]T J j l + I2’* - *l)“ 2,

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Sena III: MATEMATYKA STOSOWANA XXXVIII (1995)

Te r e s a Re g i ń s k a

Warszawa

Function representation in wavelet bases

(Received 10.05.199Ą )

A b str a c t. The paper concerns wavelet bases in the space L2(M). We give a review of elements of the wavelet theory selected from the point of view of numerical analysis applications. Our purpose is to show properties of a function representation in a wavelet basis, among others, to formulate criteria of global and local function smoothness in terms of its wavelet decomposition coefficients.

1. Introduction. Let us take into account the Haar function w{x) (Fig. 1)

f l for x G [0, | ), (F I) w(x) = < _ i fo r .r6 [| ,l),

v 0 otherwise.

The set of functions {wj,k}j,ke% obtained from this single function w by the following formula

Wjtk(x) = 2*w{2j x — k), for any j,k G Z

forms a basis (Haar basis) of the space L2(K.) of square integrable functions on R. It is easy to verify that J^Wjyk(x')wi!n(x)dx = SjiS^n (where 6 is the Kronecker symbol), i.e., this basis is orthonormal. In order to show an example of a representation of a function in the Haar basis let us take the

“hat” function f ( x ) (Fig. 2) given by the formula

(1.2) f(r) = ! 1 “ M for G I- 1 ’ 1!’

^ [0 for x g [—1,1].

Computing coefficients tVj/. = f{x )iv hk{ x)dx we obtain the series rep-

Fig. 1 Fig. 2

reservation in the form

— 1 co 2 ' — 1 — 1

(1.3) / = 2 ^ [wj-,0 - + Y2 2“ ^ * [ Y , '

j ——oo j —0 k=0 Ai= — 2J

Taking into account only finite numbers of components in the above sum we get different approximations of the “hat” function. In Fig. 3a there is shown a plot of the finite sum for j from -1 to 0 (denoted by /( —1, 0, .?))• The sum for j from -2 to 1 is presented in Fig. 3b. The hat function is marked by dotted line.

The Haar basis is the first example of a wavelet basis known in the function theory many years before the wavelet notion appeared. Generally, under the notion of wavelet basis we understand a basis generated by a single function ip E X2(R) by its binary dilations and integer translations, i.e., basis of the form

(1.4) 'ipjk(x) = 2^ip(2^x — k), for any j,k E Z.

If the function ip E iT (R ) and generates a wavelet basis, then it can be proved (cf. [1])

OO

(1.5) j ip(x)dx = 0.

— CO

The existence of smooth wavelet bases with a compact support was not self- evident. The concept of the multiresolution analysis formulated by Mall at and Meyer in 1986 stimulated a very intensive development of the wavelet, theory. The multiresolution scheme allowed to construct wavelet basis with required properties such as smoothness, compactness of the support of 0 or its Fourier transform ip, etc.

At the beginning, wavelet transforms were applied to the signal analy- sis. Together with the progress in the wavelet theory there appeared new applications, among others in numerical analysis.

Function representation in wavelet bases

Fig. 3a

Fig. 3b

In this paper, we summarize some elements of the wavelet theory se- lected from the point of view of numerical analysis applications. The start- ing point is an approximation of L2{ R) space. After preliminary remarks on the bases (Part 2) we introduce, according to Meyer [5], so called mul-

tiresolution approximation and illustrate it by simple examples (Part 3). For regular approximations the interesting estimates for seminorms of functions from considered subspaces are quoted in Part 4.

A multiresolution approximation generates a wavelet basis (Part 5). We consider an orthonormal wavelet basis as well as the semi-orthogonal one.

Relations between regularity of a semi-orthogonal bases and its dual are considered. More details can be found in monographs [1], [5].

In Part 6, following methods presented in [4], we show how the co- efficients of function decomposition in the wavelet basis allow us to esti- mate smoothness of the function. It concerns the global as well as the lo- cal smoothness. The resrdts are formulated under assumptions weaker than those in [4].

2. Bases. In the real space X2(0,27t) we can take the set of functions {1, cos nt, sin n t}^ ^

as a. basis. Any function / from L'2(0,27t) has a Fourier series representation f(t) = -bo + J ^ [a n sin nt + bn cos nt]

n— 1 with the coefficients given by the formulas

an = — I f{t) sin ntdt 1 n = 1,2, . . . , 7T J 'o

] 2f

bn = — I f{t) cos ntdt TT J n — 0,1,2,..

and the series is convergent in X2(0, 2t t).

