fgk_2011

Communicated by

P. Sablonniere

ReceivedNovember 17, 2010AcceptedJuly 12, 2011

onApproximation

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c 2011 Universidad de JaenISSN: 1889-3066

Jaen J. Approx. 3(1) (2011), 117{133

Unconditional convergence of wavelet

expansion on the Cantor dyadic group

Yuri Farkov, Ushangi Goginava and Tengiz Kopaliani

Abstract

In this paper we prove that wavelet expansions on the Cantor dyadic groupG converge unconditianally in the dyadic Hardy space H1 (G). We will do it forwavelets satisfying the regularity condition of Holder-Lipshitz type.

Keywords: wavelet expansion, Cantor dyadic group, unconditional conver-gence.

MSC: Primary 42C40; Secondary 42B30.

x1. IntroductionLet us recall denitions of the dyadic eld F and the Cantor dyadic group G. Denoteby F2 the eld of order 2, with elements f0; 1g. Then the dyadic eld F is the subset ofQ

j2Z F2 consisting of sequences

x = (xj) = (: : : ; x2; x1; x0; x1; x2; : : : );

for which xj ! 0 as j ! 1. Addition on F is the coordinate-wise addition modulo 2 :(zj) = (xj) + (yj) () zj = xj + yj (mod 2);

117

118 Yuri Farkov, Ushangi Goginava and Tengiz Kopaliani

while multiplication on F follows the rule

(zj) = (xj) (yj) () zj =Xl+k=j

xlyk (mod 2):

Denote by the zero sequence in F. We see that for each x 2 F with x 6= there exists s(x) 2 Zsuch that

xs(x) = 1 and xj = 0 for j < s(x):

There is a non-Archimedean norm on F. Indeed, set kk = 0 and for each x 2 F with x 6= setkxk = 2s(x). Then

kx+ yk maxfkxk; kykg and kx yk = kxkkykfor x; y 2 F. The Cantor dyadic group G can be dened as the additive group of the dyadic eld F,with the topology induced by k k. See Hewitt and Ross [8], Schipp, Wade, Simon [15], Taibleson [16],Golubov, Emov, Skvortsov [6] for development of the harmonic analysis on G.

The setsUl := f(xj) 2 G j xj = 0 for j < lg; l 2 Z;

form a complete system of neighbourhoods of and they satisfy the following properties:

Ul+1 Ul,TUl = fg,

SUl = G;

each Ul is a compact open subgroup of G; each Ul is homeomorphic to the Cantor ternary set.The group G is a locally compact Abelian group. Denote by the Haar measure on G normalized

so that (U) = 1 where U := U0. As usual, Lp(G), 1 p

Unconditional convergence of wavelet expansion on the Cantor dyadic group 119

For any nonzero a, the multiplication-by-a map is an automorphism of G, with adjoint also multiplica-tion-by-a. Let be the automorphism of G which coincides with the multiplication-by-e1 (i.e.,(x)j = xj+1 for x = (xj) 2 G). Notice that takes U to the larger subgroup U1 and thatl(U) = Ul for all l 2 Z.

For any function f 2 L1(G) \ L2(G) the Fourier transform bf , dened bybf(!) = Z

G

f(x)(x; !) d(x); ! 2 G;

belongs to L2(G). The Fourier operator

z : L1(G) \ L2(G)! L2(G); zf = bf;extends uniquely to all functions in L2(G).

As usual, let R+ = [0;+1). We dene a map : G! R+ by

(x) =Xj2Z

xj2j1; x = (xj) 2 G:

Take in G a discrete subgroup H = f(xj) 2 G j xj = 0 for j 0g. The image of H under is theset of non-negative integers: (H) = N. For every s 2 N, let h[s] denote the element of H such that(h[s]) = s: Note that h[1] = e1. As in Schipp, Wade, Simon [15] we set jxj := (x) for all x 2 G.

