De l`analyse de Fourier aux ondelettes Plan :
Transcription
De l`analyse de Fourier aux ondelettes Plan :
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"#$ *''"0(" #',*&)$ 0#'1.$&"%$!!$ 7% 80$)" H3 ! "#$%&$' ()*+,! -.'+/ 0$/1.'2$/3 0$ 4#5'+$' .56 #*7$8$22$/ 9:;:<;; = > ?= !"#$%&'"($! ) *$&#(+#, (!"+#-#."/"($! 9! '"%$ (-),)" 0ˆ(ν) = 2! (ν)$ " Φ <&!- (ν) , &5 2! (ν) ≥ : "$ Φ! (ν) ∈] − π, +π], (-*+ .;π) Z = 2 (ν)$ +" (Φ +ν # ) + ν . ;π R ! 0 #>(-),$ -&??" @ #&??" ,!8!," A 7"# B&!-$,&!# $),/&!&?($),+%"# 7" $&%$"# 0"# B)(+%"!-"# +%, B&)?"!$ #&! ),$3"#$3 4, /̂(ν) = 2$ (ν)$ " Φ (ν) 1 0 (" ) = ! C(),8" 4 =/∗0 D"-, '")?"$ 7" -&?'&#"!$ 0 G Z H&$" 2 0ˆ(:) = !"#$# 2% (ν) = 2$ (ν)2! (ν) "$ Φ% = Φ$ + Φ! . E"!0"C") &% *?'0,8")F -")$*,!"# 7"# B)(+%"!-"# +%, 0 (" ) +" R ! "#$%&$' ()*+,! -.'+/ 0$/1.'2$/3 I-&?'&#*!$" J*##" B)(+%"!-" 7" 0 3 0$ 4#5'+$' .56 #*7$8$22$/ 9:;:<;; @ > ?= !"#$%&'"($! ) *#+!,-$#./0 %0 1$&#(0# #+2(%0 311*4 ! "#$% (!) = " (# + !∆# )& ' ≤ ! ≤ $ − (& %&'!() *&%+,-# *- *.,/- 0!&-) !"#$%&!'() *) +&,!-)! *-$.!/0) *!*+,$% - ˆ(1 ) = $,!./0$% - (!) = −! X != ( $ (!) - −" "π#!/ , −! 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'# '( *,01)'(2'.< !"#$%$" & !"# )( .">($5 %' *'(+#,' #'&-./*,01)'(2' 3!,(0'< D!), "! ∈ R4 %0#',&"(', 5$ *'(+#,' #'&-./*,01)'(2' %' " (" ) = (" − "! )< D!), ν! ∈ R4 %0#',&"(', 5$ *'(+#,' #'&-./*,01)'(2' %' ν" (" ) = (" )# ! ν" < #$ %&'()'* +,-./$ 01*.2 3'241*5'26 3' 7&8*.'* 189 &-:';'55'2 !" "< = > ?@ .-$!+/0-1") () 203-&)- 4 /)!5'-) #%&++$!') !"# $ ∈ ! (R) %' *'(+#,' #'&-./*,01)'(2' 3!,(0' '# 2'(#,0' '( (E, E)< )"*+,-,#+ ( 8( $--'55' '#()*+,#-." /" 0,1#%"# 2 +")3'#" 45%**()'" %?)( .">($5 ∈ ! (R) 5$ *!(2#"!( %" : R −→ C (&, ν) 7−→ ( , ν) = %" & Z R (" )$ (" − &)# −! ν '" 6"-(#71"* & ! F( -,$#"1)'4 !( -,'(% $ ,0'55' '# -$",'< =?"(*!,&$#"!( %) .">($5 (" ) : 5?"(.#$(# " = & '.# 2!(#'()' %$(. " 7→ (" )$ (" − & ) G !( B ,'.#,'"(# C : 5$ *'(+#,' $ (" − &) '# !( 2$52)5' .$ #,$(.*!,&0' %' H!),"',< #$ %&'()'* +,-./$ 01*.2 3'241*5'26 3' 7&8*.'* 189 &-:';'55'2 !" "< !" > ?@ !"#$%&!'() *) +&,!-)! . %)#/0!) 12-$$"#0) ! "#$% (! − ")# ! ν " & '(#)$ Z $# (", ν) = % (! ) ,ν (! ) &! = !) = ,ν ( * Z %ˆ(ω) d (ω) & ω ,ν +π R R ,#!- $# (", ν) -#!./%!. (0/!1#)2'./#! 34!%)5/%6 7 8% % '9 :#/$/!'5% 8% " %. 8% %ˆ '9 :#/$/!'5% 8% ν ; Z !"#$%&'%())" |$# (" , ν)| = ,ν (! ) &! % (! ) <% R 2%$9)% (04!%)5/% 8% % 8'!$ (% :#/$/!'5% .%2"$=1)4>9%!-% 8% (", ν); ?! *@AB& C'D#) ' 2#!.)4 >9% (' .)'!