From the well known integral formulas 27T

J sin nt cos mtdt = 0 for n,m = 0,1, . . . , 2^tt o

j sin nt sin mtdt o

2ir

j cos nt cos mtdt = o

0 when n / m 7T when n = m

for n, m = l, 2, . .., it follows that the Fourier basis is orthogonal. In such a case it is very easy to get a relation between the norm of / £ X2(0,27r) and the norm of the sequence {6o, { am in the space I2 (Plancherel theorem):

l | /l l l 2m,27r) = ^ + X ^ fl" + ■

' n=l '

Generally, for arbitrary orthonormal basis of the space L2(Q).

we have Parseval equality, i.e., V / £ L2(Q)

ii/Hl2(q) - XZ i ° - ’ 71 — 1 where / = a n0n-

Often, for many reasons, nonorthogonal bases are used, but then usually it is assumed that 3Ci,C2 < oo, such that V / £ X2(0 )

OO OO CO

(2.1) Ci |on|2 < ||/||2 < C2 |o'n|2, for / = XZ ttn<^n

n = l n = l n= l

where {4>n} is a basis. A basis satisfying (2.1) is called a Riesz basis. Only such bases will be considered here.

Let {(f>k}kLi be a given Riesz basis of X2(fl). Then we can approximate a function / £ L2(Q) calculating finite number of coefficients of this basis

Function representation in wavelet bases

representation of / (for instance Fourier coefficients for trigonometric basis) and taking

j

fj - X / a k<Pk- k=\

as an approximation of / . Since V / E L2(£l) ||/ — fj\\ — 0 if' j —- oo, such an approximation is convergent, but the convergence may be very slow.

Let

Xj = span{(j)kyk=1.

In this way we define an infinite sequence of subspaces Ah C AA C . ■ • C Z2(ft),

for which (J A ; = Z2(fl). If the basis is orthogonal, then fj de- fined above is the orthogonal projection of / onto the subspace X j. In this construction computational convenience consists in the fact that for get- ting a better approximation of / all previously calculated coefficients can be used. Namely, if an approximation of / is given by a sequence of j numbers o A. k = 1, . . . , jf, which are coefficients of / in the basis, then in order to ob- tain j -f 1 approximation it is sufficient to compute and add to the previous sequence the coefficient Q ,-)-i, only.

3. Multiresolution approximation o f Z2(R ). In numerical methods we often deal with sequences of subspaces {F j} which approximate L2 in the following sense:

V f E L2 inf ||/ - v\\L 2 — 0 as j —►v£Vj oo,

where infu£Vj \\f — l»|| is called the error of approximation of / in Vj while the function fj realizing this inhmum is called the best approximation of / in Vj.

The simplest example of such an approximation of Z 2(R ) is the sequence of subspaces Vj,

(3.1) Vj = { / G Z2(M); f\[kh3,(k+i)h}) = const Vfc G Z }

corresponding to the sequence hj — 0 as j oo. A suitable choise of /?/

gives the nested sequence of spaces . . . , Fj- C Fj+i C . . . From now on we put hj = 2~Ń An example of a piecewise constant function from Fq is given in Fig. 4.

The natural basis of Fj is the set of characteristic functions {4>j,k}kei of the intervals [k‘2~:\ (k + 1 )'2~] ) multiplying by normalization constants

<f>jM*) = ^~x('2Jx - k) k G Z,

Fig. 4

where \ is the characteristic function of [0,1), i.e.,

\ ( - c ) 1 for xE [0,1), 0 for x<£ [0,1).

It is the orthonormal basis of Vj, since for any fixed j J_^°. <j>j,k{%)4>j,i[x)dx = Sk{. Nevertheless, the set

is not a basis of L2{M) since the set is too numerous - its elements are linearly dependent (for example (f)jjk = -^ [ ę J+i;2k ! <^j+i,2*+i])-

The characteristic property of the above approximation of L2{R) is that the basis of every subspace Vj is obtained by binary dilations and transla- tions of the one function - the characteristic function of the interval [0,1).

Such a type of approximations will be called multiresolution approximation according to the following definition:

Defi nit io n 1. A multiresolution approximation of the space h2(K.) is an increasing sequence of closed linear subspaces { V)} satisfying the fol- lowing properties:

1) Vi € Z V C VJ+1- n _ « VJ = {0}; U -,* Vj is dense in I 2(R);

2 ) V j £ Z / G V ?- 4=> / ( 2 - ) G V j + i ;

3) V k e Z f e V o & /(•-/,•) G v„;

4) 3g 6 Vo such that the family {gk}kez, 9k{%) = fj{x - k), is a R.iesz basis of Vo.

