Week7Student2009

# Week7Student2009 - Week 7 Lecture 12 Fourier estimation and...

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Unformatted text preview: Week 7 Lecture 12 Fourier estimation and Linear Minimaxity An orthonormal basis for L 2 ([0 ; 1]) is & 1 ( x ) = 1 & 2 k ( x ) = p 2 cos (2 ¡kt ) ;& 2 k ( x ) = p 2 sin (2 ¡kx ) ; k & 1 . The periodic Sobolev class f W & 2 ( M ) is de&ned as F = & f : Z 1 ¡ f ( m ) ¢ 2 ¡ M; f ( j ) (0) = f ( j ) (1) , j = 0 ;:::;m ¢ 1 £ Let ¢ j = ¤ f;& j ¥ . It is known that F = ( f : 1 X k =1 (2 ¡k ) 2 m ¦ ¢ 2 2 k + ¢ 2 2 k +1 § ¡ M ) . Why? For any f 2 F , we have f ( m ) ( x ) = (2 ¡k ) m 1 X k =1 h ¢ 2 k p 2 cos (2 ¡kt ) + ¢ 2 k +1 p 2 sin (2 ¡kt ) i , m & 1 integers for m = 0 ; 1 ;:::;£ ¢ 1 . Apparently f ( m ) (0) = f ( m ) (1) , m = 0 , 1 , ::: , £ ¢ 1 , and R ¦ f ( & ) § 2 ¡ M . Remark : The Sobolev class W m 2 ( M ) is de&ned as F = & f : Z 1 ¡ f ( m ) ¢ 2 ¡ M £ (without periodic boundary conditions). The representation of the model in the ellipsoid form can be achieved using the Demmler-Reinsch spline. For the model dy ( t ) = f ( t ) dt + ¤dW ( t ) , ¤ = n & 1 = 2 or an orthogonal transformation of the white noise model, Z & j dy ( t ) = Z & j f ( t ) dt + ¤ Z & j dW ( t ) i.e., y i...
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Week7Student2009 - Week 7 Lecture 12 Fourier estimation and...

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