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Unformatted text preview: Week 11 Lecture 20 An Introduction to Wavelet regression De&nition: Wavelet is a function such that f 2 j= 2 & 2 j & Â¡ k Â¡ ;j;k 2 Z g is an orthonormal basis for L 2 ( R ) . This function is called &mother wavelet&, which can be often constructed from Â¡father wavelet& â€™ . The Â¡father wavelet & â€™ is not a wavelet, but we can construct wavelets from it, so it is equally important as mother wavelet. 1. Example : Haar wavelet (A. Haar, Math. Ann. (1910)) ( x ) = Â¢ 1 x 2 [0 ; 1 = 2) Â¡ 1 x 2 [1 = 2 ; 1) and 2 j= 2 & 2 j & Â¡ k Â¡ = Â¢ 2 j= 2 x 2 Â£ 2 & j k; 2 & j k + 2 & ( j +1) Â¡ Â¡ 2 j= 2 x 2 Â£ 2 & j k + 2 & ( j +1) ; 2 & j k + 2 & j Â¡ . Let â€™ ( t ) is the indicator function over the interval [0 ; 1) . LetÂ¢s deÂ£ne V j = span n â€™ jk = 2 j= 2 â€™ & 2 j & Â¡ k Â¡ , k 2 Z o Since â€™ ( x ) = â€™ (2 x Â¡ 1) + â€™ (2 x ) , the following four properties are satisÂ£ed (i) ... Â¢ V & 2 Â¢ V & 1 Â¢ V Â¢ V 1 Â¢ V 2 Â¢ :::; (ii) f ( x ) 2 V j () f (2 x ) 2 V j +1 ; (iii) [ j 2 Z V j = L 2 ( R ) , (iv) there is a function â€™ such that f â€™ ( & Â¡ k ) ;k 2 Z g is an orthonormal basis for V . An equivalent form of property (iii) : Recall that b f ( & ) = Z 1 &1 f ( x ) exp ( Â¡ i&x ) dx It can be shown that, for any â€™ the property (iii) is equivalent to b â€™ (0) 6 = 0 and j b â€™ j is continuous at . A sketch of the proof is as follows: First, [ j 2 Z V j is invariance under all translations. Second, if g is orthogonal to [ j 2 Z V j , which implies g ( x ) is orthogonal to â€™ j ( x + t ) for all t , then Plancherel formula implies b g ( & ) b â€™ & 2 & j & Â¡ = 0 a.s.. We know b â€™ 6 = 0 around . Let j ! 1 , we then conclude b g ( & ) = 0 a.s.. 1 From â€™ to : Observe that ( x ) = â€™ (2 x ) & â€™ (2 x & 1) and h â€™; i = 0 . If we de&ne W j = span n 2 j= 2 & 2 j Â¡ & k Â¡ , k 2 Z o then V L W = V 1 and more generally V j L W j = V j +1 We also see that J L j = &1 W j = V J +1 ! L 2 ( R ) , J ! 1 , in other words, f jk = 2 j= 2 & 2 j Â¡ & k Â¡ , j;k 2 Z g is an orthonormal basis for L 2 ( R ) : 2. Multiresolution analysis (MRA) &a general framework to con struct wavelet functions More generally, if there is â€™ such that f â€™ ( Â¡ & k ) ;k 2 Z g is an orthonormal system, and 1 p 2 â€™ Â¢ x 2 Â£ = X h k â€™ ( x & k ) and j b â€™ j is continuous at with b â€™ (0) 6 = 0 . LetÂ¡s de&ne V j = span n 2 j= 2 â€™ & 2 j Â¡ & k Â¡ , k 2 Z o Then the following four properties are satis&ed i) ... Â¢ V & 2 Â¢ V & 1 Â¢ V Â¢ V 1 Â¢ V 2 Â¢ :::; ii) f ( x ) 2 V j () f (2 x ) 2 V j +1 ; iii) [ j 2 Z V j = L 2 ( R ) iv) there is a function â€™ such that f â€™ ( Â¡ & k ) ;k 2 Z g is an orthonormal basis for V ....
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 '09
 HarrisonH.Zhou

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