It should be stressed that the first assumption means that { Vy} /es is nested and convergent approximaton of Z2(R) in the sense described at the begining of this section. Of course, all these assumptions are satisfied by the sequence of the subspaces defined by (3.1) where ry = y.

Function representation in wavelet bases 9 Since for any fixed j Vj C Vj+1, we can introduce a subspace Wj accord- ing to the following definition:

(3.2) Wj = { / G Vj+1 : j f(x)g(x)dx = 0 Mg £ L j}.

Obviously, Wj is the orthogonal complement of Vj in Vj+i, i.e.

Vj ® Wj = hj'+i l°r j £ Z.

Thus, Vj > jo we have

Vj ~ VjQ © Wjo © • • • © W j-1 and

( 3 .3 ) I 2 ( R ) = ® W ' j .

j e z

In the case of piecewise constant approximation (3.1) it is easy to find that any subspace Wj is spanned by the functions {vĄ2Jx + k )}^ ^ which are generated by the Haar function w (the formula (1.1)).

For the proof it is enough to observe that the subspace IT = { f ( x ) = ^ a ' fcu;(r - k), |afc|2 < oo j

k 7L k£Z

is equal to LI©. Since w(x)

Moreover, for any k,l G Z = \(2r) — \(2.r — 1), thus w G Vj, and IF C Vj

CO k-\-1

j w{x - k )}©,r — l)dx = Sk,ł J w{X — k)dx = 0.

— CO k

thus V / G VI© V/? G Vo f{x)h (x)d x = 0, which means, that Vo ©IF the end observe that

,Y'(2.r) = + ^\'(Vj X(2.t - 1) = - ^ »’(:r) + ^ \ (x )

At

hence any element j j G Vj is a, sum of f 0 G Vo i w o G IT. Thus IT = IV©.

Similarly, for any j G Z the set {'Wj^x) = w{2Jx - k ) ) k(zi constitutes the basis of Wj. Due to (3.3) the family {wjkjjkez is a wavelet basis of Z2(R).

So, in this case we get the Haar basis.

Let us illustrate the decompositon I© = V_i © IV_|. The function from I© given in Fig. 4 is the sum of jj G V_i shown in Fig. 5a and j> G IT_i shown in Fig. 5b. Evidently, the scalar product ( j i , / 2) = 0. The function f -2 represents details corresponding to the scale h = 2°.

Let us return to the piecewise linear approximation of the hat func- tion considered in Section 1. Applying introduced notation we state that in Figures 3a and 3b we have the approximation of / in IT_i © IT0 and IV_2 © IT_i © LI© © ITi, respectively. To get better approximation of /

Fig. 5a Fig. 5b

more subspaces Wj should be taken, particularly for j from — oo to jo. It is realizable, since

- l

J - — ^OO

A drawback of the example of multiresolution approximation considered above is its low regularity. An approximation of a regular function by piece- wise constant ones does not give an approximation of its derivatives and the error of approximation decreases slowly. So, we want g to have some smoothness and a good localization.

Thus, among different multiresolution approximations of T2(M) it is use- ful to have classes of regular approximations. We introduce the following notation

F0 = {g 6 X°°(K) : Vm e N, 3Cm < oo |ff(*)| < C „( 1 + |.r| ) " m}

<3'4) Fr = {</ e W ^ (R ): <,<*> 6 F „,0 < .s < r}

Defi ni tio n 2. A multiresolution approximation of T2(R) is called r- regular for a certain r G N if there exists g E Fr such that its integer translations {g{x — k)}ke% forms a Riesz basis in Vo-

The assumption on a rapid decay of g as \x\ oo means that the function g is very small outside a short interval. In such a case we say it has a good localization. This assumption is satisfied for example by or by compactly supported functions. Of course functions of the form (c + dx)~k do not satisfy this assumption.

Looking for an example of regular multiresolution approximation we can take nested spaces of splines of order r. Let us consider the simplest case of piecewise linear approximation. Let

(3.5) Vo = { / e £2(R)nC’(R);Vk 6 z / |M+1) e Pi([k, k + 1))}

where Pi{0.) is the set of polynomials of degree 1 on D. If / E Vo and

Function representation in wavelet bases 11

f{k ) = a k, then

f{ x ) = akCJ(x ~ k)

&EZ

where g is the hat function given by (1.2)

The family of functions {</(• — k)}kei is linearly independent. Since

||/||“ = - ^][|Q'fc+il2 + la'fc|“ + Reak®k+i]

Ar£Z

and

— ( |oA;|" + |a'A;+i|2) < |a'A,-+i|“ + |afc|“ + R eaka k+i < iafc+il- )ł thus

< ii/n2 < I ] k i2

^ jeż jez

which means that the set {</(• — k )}k£i satisfies the condition (2.1), i.e., this is a Riesz basis of the space Vo.