The Walsh functions on G can be dened by

Ws(x) = (x; h[s]); x 2 G; s 2 N:

It is well known that fWsg is an orthonormal basis in L2(U).We recall that a collection of closed subspaces Vj L2(G), j 2 Z, is called a multiresolution

analysis (an MRA) in L2(G) if the following hold:(i) Vj Vj+1 for all j 2 Z ;(ii)

SVj = L

2(G) andTVj = f0g;

(iii) f() 2 Vj () f( ) 2 Vj+1 for all j 2 Z ;(iv) f() 2 V0 =) f(+ h) 2 V0 for all h 2 H;(v) there is a function ' 2 L2(G) such that the system f'(+h) j h 2 Hg is an orthonormal basis

of V0:The function ' in condition (v) is called a scaling function in L2(G):


For arbitrary ' 2 L2(G) we set'j;h(x) := 2

j=2'(jx+ h); j 2 Z; h 2 H:We say that a function ' generates an MRA in L2(G) if the family f'(+h) j h 2 Hg is an orthonormalsystem in L2(G) and, in addition, the family of subspaces

Vj = closL2(G)span f'j;h jh 2 Hg; j 2 Z;is the MRA in L2(G). If a function ' generates an MRA in L2(G), then it is a scaling function inL2(G). In this case the system f'j;h j h 2 Hg is an orthonormal basis of Vj for every j 2 Z and onecan dene an orthogonal wavelet in such a way that f j;h j j 2 Z; h 2 Hg is an orthonormal basisof L2(G).

In the sequel, 1E stands for the characteristic function of a subset E of G.

Example 1.1. The Haar wavelet on G can be dened by

H(x) =

8


If a 6= 0, then ' generates an MRA in L2(G) ([11]). In particular, for a = 1 and a = 1 the Haarfunction 1U and the displaced Haar function 1U ( + h[1 ]) are obtained respectively. If 0 < j aj < 1,then ' can be written in the form

'(x) = (1=2)1U (1x)(1 + a

1Xj=0

bjW2j+11(1x)); x 2 G:

In this case,

(x) = a0 '(x+ h[1]) a1 '(x) + a2 '(x+ h[3]) a3 '(x+ h[2]):Also, when 0 < j bj < 1=2 the system f j; hg is an unconditional basis in all spaces Lp(G), 1 < p


It is known that if ac 6= 0 then f j; hg is an orthonormal basis in L2(G), and if ac = 0 then f j; hg isa Parseval frame for L2(G) (see [4, 5]).

The dyadic modulus of continuity of the scaling function ' satisfying the equation

'(x) =2n1Xs=0

as'(x+ h[s]) (1.1)

is dened by the equality

!('; ) := supfj'(x+ y) '(x)j : x; y 2 G; (y) 2 [0; )g; > 0:If ' satises !('; 2j) C2j , j 2 N, for some > 0, then there exists a constant C('; ) suchthat

!('; ) C('; ) : (1.2)Denote by ' the supremum for the set of all values > 0 for which inequality (1.2) holds. Accordingto [3], if n = 2 then ' = log2(1=jb j), with b as in Example 1.2. Recently, for the cases n = 3 andn = 4 some values of ' have been calculated (see [13]). In particular, for n = 3 we have ' = log2 bwhere

b =8


Meyer (see [2], [12] or [7]) proved that expansions in the wavelets on the real line convergeunconditionally in H1 (R) and Lp (R) (1 < p


Theorem 1.5. Let be an orthogonal wavelet in L2(G). Suppose that satises the regularityconditions (1.3) and (1.4). Then convergence of wavelet expansion is unconditional in all spacesLp (G), 1 < p 0 and f be a function dened on G such thatZG

f(x)d (x) = 0; (1.6)

ZG

jf(x)j2d (x)1=2

C1and

jf(x)j C2jxj(1+); x =2 U:Then f 2 H1(G) .Proof. Dene the functions

fn(x) =f(x) fn(U)

1n(U)(x); n = 0; 1; 2; ::;

where

fn(U) =1

jn(U)jZn(U)

f(t)d(t)

is the average of f on n(U).First, we prove that

kf fnk1 ! 0; (1.7)where k k1 := k kL1(G). Indeed, it is easy to show that


kf fnk1 kf fn(U)k1 + kfn(U)1n(U)k1:Since

kf f1n(U)kL1 =ZGnn(U)

jf(t)jd (t) C2ZGnn(U)

jxj1d (t)! 0 as n!1: (1.8)