$1#)24% 8% E#9)/%) F 1%!G.)% 5(/$$'!.% %$. $.'D(% %. "%)2%. 8% )%-#!$.)9/)% (% $/5!'( % ; ! "#$%&$' ()*+,! -.'+/ 0$/1.'2$/3 0$ 4#5'+$' .56 #*7$8$22$/ 9:;:<;; 9; = >? !"#$%&!'() *) +&,!-)! . %)#/0!) 12-$$"#0) !"#$%&' ()*+#$, H#/. ∈ '! (R) ∩ ' (R)& )4%((% ':%- ($ = *; I(#)$ "#9) .#9. % ∈ ' (R) 7 ZZ * % (! ) = +π R9 $# (", ν) ,ν (! ) & ν &", 1#)29(% 8% )%-#!$.)9-./#! J ZZ R9 |$# (", ν)| & ν &" = +π (# = (#ˆ , -#!$%):'./#! 804!%)5/%. <' KEFEC %$. 9!% '""(/-'./#! !#! $9)L%-./:% 8% ' (R) 8'!$ ' (R ) 7 % ∈ ' (R) ←→ $# ∈ ' (R ) . ! )%")4$%!.% 9! $/5!'( *, 5)M-% F 9!% 1#!-./#! 8% 8%9N :')/'D(%$; O! 4(42%!. Φ ∈ ' (R ) 8#/. :4)/P%) 8%$ ")#")/4.4$ 8% %"*&+*(+#" "#9) "#9:#/) G.)% (' KEFEC 809! 4(42%!. % 8% ' (R); ! "#$%&$' ()*+,! -.'+/ 0$/1.'2$/3 0$ 4#5'+$' .56 #*7$8$22$/ 9:;:<;; 99 = >? !"#$%"& '" ()*+&,-)#." '" /-0)1") 2 ,"+3()" 4%1&&*+(" !"# $%& '(($)*'+)!,& (#'+)-"%&. /)01#%,+%& 2%,3+#%& (! ) = " (! ) = (α/π) ! *5%&+ $' (! ) = &!,+ "+)$)&1%& 4 #− " " . α !"#$%&!'() *) +",&! 6 7 − |! |/$ I[−!,+!] (! ). $ > 8. 2%,3+#% +#)',9"$')#% 6 (! ) = α + (7 − α) *!&(π ! /$) I[−!,+!] (! ). $ > 8. α ∈ [8, 7] 6 (!"# α = 8. !, ' ", *!&),"& +#!,-"1 &"# [−$, +$] 6 (!"# α = 7. !, ' ",% 2%,3+#% #%*+',9"$')#%. ν" ('& /1:,)% 6 (!"# α = 8.;<. !, ' $' 2%,3+#% /% ='>>),9? #$ %&'()'* +,-./$ 01*.2 3'241*5'26 3' 7&8*.'* 189 &-:';'55'2 "< <= "> ? @A 5*%60%& @, (!&% % (&) = ' (! + &∆! ). 8 ≤ & ≤ ( − 7. )*"&$+ ,*)-.#! ,# ,/.0# 1&*#. %+ (&). −( + 7 ≤ & ≤ ( − 7. $%& *!%A*)%,+& /)&*#%+& /% $' 2%,3+#% #1%$$% %+ &B>1+#)-"%. *2#2 (−&) = (&) (!"# 7 ≤ & ≤ ( − 7. %+ # −! X (&)" = 7? $= !"#$%&!'() *) +&,!-)! . %)#/0!) 12-$$"#0) *-$3!40) 5 3% (4 , + ) = # −! X $= % (&) = 7 ( % (&) (& − 4 ) # −& "π'$/# , # −! X (= (& − 4 ) # −! X '= 4 , + = 8, 7, . . . , ( − 7; 3% (4 , + ) # & "π'$/# , & = 8, 7, . . . , ( − 7. !"# +!"+ 8 ≤ 4 < ( . 3% (4 , + ) &5!C+)%,+ ('# $' DDE /% % (&) (& − 4 ). $' /1*!>(!&)+)!, %&+ /!,* %, 5 (( " $,(( ))? F% >3>% (!"# $' #%*!,&+#"*+)!,? #$ %&'()'* +,-./$ 01*.2 3'241*5'26 3' 7&8*.'* 189 &-:';'55'2 "< <= "! ? @A !"#$%&'!"& ! "#!$%&'#()* +* ,'-#.*# / &*$0"#* 12.%%!$"* 3*#(*" +4!$!25%*# -$ %.1$!2 +!$% -$* &*$0"#* "*(3%6&#)7-*$8* [ − ∆ (! ), + ∆ (! )] × [ν − ∆ν (! ), ν + ∆ν (! )]9 :*% &*$0"#*% %'$" ;<*% = '$ ! "'->'-#% 2! (0(* #)%'2-".'$ *$ "*(3% *" *$ &#)7-*$8*%9 :*#"!.$% %.1$!-< %'$" -$* %-88*%%.'$ +* "#$%&% %'$'()**$(+&%? '@ 2! &#)7-*$8* *%" %"!A2* %-# -$* 2'$1-* +-#)*? *" +* "#$%&% '+$*%(')(+&%? 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