Let us define subspaces F?, j / 0 according to the point 2) of Definition 1, i.e.

(3.6) Vj = { / G f 2( l ) n C ( l ) ;

VA; G Z f\[k2~j ,(k+i)2~i] £ -Pi([^2 2, (A: + 1)2 J])}-

Since g(x) - \g{2x + l)+ g (2 x ) + \g{2x - 1)., thus F0 C Vi and analogously

• • C V0 C Vi C • • ■ C f 2( l ) .

If / G n jezVj then 3a, b G 1 Vj G Z \fx G [0,2_J] f( x ) = ax + b. Hence / G X2(R ) iff a = b = 0. Density of Uj^iVj in X2(R ) follows from the fact that V / G X2(R) the sequence {fj}j£ Z ,fj G Vj given by the formula

(H -D 2 --1

fj = ^ a kg{23x - k), where a k = 2:l j f(x)d x

k E Z 2 — /

converges to / in the norm, i.e.,

11/ - f j11 < ^L2( i ) ( / ,2_J) — 0 as j — oo,

where l oj j i(®)(/, h) is the continuity modulus of / in X2(R). Thus, according to Definition 1 the sequence of subspaces (3.6) constitutes a multiresolution approximation of X2(R). Moreover, due to Definition 2, it is 1-regular ap- proximation since g as well as g' belong to X2(R ) and have compact support.

In this example a description of an orthonormal basis of the subspace W j is more complicated than in the previous case of piecewise constant approximation. The corresponding wavelets, called Battle-Lemarie wavelets, have infinite support.

Nevertheless, it can be proved (cf. [2], clip. 5) that there exists so called spline wavelet function ip with a. compact support such that a family {ipjk(%) = 22i/'(2ji.t — A-)}a:g s f°r any fixed j is an orthogonal basis of T-Tj, where W'- is another subspace of T } +i such that \ j+1 is a direct sum \}+ !¥ '.

The plot of ip is presented in Fig. 6. Of course, the set {ipjk}j,ke% 18 n°f an orhonormal basis of L2{R).

Fig. 6

In [1] there is described a construction of an orthonormal wavelet bases- with arbitrary smoothness and with compact support. The wavelets gener- ating such bases are called Daubechies wavelets.

The definition of a multiresolution approximation can be extended to the multidimensional case of L2(R n) and, moreover, to other function spaces. It was done in [5] for the spaces Xp(Mn) for p £ [1, oo],

4. An r-regular multiresolution approximation. In this part we present two results concerning estimates of seminorms of functions belong- ing to subspaces V) and Wj corresponding to r-regular multiresolution ap- proximation of L2{R). It should be mentioned that the results formulated below can be extended to multiresolution approximation of L2(R n) as well as of Lp{R n), for an exponent p £ [1, oo]. The detailed proofs of these results are contained in the Meyer’s monograph [5], clip. 2.

For r-regular multiresolution approximation the following inequality of Bernstein’s type holds:

Theo rem 1. If {Vj} jez is an r-regular multiresolution approximation of L2{R ), then 3C < oo such that Mj £ Z V / £ Vj Vo' < r

(4.1)

Proof (cf. [5], clip. 2 Th. 3)

< C'2aj\\f\\.

13 Function representation in wavelet bases

Let j = 0. A function / from the space F0 lias the form f(x ) ]Ta-€Zctkg(x — k) where g satisfies the condition (3.4). Hence, for a < r

dc

dx fix] < Y i"*/f £ Ej

dc

dx1■g{x - k) da fees

v . dc

A-es Y0ix ~ A')|

due to the Schwartz inequality. From localization property (3.4) with any fixed natural m > 2 we get

E

E

for a certain constant qm, since the sum appearing in the right hand side of the inequality is a 1-periodic continuous function, so it is bounded. Thus for m > 2

I dxc-fix,^dc dx < qm K |2 J k&

dc

dx zdix ~ k) dx E 2f/m ^ " I^A;|"" j

A’GS 0

1

(i + 3-r -dx 2 <h

m ^— 1 E l “ 4 -kei which gives (4.1) for j = 0, since {g{ • - &)}a:<=z by the assumption is a Riesz basis of F0.

