Hn ; . . . ; Hn
n ¼ 1;
Hð0Þ 1
2 þ
þ
þ
þ
Hð1Þ 1 Hð2Þ 1 Hð3Þ 1
2 þ
þ
2 þ
þ
2 þ
þ
where we omit 1 in these elements of the Hadamard matrix.
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2 3
1
4 (2)
(1)
(0)
H1
H1
H1
(3)
H1
Fig. 1.
Figs. 1 and 2 show the Walsh functions associated with the Hadamard matrix. Black and white areas represent 1 and 1, respectively. The triangle above in Fig. 1 shows a certain order.
n ¼ 2;
Hð0Þ 2
2
þ þ
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Hð1Þ 2 Hð2Þ 2 Hð3Þ 2 Hð4Þ 2 Hð5Þ 2 Hð6Þ 2 Hð7Þ 2 Hð8Þ 2 Hð9Þ 2 ð10Þ H2 H2ð11Þ H2ð12Þ H2ð13Þ H2ð14Þ H2ð15Þ
2
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Our next theorem dealing with the relation between the W and the H has an exact analogue in the one-dimensional case [17]. Theorem 7. Suppose 0rkr4n 1, and write k ¼ k2n1 22n1 þ k2n2 22n2 þ þ k1 2þ k0 , where ki 2 f0; 1g; i ¼ 0; 1; . . . ; 2n1. Set k0 ¼ k0 22n1 þ k1 22n2 þ þ k2n2 2þ k2n1 , then HðkÞ n ¼ W k0 .
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(1)
H2
(2)
H2
(4)
H2
(5)
H2
(6)
H2
(8)
H2
(9)
H2
(10)
H2
(12)
H2
(13)
H2
(14)
H2
H2
(3)
H2
(7)
H2
(11)
H2
(15)
H2
Fig. 2.
Proof. If j ¼ j2n1 22n1 þ j2n2 22n2 þ þ j1 2 þ j0 , then the element hkj of the Hadamard matrix H4n , that is, the value of HðkÞ n over nn;jþ1 , can be expressed as P2n1 hkj ¼ ð1Þ s¼0 ks js : Q On the other hand, we have W k0 ¼ ki ¼1;i2f0;1;...;2n1g f2n1i from the definition of the W. In order to prove the theorem, we only need to show that P2n2lþ1 Y f2n1i ðPÞ ¼ ð1Þ s¼2n2l ks js ð2Þ ki ¼1;i2f2n2l;2n2lþ1g
for l ¼ 1; 2; . . . ; n and P 2 nn;jþ1 . Let (u,v,w) be the area coordinate of P with respect to n, and in binary notation u ¼ 0:u1 u2 u3 . . . ;
v ¼ 0:v1 v2 v3 . . . ;
We notice the fact that 8 ð0; 0; 0Þ > > > > < ð1; 0; 0Þ ðu1 ; v1 ; w1 Þ ¼ ð0; 1; 0Þ > > > > : ð0; 0; 1Þ
w ¼ 0:w1 w2 w3 . . . :
if j2n1 ¼ 0 and j2n2 ¼ 0; if j2n1 ¼ 0 and j2n2 ¼ 1; if j2n1 ¼ 1 and j2n2 ¼ 0; if j2n1 ¼ 1 and j2n2 ¼ 1:
If k2n1=1 and k2n2=1, then f0 ðPÞ f1 ðPÞ ¼ ð1Þv1 þw1 ð1Þu1 þw1 ¼ ð1Þu1 þv1 ¼ ð1Þj2n2 þj2n1 :
R.-h. Wang, W. Dan / Journal of the Franklin Institute 347 (2010) 1782–1794
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If k2n1=1 and k2n2=0, then f0 ðPÞ ¼ ð1Þv1 þw1 ¼ ð1Þj2n1 : If k2n1=0 and k2n2=1, then f1 ðPÞ ¼ ð1Þu1 þw1 ¼ ð1Þj2n2 : If k2n1=0 and k2n2=0, then the equality is trivial. Therefore, (2) holds for l=1. Similarly, we can prove (2) for l ¼ 2; 3; . . . ; n. The theorem is true. & 5. Conclusion This paper deals with piecewise constant orthogonal functions over nested triangular domains. Firstly, we have constructed the improved Haar functions and proved the uniform convergence of the corresponding Haar–Fourier series. Secondly, the Walsh functions in Paley order and Hadamard order have been designed, respectively. Based on the relation between the improved Haar and Walsh functions, we have shown the uniform convergence of the Walsh–Fourier series. Finally, the relation between the Walsh functions in the two different orders has been proposed. Acknowledgements The authors would like to thank the reviews for their helpful comments and suggestions which have improved the presentation of the paper. This work was supported by the National Natural Science Foundation of China (Nos. U0935004, 11071031, 11001037, 10801024 and the Fundamental Research Funds for the Central Universities (Nos. DUT10ZD112, DUT10JS02. References [1] N. Ahmed, K.R. Rao, Orthogonal Transforms for Digital Signal Processing, Springer-Verlag, Berlin, 1975. [2] K.G. Beauchamp, Walsh Functions and their Applications, Academic Press, London, New York, San Francisco, 1975. [3] C.F. Chen, C.H. Hsiao, A Walsh series direct method for solving variational problems, J. Franklin Inst. 300 (1975) 265–280. [4] C.F. Chen, Y.T. Tsay, T.T. Wu, Walsh operational matrices for fractional calculus and their application to distributed systems, J. Franklin Inst. 303 (1977) 267–284. [5] Y.Y. Feng, D.X. Qi, On the Haar and Walsh systems on a triangle, J. Comput. Math. 1 (1983) 223–232. [6] A. Haar, Zur Theorie der orthogonalen Funktionen systeme, Math. Ann. 69 (1910) 331–371. [7] H.F. Harmuth, Transmission of Information by Orthogonal Functions, second ed., Springer-Verlag, Berlin, Heidelberg, New York, 1972. [8] R.E.A.C. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932) 241–279. [9] D.X. Qi, Walsh function on the triangular area, Chinese Sci. Bull. 33 (1988) 715–716. [10] D.X. Qi, New definition for a kind of self-similar structure and Walsh function, Chinese Ann. Math. Ser. A 12 (1991) 103–105. [11] G.P. Rao, Piecewise Constant Orthogonal Functions and their Application to System and Control, SpringerVerlag, Berlin, 1983. [12] B.G. Sloss, W.F. Blyth, A priori error estimates for Corrington’s Walsh function method, J. Franklin Inst. 331 (1994) 273–283.
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[13] B.G. Sloss, W.F. Blyth, A Walsh function method for a non-linear Volterra integral equation, J. Franklin Inst. 340 (2003) 25–41. [14] G. Szego, Orthogonal Polynomials, American Mathematical Society, New York, 1937. [15] R.H. Wang, Numerical Approximation, Higher Education Press, Beijing, 1999. [16] J.L. Walsh, A closed set of normal orthogonal functions, Am. J. Math. 45 (1923) 5–24. [17] W.X. Zheng, W.Y. Su, F.X. Ren, Walsh Function Theory and its Application, Shang Hai Science and Technology Press, Shang Hai, 1983.