On the other hand from (1.6) we obtain

fn(U)1n(U) =1n(U)

jn(U)jZGnn(U)

f(t)d (t) ;

kfn(U)1n(U)k1 ! 0 as n!1: (1.9)Combining (1.8) and (1.9) we obtain (1.7). Consequently, we can write

fL1= f0 +

1Xn=1

(fn fn1):

Now we prove that (2C)12n (fn+1 fn) is an atom, where the positive constant C will be denedbelow. It is evident that

supp(fn+1 fn) n+1(U)and Z

n+1(U)

(fn+1(x) fn(x))d (x) = 0:

Let Qn = n(U)nn1(U) for n = 1; 2; 3; :::: Then we have

fn+1 fn = f1Qn+1 fn+1(U)1n+1(U) + fn(U)1n(U): (1.10)

Also, it is easy to show that

jfn(U)j = 1jn(U)j

Zn(U)

f(t)d (t)

2nZGnn(U)

jf(t)jd (t)

C22nZGnn(U)

jxj1d (t) C212(1+)n: (1.11)


From (1.10) and (1.11) we obtain

kfn+1 fnk1 C22(1+)n + jfn+1(U)j+ jfn(U)j C22(1+)(n) + 2C212n(1+)= (21 + 1)C22n(1+) = C2n(1+): (1.12)

Consequently, the functions

1

C2n(1+)1

2n+1(fn+1(x) fn(x)) = (2C)12n(fn+1(x) fn(x))

are atoms.Now we prove that f0 2 H1. Note thatZ

U

f0(x)dx = 0

andkf0kH1(U) C 0kf0kL2(U) 2C 0C1:

Hencekf0kH1(G) kf0kH1(U) 2C 0C1: (1.13)

Combining (1.12) and (1.13) with the equality

f = f0 +1Xn=1

12C 2

n

12C 2

n(fn fn1)

we obtain

kfkH1(G) 2C 0C1 +1Xn=1

2C

2n C 00:

Lemma 1.6 is proved.

Corollary 1.7. Suppose that f is a function on G such thatZG

f(x)d(x) = 0;

ZG

jf(x)j2d(x)1=2

C12N=2

andjf(x)j C22Njx+ y0j(1+) for x 2 Gn

y0 +

N (U

for some > 0 and N 2 N. Then f 2 H1 and kfkH1 C:


Proof. Let g(x) = f(x+ y0): Then we haveZG

g(x)d(x) = 0;

ZG

jg(x)j2d(x)1=2

C12N=2

and

jg(x)j C22Njxj(1+) for x =2 N (U):Applying Lemma 1.6 for the function 2Ng(N (x)) we obtain that kfkH1 C which completes theproof.

x2. Proof of the Main ResultProof of Theorem 1.4. Let = fj;k; j 2 Z; k 2 Hg be a sequence of numbers such that j;k 2f1; 1g : Denote

Tf (x) :=X

(j;k)2ZHj;k < f; j;k > j;k (x) ;

where, as before,

j;k (x) = 2j=2

jx+ k

:

We prove that there exists a constant C independent on and f 2 H1(G) such that kTfkH1 CkfkH1 : It is clear from the denition of H1(G) that the proof of Theorem 1.4 will be complete if forany dyadic atom a the following inequality

kTakH1 c N ; j(k) 2 2Qg and A3 = f(j; k) : j > N ; j(k) =2 2Qg:


We can write

Ta (x) =

0@ X(j;k)2A1

+X

(j;k)2A2+

X(j;k)2A3

1A (j;k < a; j;k > j;k (x))= I + II + III: (2.3)

It is easy to show that

j< a; j;k >j =ZQ

a(x) j;k(x)dx

= ZQ

a(x)( j;k(x) j;k(y0))dx

kakL1 supx2Q

j j;k(x) j;k(y0)j supx2Q

j j;k(x) j;k(y0)j: (2.4)

For any x; y0 2 Q we have

j(j(x) + k) + (j(y0) + k)j 2j jx+ y0j 2j (Q) :