The inequality (4.1) for j ^ 0 can be established by a simple change of variables. Indeed, if / £ Vn then f(x) ⁼ h{2Jx) for a certain h £ Lq. Thus there exists a constant C such that

O O O O O O

J \f'{x)\“dx = ^2:/ J \h'{y)\2dy < C22J J \h{y)\2dy

— OO — '>0 — OO

= C H 2i f |/(*)|2d*. A

— OO

In general, an inverse inequality to (4.1) is not true, i.e., does not exist, a constant C > 0 such that V / £ Vj ||^r/|| > C2ai\\f\\. Nevertheless, it was observed (cf. [5], clip. 2 Tli.7) that similar estimates from below can be obtained for seminorms of functions belonging to the subspaces Wj = Vj 0 V j-i. Such a result seems to be very important for different numerical applications.

Th e o r e m 2. If { V j } j ^ z is an r-regular multiresolution approximation of L2(R) and Wj is an orthogonal complement Vj in Vf+i, then 9C'i,(72,0 <

C\ < C-2 < oo independent of j such that VO < a < r V / G IF, Ci2“J||/ll < dxr < Co2aJ

The proof of this theorem based on a construction of wavelet basis is more complicated and can be found in [5], clip. 2, Tli.7.

5. Wavelet bases. For examples of multiresolution approximation con- sidered in section 3 one was able to find a function V’ such that 2H2ip(23x — k)

for k ^€ Z form a basis of ^V 7-+1© Vj (or W-), which means that {2J/ 2 ^(2J;r ^—

k)f ■jtkez is a wavelet basis of L2(R). It can be proved that any multires- olution approximation of L2(R) allows to construct corresponding wavelet bases of L2(R), among others an orthogonal basis.

Let {V jh e z be a multiresolution approximation of L2( R), and let the functions g(x — k) for k G Z form a R.iesz basis of Tfi. This basis can be

“orthogonalized” in the following way (cf.[5]): if the Fourier transform of a function <f> is given by the formula

(5.1) <£(£) = £ (£ )( Y + 2ttAj)|2) k= — oo

then <j)(x — k) for 1' G Z form an orthonormal basis of Fo. By a similar procedure we shall obtain a function ip generating an orthonormal wavelet basis.

Th e o r e m 3 . Let {V j}jez be r-regular multiresolution approximation of the space L2(R), and let {f>(x — k)} be an orthonormal basis of Vq defined by (5.1). Then there exists 27t periodic function uiq such that

m f ) = m0({)<frO.

If A __________

i i 2 f ) := + *)$(£),

then ip G L2(R), ip^l G L°°(R) for q < r, and the functions ipjkix) = 2:l/2ip(2Jx - k), j,k G Z,

form an orthonormal basis of L2(R) which has the following properties:

( 5 .2 ) dq

( 5 .3 )

Vm G N 3Cm such that

d x v ip(x) < Cm( \ -f |x

oo

J x lip(x)dx = 0 when / = 0 ,1, . . . , q.

The proof can be found in the monographs devoted to wavelets theory

[1, 2 , 6].

Function representation in wavelet bases 15 In the sequel we will apply the following definitions introduced in [2] (Def. 3 and 4) or in [5] (Def. 5).

De f i n i t i o n 3. A function ip G L2(R )fl X°°(R) is said to be an orthog- onal wavelet when the set of functions 2^ip(2^x — k), j,k E Z forms an orthonormal basis of X2(R).

A standard way to construct orthogonal wevelets is described in Theo- rem 3.

Now, let us assume that we have a wavelet ip such that the set of functions

■ipjf. = 2?ip[2j x — k), j, k E Z forms a noil-orthogonal R.iesz basis of X2(R).

Then there exists a second basis Ui U2(R) such that

OO

(5.4) j ipjk(x)4’ls(x)dx = SjiSks

— C O

The basis is called dual to {4>jk}jkez- Unfortunately, in general, the dual basis is not of a wavelet type (cf. [2]). But, there exists an important class . of wavelet bases for which their dual ones are wavelet bases, too. They are

generated by semi-orthogonal wavelets defined as follows

De f i n i t i o n 4. A function ip E X2(R )n X°°(R) is said to be a semi- orthogonal wavelet if the set of functions ^jk(x) = 2^ip(2^x — k ), j.k E Z is a R.iesz basis of X2(R) satisfying

OO

(5.5) J 4’j k{x)ipia(x)dx = 0 for j ^ l

— 'DC'

