Multiresolution

Definition

En sekvens af lukkede vektor underrum af L 2 ( R ) er en multiresolution tilnærmelse, hvis den opfylder følgende fem egenskaber: (Vj)j∈Z{\ displaystyle (V_ {j}) _ {j \ in \ mathbb {Z}}}

• ∀j∈ZVj+1⊂Vj{\ displaystyle \ forall j \ in \ mathbb {Z} \ quad V_ {j + 1} \ subset V_ {j}}
• ⋃j∈ZVj¯=L2(R) og ⋂j∈ZVj={0}{\ displaystyle {\ overline {\ bigcup _ {j \ in \ mathbb {Z}} V_ {j}}} = L ^ {2} (\ mathbb {R}) {\ text {and}} \ bigcap _ { j \ in \ mathbb {Z}} V_ {j} = \ {0 \}}
• ∀j∈Zf∈Vj⟺f(2⋅)∈Vj+1{\ displaystyle \ forall j \ in \ mathbb {Z} \ quad f \ in V_ {j} \ iff f (2 \ cdot) \ in V_ {j + 1}}
• ∀(j,k)∈Z2f∈Vj⟺f(⋅-2-jk)∈Vj{\ displaystyle \ forall (j, k) \ in \ mathbb {Z} ^ {2} \ quad f \ in V_ {j} \ iff f (\ cdot -2 ^ {- j} k) \ in V_ {j }}
• Der findes sådanne, der er en base af Riesz fra .θ∈V0{\ displaystyle \ theta \ i V_ {0}}(θ(⋅-ikke)ikke∈Z){\ displaystyle \ left (\ theta (\ cdot -n) _ {n \ in \ mathbb {Z}} \ right)}V0{\ displaystyle V_ {0}}

Reference

1. (in) Stéphane Mallat , "  Multiresolution approximations and orthonormal wavelet bases of L 2 ( R )  " , Trans. Bitter. Matematik. Soc. , Vol.  315, nr .  1,1989, s.  69-87 ( DOI  10.2307 / 2001373 , læs online ).

Bibliografi

Yves Meyer , Wavelettes and operators , vol. Jeg, Hermann, 1990