ESTIMATES FOR WAVELET COEFFICIENTS ON SOME CLASSES OF FUNCTIONS

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1 ESTIMATES FO WAVELET COEFFICIENTS ON SOME CLASSES OF FUNCTIONS V F Babeo a S A Sector Let ψ D be orthogoal Daubechies wavelets that have zero oets a let W { } = f L ( ): ( i ) f ˆ( ) N We rove that li su D ( ψ f ) : f W D ( ψ ) = q ( π) π ( π) q Let L = L ( ) be the sace of easurable fuctios f : C with fiite or f where f = f L ( ) = f x ( ) x if < a f = f L ( ) = vrai su f( x) x For f where L ( ) a g L q ( ) where q [ ; ] + = we set q We cosier the followig classes of fuctios f W Deroetrovs Natioal Uiversity Deroetrovs ( f g ) = f( x) g( x) x L ( ): For N a ( ) we set { } = f L( ): ( i ) fˆ( ) Istitute of Alie Matheatics a Mechaics Uraiia Natioal Acaey of Scieces Doets Traslate fro Urais yi Mateatychyi Zhural Vol 59 No Deceber 7 Origial article subitte Jue 9 6

2 79 V F BABENKO AND S A SPECTO f ˆ( ) = π ( ) i x f x e x is the Fourier trasfor of a fuctio f For = we obtai the staar Sobolev classes ( = f L f ) ( ): W { } For a fuctio ψ( t) L ( ) a ubers j Z we set f j/ ψ j ( t) = ψ( j t ) If the syste of fuctios { ψ j } j Z fors a orthooral basis of the sace L ( ) ie ay fuctio L ( ) ca be reresete i the for of the su of the series f ( t ) = ( ψj ν f ) ψj ν i Z j Z ( t) () coverget i L ( ) the the fuctio ψ ( t ) is calle a orthogoal wavelet May alicatios of orthogoal wavelets are base o the ivestigatio of the value of the wavelet coefficiets i reresetatios of the for () eeig both o roerties of the wavelet ψ ( t ) a o the soothess of the fuctio f Assue that the wavelet ψ ( t ) has zero oets or which is equivalet that ψ ˆ ( ) has a zero of ultilicity at zero We efie a fuctio ψ ( t) by the relatio ( ψ() t ) ( ) = ( i) ψˆ ( ) We set ; ( ψ) = su C κ q ( ψ f ) ψˆ f W q where = It is easy to see that C κ ; q ( ψ) ca be reresete i the for C κ q ; ( ψ) = ( ψ) ψˆ q ()

3 ESTIMATES FO WAVELET COEFFICIENTS ON SOME CLASSES OF FUNCTIONS 793 Note that C κ ; q ( ψ) is the exact costat i the iequality ( ψ j ν f ) j + q C q f i q κ; ( ψ) ψˆ ˆ( )( ) Let N The trigooetric olyoials which satisfy the equalities where / H ( ) = h() l e l = are calle Daubechies filters (see eg [] Sec 6) il h l ( ) H ( ) = cos si P P ( x ) = + = A fuctio ϕ D whose Fourier trasfor has the for x ( ϕ D ) ( ) = π H l = l ( ) is the orthogoal scalig fuctio A fuctio whose Fourier trasfor has the for i ( ψ D ) ( ) = e H D + ( ) π ϕ is calle a orthogoal Daubechies wavelet ψ D The wavelet ψ D ossesses the followig roerties (see [] Cha 6; [] Sec 6): (i) su ψ D = [ ( ) ]; (ii) ψ D has zero oets; (ii) there exists λ > such that ψ D C λ where C α = f : fˆ( ) α ( + ) < α > (3)

4 794 V F BABENKO AND S A SPECTO Furtherore (see eg [3] Sec 55) D = D H + π ϕ ( ψ ) ( ) ( ) = H π H l + π ( ) (4) l = a oreover H ( ) = c si uu (5) The followig theore is the ai result of the reset aer: Theore Let be a fixe iteger The For = q = this theore was rove i [4] li C q ; ( ψ D ) = ( ) / / q π π To rove this theore we ee the lea resete below which was also rove i [4] for = Let ˆΨ = ( ) π χ [ π π] + χ [ π π] where χ is the characteristic fuctio of the iterval I Lea Suose that < < is a iteger ( ψ ) is a sequece of fuctios with coact suort a the followig coitios are satisfie: (i) for a certai ε ieeet of < ε < π oe has < ε ( ψ ) ( ) as ; (ii) ( ) Ψ ˆ as The ψ li ( ψ) = ( ) / / π (6) π