If 'ip is a semi-orthogonal wavelet, then it has a dual ip in the sense that the family ipjk{%) = 2j/2'0(2Xt — Ar) is biortliogonal to {'4>jk}jkEZ

O O ______________

J ipjk(x)ipis{x)dz = SjiSks-

It is proved ([2], Th. 3.25) that the Fourier transform of the function F is explicitly given by the formula

55.6) 4P0 = ^ (0

E r = - c o W ( +2ri-)p

In the next part we will assume certain regularity of considered semi- orthogonal wavelet such as differentiability, localization, and vanishing the moments, i.e.,

(5.7) i/Ud € X°° for 0 < q < r,

'5.8' dxqdq'ip[x) < Cm (1 + | .i*|) m for 0 < q < ?*, m E Z ,

(5.9) I x q tp( x )dx = O for 0 < q < r.

— CC'

De f i n i t i o n 5. A function 0 G L2(R )fl X°°(R) is said to be an orthogo- nal (semi-orthogonal) wavelet of class r if the set of functions 2? 0(2:/.r — k), j,k G Z forms an ort honorni al (semi-orthogonal) basis of L2( R) and 0 sat- isfies the conditions (5.7), (5.9) and for any m G Z there exists a constant Cm such that, (5.8) holds.

Existence of an orthogonal wavelet of class r follows from Theorem 3.

Such wavelets arise from r-regular multiresolution approximation. For exam- ple, for every r there exists the Battle-Lemarie wavelet of class r associated with multiresolution approximation consisting of spline function spaces of desired regularity (the exact definition can be found in [1] clip. 5). The case r = 1 is considered in Section 3.

Le m m a 1. If a semi-orthogonal wavelet 0 is of the class r then the dual wavelet 0 is of the class r, too.

P r o o f. The proof is based on the formula (5.6) and on the facts that YltL-oo + 27T/012 is bounded from above and below by positive con- stants (cf. [5], clip. 2 T li.l) and, moreover, it is a C 00-function ( [5], clip. 2 Lemma, 7).

Since 0 ^ G X2(R ) implies |£|90(£) G £2(R), thus, by (5.6), |£|90(£) G X2(R ). Therefore, 0^-* G X2(R) for 0 < q < r.

The assumption (5.9) is equivalent to 0 ^ (0 ) — 0 for 0 < q < r. Due to (5.6) and the differentiability of oo l'0(^ + 27rA?)|2, we easily find that, also ^70(0) = 0 for 0 < q < r, which means that r + 1 moments of 0 vanish

OO

(5.10) j x qip(x )dx = 0 for q = 0,1,. . ., r.

— OO

The proof of localization property (5.8) for 0 is the same as the proof of this property for <f>(x) in Theorem 2 in [5], clip. 2. A

6. Wavelet criteria of function smoothness. Let 0 be a given semi- orthogonal wavelet and 0 be its dual and let {'f jk}j,kei-> {f'jk}j,kez be the corresponding wavelet bases.

Let Cjk be coefficients of the representation of a function / G X2(R ) in the considered wavelet basis, i.e.,

f( x ) = Cjktfjk where cjk = (f,4 ’jk)•

jk£l (6.1)

17 Function representation in wavelet, bases

The error estimate of the approximation of / by f :j{x) = ■ YLkez cjk4'jk is expressed by c jk. namely

ll/-/j| |2< e ^ ^ M 2

./ > jo

Behaviour of wavelet coefficients is related to smoothness of the considered function, and moreover, the smoothness of / can be estimated in terms of decay estimates of its wavelet coefficients. For a function representation in an orthonormal wavelet basis such results were obtained in [4]. Here we extend these results onto the case of semi-orthogonal wavelets using weaker assumptions on their asymptotic vanishing. The proofs are similar to those included in [4]. In the sequel different constants will be denoted by the same letter C when no confusion can arise.

Th e o r e m 4. If ip is a semi-orthogonal wavelet such that (6.2) 3C < oo |y>(f/)(.T)| < C(1 + |x|)~2, for q = 0,1, (6.3) 3C < oo |y;(.T)| < C{ 1 + |:c|)-2 ,

then 3C\ < oo V.t, y E R, \f(x) — f(y)\ < C\ \x — y\a with a E (0,1) if and only if

(6.4) M < C32 - 0 + “ >i.

P r o o f. (The idea of the proof is the same as for an orthogonal wavelet [4]) (=>) Using the fact that, f f f 'ij)(x)dx = 0 and next applying the Holder continuity property for / arid the localization property for ip we get

O O ____________ _

kjfcl = | j { f { x ) - f(k;2~J)ypjk{x)dx\

— oo oo

< C f jx - k 2 -Jja2 i(l + 12:)x - k\)~2dx

— OO

The exchange of variable y = 2Jx — h gives

\cjk\ < /'

(( + !/ -dy, which gives (6.4), since a 0 (0,1).