Thus from (1.4) it follows that

j j;k(x) j;k(y0)j 2j=22j (Q)minf(1 + jj(x) + kj)1; (1 + jj(y0) + kj)1g: (2.5)

Since j N we havejj(x) + j(y0)j 1

and

supx2Q

(1 + jj(x) + kj)1 C(1 + jj(y0) + kj)1: (2.6)

Using the fact

k j;kkH1 = 2j2j=2k kH1 = 2j=2C; (2.7)from (2.4)-(2.7) we obtain

kIkH1 XjN

Xk2H

j < a; j;k > j k j;kkH1

C (Q) XjN

Xk2

2j=22j2j=2 supx2Q

1

(1 + jj(x) + kj)1+


C (Q) XjN

2jXk2

C

(1 + jj(y0) + kj)1+

C (Q) XjN

2j C: (2.8)

Now, let us estimate III: From (1.3) we have

j< a; j;k >j =ZQ

a(x) j;kd (x)

1 (Q)ZQ

2j=2(1 + jj(x) kj)1d(x):

Using the last inequality, from (2.7) we get

kIIIkH1 X

(j;k)2A3j < a; j;k > j k j;kkH1

CXj>N

Xj(k)=22Q

2j=22j=21

(Q)

ZQ

(1 + jj(x) + kj)1dx

CXj>N

1

(Q)

ZQ

Xk: j(k)=22Q

(1 + jj(x) + kj)1dx:

Since x 2 Q and j(k) =2 2Q we have jj(x) + kj 2j (Q) andXk: j(k)=22Q

(1 + jj (x) + kj)1 CX

k: k>2j(Q)

jkj1 C(2j (Q)):

Consequently,

kIIIkH1 CXj>N

1

(Q)

ZQ

C(2j (Q)) = C ( (Q))Xj>N

2j = C: (2.9)

Finally, we estimate II: Note that

kIIkL2 kakL2 (Q)1=2 = 2N=2 (2.10)and Z

Q

II = 0: (2.11)


On the other hand, from (1.3) we have the following pointwise estimate

jIIj CXj>N

Xk: j(k)22Q

j < a; j;k > j2j=2(1 + j2jx kj)1

0@Xj>N

Xk: j(k)22Q

j < a; j;k > j21A1=20@X

j>N

Xk: j(k)22Q

2j(1 + jj(x) + kj)221A1=2 :

The rst factor is at most kakL2 (Q)1=2 and we have

jIIj (Q)1=20@Xj>N

Xk: j(k)22Q

2j(1 + jj(x) + kj)221A1=2 :

Note that for xed j > N there are at most C2jN points k 2 H such that j(k) 2 2Q: On theother hand x =2 4Q and j(k) 2 2Q imply

jj(x) + kj = 2j jx+ j(k)j C2j jx+ y0j:Thus

II C(Q)1=20@Xj>N

2jN2j(2j jx+ y0j)221A1=2 (2.12)

Cjx+ y0j1((Q)1Xj>N

2N22j)1=2

Cjx+ y0j10@Xj>N

22j

1A1=2 Cjx+ y0j12N :

From (2.10)-(2.12) and Corollary 1.8 we have

kIIkH1


Acknowledgements

The authors would like to thank the referee for helpful suggestions.

References

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[10] Lang W. C. (1998)Fractal multiwavelets related to the Cantor dyadic group, Internat. J. Math. Math. Sci. 21(2),307{314.

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[14] Rodionov E. A. and Yu. A. Farkov (2009)Estimates of the smoothness of dyadic orthogonal wavelets of Daubechies type, Mat. Zametki86(3), 429{444 (English version: Math. Notes 86(3-4) (2009), 407{421).

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Yuri Farkov,Russian State Geological Prospecting University,23 Ulitsa Miklukho-Maklaya,Moscow 117997, [email protected]


U. Goginava and T. Kopaliani,Institute of Mathematics,Faculty of Exact and Natural Sciences,Tbilisi State University, Chavchavadze str. 1,Tbilisi 0128, [email protected]

[email protected]

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