5 ESTIMATES FO WAVELET COEFFICIENTS ON SOME CLASSES OF FUNCTIONS 795 Proof We rereset ( ψ) i the for ( ψ) = ( i ) ( ψ ) ( ) = ( ψ) ( ) Usig the Miowsi iequality we get = ( ψ ) / ˆ ( ) ˆ ( ) ( ) Ψ Ψ ( ) / (( ψ) ( ) Ψˆ ( ) ) + Ψˆ ( ) ψ + ˆ ( ) ( ) Ψ ( ) Ψˆ ( ) = I + I O the other ha / (( ψ) ( ) Ψˆ ( ) ) + Ψˆ ( ) ψ ˆ ( ) ( ) Ψ ( ) Ψˆ ( ) = I I For I we have = + π π = ( ) / / π π I π π π π Let us show that I = ( ψ ) ( ) Ψˆ ( ) as We fix ε ( ; π ) Diviig the iterval of itegratio ito two arts we obtai < ε > ε I = ( ψ ) ( ) Ψˆ ( ) + ( ψ ) ( ) Ψ ˆ ( ) = I + I

6 796 V F BABENKO AND S A SPECTO Cosier I Taig ito accout coitio (i) a the fact that Ψ ˆ ( ) = for ( π; π ) we establish that I as Furtherore by virtue of coitio (ii) we have I ε ( ψ ) ( ) Ψ ˆ ( ) as > ε Thus ( ψ) I = ( ) / / / π π as The lea is rove We also ote the secial case of the lea For = we get li ( ψ ) = ( π) q (7) q Proof of Theore It is ecessary to verify coitios (i) a (ii) of Lea for orthogoal Daubechies wavelets We use relatios (4) a (5) a the fact that c = π si = Γ ( + ) π Γ( ) π (8) To rove coitio (i) we choose < ε < a ote that H ( ) for ay We obtai < ε ( ψ ) ( ) D / π π + H < ε c π / / / < ε si tt Furtherore sice si we get

7 ESTIMATES FO WAVELET COEFFICIENTS ON SOME CLASSES OF FUNCTIONS 797 c π / / / < ε si = tt ( + ) / / / π < ε si tt ( + ) / / π < ε si + ( + ) / / / π < ε si ( + ) / / π < ε si For > we have ( + ) / / π < ε si ( ) / / + c εsi π ε c π ( / + )+ / + ε Sice c π the last exressio tes to zero as elatio (i) is rove To rove coitio (ii) we set I = [ π; π ] a Let us rove that I δ = [ π π+ δ) ( πδ π+ δ) ( π δ π+ δ) ( πδ π] I \ Iδ ( ψ ) ( ) ˆ D Ψ ( ) as We have

8 798 V F BABENKO AND S A SPECTO I \ Iδ ( ψ ) ( ) ˆ D Ψ ( ) ( ψ D ) ( ) H + π π I \ Iδ + Ψ ˆ ( ) + π H π I \ Iδ (9) For fixe δ the sequece H + π π coverges uiforly as to Ψ ˆ ( ) i I\ Iδ Therefore the seco ter o the right-ha sie of (9) tes to zero as The first ter o the right-ha sie of (9) ca be rewritte i the for Taig ito accout that the relatio H π l + H π ( ) () I \ I l δ H π + a the estiates H 4 { } / c π π δ si 4 H 8 / π c 4 a l H l 3 ( ) ( ) ( ) establishe i [4] for I\ I we ca estiate itegral () for all sufficietly large as follows: δ H π l + H π ( ) I \ I l δ { } / / π π δ π ( 4π) si c c π 4 4 ( ) It is clear that the right-ha sie of the iequality obtaie tes to zero as ( )

9 ESTIMATES FO WAVELET COEFFICIENTS ON SOME CLASSES OF FUNCTIONS 799 Sice I δ = 6 δ we have ( ψ ) ( ) ˆ D Ψ ( ) Iδ π / I δ π / 6δ () We ow verify that for all > we have ( ψ ) ( ) π D as Accorig to ow results cocerig the regularity of Daubechies wavelets (see eg [5] Sec 4) there exist ositive costats C a C such that the followig iequality hols for all > π : ( ψ D ) ( ) Clog C The > ( ψ ) ( ) π D C ( ) > π Clog = C ( ) > π ( Clog ) ( Clog ) ( π) ( ) > π The last exressio tes to zero as + Thus the coitios of Lea for orthogoal Daubechies wavelets are satisfie Usig () (6) a (7) we obtai C ; D q li ( ψ ) = li D D = q ( ψ ) ( ψ ) ( π) π q ( π) li C q ; ( ψ D ) = ( ) / / q π π The theore is rove EFEENCES I Ya Noviov a S B Stechi Mai theories of slashes Us Mat Nau 53 No (998) I Daubechies Te Lectures o Wavelets Society of Iustrial a Alie Matheatics Philaelhia (99) 3 G Strag a T Nguye Wavelets a Filter Bas Cabrige Press Wellesley (996) 4 S Ehrich O the estiatio of wavelet coefficiets Av Cout Math () 5 A K Louis P Maab a A ieer Wavelets: Theory a Alicatios Wiley Chichester (997)

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