(<= ) From the inequality (6.4)) for x ^ y

If( x ) ~ f i d ) | < C [2 ( * + a» Y IiPjk{z) ~ 4'jk{y)\

k£Z

Let, J be a natural number such that 2 J < |.t — y\ < 2 J+1. We are going to estimate separately the sum over j < J and over j > ./. Tlie first sum.

after using mean value theorem for ip and the assumption (6.2) for V;/, is estimated as follows:

] T |2-a j ^ 2 K’(2‘?a: - k ) - i i 2Jy - *01 j ^ J /cGZ

< C2- " '72-, |3; - j / | ^ 2(o^1)<J- i) ^ m a x { ( l + |2Ja :-/r| )--,(l + |2''2/-fc | )^ }

j < j k e i

where, by the definition of J, 2^1_cv^jr|.r — y\ < 2\x — y\a.

Estimating the second sum over j > J we get the expression C\x - y\a

]T J j l + I2’* - *l)“ 2,

j > J k e i

since 2~aJ < |.t — y\°.

It remains to show that in both the cases ( j < J and j > J) the functions (6.5) 9j(.T)‘= ^ ( i + ^ - f c l ) -2

are uniformly bounded with respect to x and j. Let us observe that for any given j , g:/(x ) is a periodic continuous function with the period 2~J, so it has finite supremum over x E R. On the other hand gi(x) = g j{2l~J x). thus sllPxGi fji(x) = supx.G lg p x ) for any /, j 6 Z. It ends the proof. A

Now, let us go to investigate the case of local smoothness of / . We say that / satisfies the Holder criterion at ,r0 with the exponent s (i.e., / 6 C* ) if there exists a, polynomial P of degree [s] such that

(6.6) / (

0) + 0

Is).

Th e o r e m 5. Let ip be a given semi-orthogonal wavelet and let ip be its dual (5.6). Assume that the following conditions

CO

(6.7) J x lip(x)dx = 0 for / = (),..., [.s],

— CO

(6.8) |^(y0| < Cn ( 1 + M )~ N f or N = [s] + 2 are satisfied for a given s £ R + . If f £ C f , then 3C < oo (6-9) |c?,| < 6'2_( 2+s)f(i + |2Lt0 - A:|s) where cjk are defined by (6.1).

P r o o f. Due to the assumption (6.7) J^° P(x — xo)rf lk(x)dx — 0 for the polynomial P of degree [s]. Thus, by (6.6)

\C 3 k \ j f{x)ip:ikix)dx < C J \x - xo\*\ipjk(x)\dx.

Function representation in wavelet bases 19 Taking into account

I* - *0|* < c ^{(I* -}

^{+ I}

- *o|*) =

fe|* + 1^*»-

and

\4’ik(x)\ < 2*C V(1 + |2J* - k \ r N which follows from (6.8), we get

CO

\cjk\ < C 2 h ~ 3 s f {}2Jx - k \ s + \Px0 - k \ s) (l + \2j x - k \ ) ~ N dx.

— oo

Now, the simple change of variable under the integral [y = 2 Ur — k) yields to the desired conclusion'. A

The direct inversion of Theorem 5 is not true. However, a certain modi- fication of such an implication was formulated in [4]. Here we get the same result under weaker assumptions on the considered wavelet. Let

^ f s - 1 , if 3 £ N;

[ [³], otherwise.

Th e o r e m 6. Let f be a given semi-orthogonal wavelet and let f be its dual (5.6). Let us assume that they satisfy the following conditions

(6.10) \f(x)\ < C ( l+ \ x \ ) -\

(6.11) |y.’^(.T)| < Cn { 1 + \x\)~N for 0 < l < v + 1, and N = [s] + 2.

If there exists (3 £ (0,1), such that f £ C 13 and wavelet coefficients of f satisfy the inequality (6.9) for a certain s > (3

then there exists a polynomial P of degree < s such that

(6.12) \ f { x ) - P { x - x 0 )| < C \ x - x 0\s ^i + log2 ^ ^ for |.T-a*0| < 1.

P r o o f. First, we introduce auxiliary functions / / ( A ‘= X ! ^ciki'jk

fees

Using inequalities (6.9) and (6.11) for / = 0 we easily get an estimate l/,U )l < C 2 - « 1 + |2V- - A f + 2^1* - *019(1 + |2U - k\)~N.

rei

The uniform boundness of the function f2 ke^\2Jx — A‘|s(l + |2ir — A;|)_aa

follows from the same arguments as the boundness of (6.5). Thus (6.13) \fj(x )\ < C2~si(l + 2-/ |.r — .t0|)s.

Since

| l W * ) = 2 = 2'i V>(0(2i.f - k), it follows from (6.11) that

dl

dxl x < CN2 h lj( l + \-2j x - k\)~N.

Thus, in a similar way as for \fj(x)\ we deduce

(6.14) | /- /) (,'h)l < C2{l~s)j{l + 2j \x - x0\Y for l < v + 1. For any j € Z let us define a polynomial

P j i y ) — fji .x o ) + fj{. x o ) y + • • • + ^ f j 1( x o )yL which is the Taylor series of fj at ;r0 of the order and put

P {x - x 0) '= £ > ( * ~ x °)' je i

A convergence of the above sum for |ar — ^x.q\ < 1 will be immediately proved when we will show that (6.12) occurs for such x. In order to get the desired bound for the expression

I/O ) - P{x - Xo )| = | ^ ( / ; 0 ) - P j ( x - x 0)) ,

jez

we are going to estimate separately the following terms:

•hi = ^ |/jO ) - Pj(x ~ x0)U

j <Jo

c = 5 3 i/i(®)i,

./ >:i o

A s = |P j { x - a,-0 )|, j>:io

where jo is such that

2-j0-1 < |,n — xq I < 2~jo.

By (6.14), 3c that for / < j 0 supjc_ a7()(< _ ^ o|\f(/ +1){0\ < cO i/+1- sO Thus we have

Ax < C|x ~ a:or +12Jo(i/+1" s) ^ 2~(l/+1~s)(jo-.j) .

j< jo

Applying now the definition of jo we get Tj < C\x — :c0|s-

Function representation in wavelet bases 21

Let ji - p 0. According to (6.13)

Y | fj (x ) | < c Ix ~ -To 15 < c(ii - jo ) |x - x 0 \s.

On the other hand, since / <E C,/J, by Theorem 4, the wavelet coefficients of / satisfy the condition |c7fc| < c2~^~f3k and thus

< c:2 -Pin

3>3i j>j i i>h

-P U -h )

Since 2~l3jl = 2~sjo < 2s|:r - ;r0|s, we get A2 < C (ji - jo )k - .r0|s, where the difference ji — jo is bounded by — l)lo g2 |b — ^o|_1-

Finally, due to the estimates (6.14), |/ji)(a;o)| < 2~{s~l)j. Thus the term A3 is bounded by

/=0 -X)

where Ci = X^ 2~^s~1^ < Cv < oo for / = 0, . . . , u.

j=i

Using the definition of j 0 we get A3 < c\x - :r0|U Summing the bounds for the terms A i, A 2, A3 we obtain the desired result. A

The assumption / E C 13 cannot be replaced by uniform continuity of / and the estimate (6.12) is optimum. In [4] corresponding counter-examples confirming these facts have been presented. For more comments see also [1]

§9.2.

The theorems presented in this section are important from approxima- tion point of view. Let f j be an approximation of / at the scale 2~J, i.e.,

fj = X ! X / ’/fc^ fc-

/ ^ j k £

Basing on the estimates of c ip presented above it can be proved that if / satisfies the Holder criterion with the exponent s for an arbitrary point ;r, then

I I / - /ill < C 2 -ja|J/||c[.,.

Moreover, in order to preserve the same order of approximation in the case when / is less smooth in some points or in some regions, a more accurate approximation should be taken in these regions only. In [4], (Tli.16) it is shown precisely how this local refinement should be performed.

References

[1] Ingrid D a u b e c h ie s , Ten Lectures on Wavelets, SIAM, Philadelphia, Pennsylvania 1992.

[2] Ch. K. C h u i, An Introduction to Wavelets, Academic Press, Inc. 1992.

[3] Ch. K. C h u i, Ed., Wavelets: A tutorial in Theory and Application, Academic Press, Inc. 1992.

[4] S. J a ffa rd , Ph. L a u r e n c o t, Orthonormal Wavelets, Analysis of Operators, and Applications to Numerical Analysis, in [3].

[5] Yves M e y e r, Wavelets and operators, Cambridge Univ. Press 1992.

[6] L. L. S c h u m a k e r , G. W e b b , Eds., Recent Advances in Wavelet Analysis, Academic Press, Inc. 1994.

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