Wavelet coorbit spaces over general dilation groups Hartmut Fhr - PowerPoint PPT Presentation

Aims of this talk Main objective Establish notion of nice wavelets for higher-dimensional wavelet transforms, with dilations coming from a suitable matrix group, the dilation group. This was previously studied for: Similitude groups ( � isotropic Besov spaces), shearlet dilation groups (Kutyniok, Dahlke, Steidl, Teschke et. al.) Strategy Verify prerequisites for coorbit theory (Feichtinger/Gröchenig). This provides access to: H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 8 / 31

Aims of this talk Main objective Establish notion of nice wavelets for higher-dimensional wavelet transforms, with dilations coming from a suitable matrix group, the dilation group. This was previously studied for: Similitude groups ( � isotropic Besov spaces), shearlet dilation groups (Kutyniok, Dahlke, Steidl, Teschke et. al.) Strategy Verify prerequisites for coorbit theory (Feichtinger/Gröchenig). This provides access to: ◮ Consistent notion of wavelet coefficient decay, associated smoothness spaces H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 8 / 31

Aims of this talk Main objective Establish notion of nice wavelets for higher-dimensional wavelet transforms, with dilations coming from a suitable matrix group, the dilation group. This was previously studied for: Similitude groups ( � isotropic Besov spaces), shearlet dilation groups (Kutyniok, Dahlke, Steidl, Teschke et. al.) Strategy Verify prerequisites for coorbit theory (Feichtinger/Gröchenig). This provides access to: ◮ Consistent notion of wavelet coefficient decay, associated smoothness spaces ◮ Useful notions of nice wavelets: Sets A w (analyzing vectors) and B w (frame atoms). H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 8 / 31

Aims of this talk Main objective Establish notion of nice wavelets for higher-dimensional wavelet transforms, with dilations coming from a suitable matrix group, the dilation group. This was previously studied for: Similitude groups ( � isotropic Besov spaces), shearlet dilation groups (Kutyniok, Dahlke, Steidl, Teschke et. al.) Strategy Verify prerequisites for coorbit theory (Feichtinger/Gröchenig). This provides access to: ◮ Consistent notion of wavelet coefficient decay, associated smoothness spaces ◮ Useful notions of nice wavelets: Sets A w (analyzing vectors) and B w (frame atoms). Additional task: Identify easily accessible subsets of the abstractly defined sets A w and B w . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 8 / 31

Aims of this talk Main objective Establish notion of nice wavelets for higher-dimensional wavelet transforms, with dilations coming from a suitable matrix group, the dilation group. This was previously studied for: Similitude groups ( � isotropic Besov spaces), shearlet dilation groups (Kutyniok, Dahlke, Steidl, Teschke et. al.) Strategy Verify prerequisites for coorbit theory (Feichtinger/Gröchenig). This provides access to: ◮ Consistent notion of wavelet coefficient decay, associated smoothness spaces ◮ Useful notions of nice wavelets: Sets A w (analyzing vectors) and B w (frame atoms). Additional task: Identify easily accessible subsets of the abstractly defined sets A w and B w . ( � bandlimited Schwartz functions, vanishing moment criteria) H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 8 / 31

Overview Introduction: Nice wavelets in dimension one 1 Square-integrability over general dilation groups 2 Outline of coorbit theory: Analyzing vectors and frame atoms 3 Wavelet coorbit spaces over general dilation groups 4 Vanishing moment conditions and coorbit spaces 5 H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 9 / 31

Setup: d -dimensional CWT H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 10 / 31

Setup: d -dimensional CWT H < GL ( d , R ) a closed matrix group H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 10 / 31

Setup: d -dimensional CWT H < GL ( d , R ) a closed matrix group G = R d ⋊ H , the affine group generated by H and translations. As a set, G = R n × H , with group law ( x , h )( y , g ) = ( x + hy , hg ) . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 10 / 31

Setup: d -dimensional CWT H < GL ( d , R ) a closed matrix group G = R d ⋊ H , the affine group generated by H and translations. As a set, G = R n × H , with group law ( x , h )( y , g ) = ( x + hy , hg ) . L 2 ( G ) denotes L 2 -space w.r.t. left Haar measure H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 10 / 31

Setup: d -dimensional CWT H < GL ( d , R ) a closed matrix group G = R d ⋊ H , the affine group generated by H and translations. As a set, G = R n × H , with group law ( x , h )( y , g ) = ( x + hy , hg ) . L 2 ( G ) denotes L 2 -space w.r.t. left Haar measure G = R d ⋊ H , the affine group generated by H and translations H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 10 / 31

Setup: d -dimensional CWT H < GL ( d , R ) a closed matrix group G = R d ⋊ H , the affine group generated by H and translations. As a set, G = R n × H , with group law ( x , h )( y , g ) = ( x + hy , hg ) . L 2 ( G ) denotes L 2 -space w.r.t. left Haar measure G = R d ⋊ H , the affine group generated by H and translations Quasi-regular representation of G on L 2 ( R d ) , acting via ( π ( x , h ) f )( y ) = | det ( h ) | − 1 / 2 f ( h − 1 ( y − x )) . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 10 / 31

Setup: d -dimensional CWT H < GL ( d , R ) a closed matrix group G = R d ⋊ H , the affine group generated by H and translations. As a set, G = R n × H , with group law ( x , h )( y , g ) = ( x + hy , hg ) . L 2 ( G ) denotes L 2 -space w.r.t. left Haar measure G = R d ⋊ H , the affine group generated by H and translations Quasi-regular representation of G on L 2 ( R d ) , acting via ( π ( x , h ) f )( y ) = | det ( h ) | − 1 / 2 f ( h − 1 ( y − x )) . Continuous wavelet transform: Given suitable ψ ∈ L 2 ( R d ) and f ∈ L 2 ( R d ) , let W ψ f : G → C , W ψ f ( x , h ) = � f , π ( x , h ) ψ � H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 10 / 31

Setup: d -dimensional CWT H < GL ( d , R ) a closed matrix group G = R d ⋊ H , the affine group generated by H and translations. As a set, G = R n × H , with group law ( x , h )( y , g ) = ( x + hy , hg ) . L 2 ( G ) denotes L 2 -space w.r.t. left Haar measure G = R d ⋊ H , the affine group generated by H and translations Quasi-regular representation of G on L 2 ( R d ) , acting via ( π ( x , h ) f )( y ) = | det ( h ) | − 1 / 2 f ( h − 1 ( y − x )) . Continuous wavelet transform: Given suitable ψ ∈ L 2 ( R d ) and f ∈ L 2 ( R d ) , let W ψ f : G → C , W ψ f ( x , h ) = � f , π ( x , h ) ψ � Dual action of H on R d , defined by H × R d ∋ ( h , ξ ) �→ h T ξ . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 10 / 31

Admissible vectors and wavelet inversion H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 11 / 31

Admissible vectors and wavelet inversion Definition H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 11 / 31

Admissible vectors and wavelet inversion Definition ψ ∈ L 2 ( R d ) is called admissible if W ψ : L 2 ( R d ) ֒ → L 2 ( G ) isometrically. H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 11 / 31

Admissible vectors and wavelet inversion Definition ψ ∈ L 2 ( R d ) is called admissible if W ψ : L 2 ( R d ) ֒ → L 2 ( G ) isometrically. π is called discrete series representation if π is irreducible and has an admissible vector. H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 11 / 31

Admissible vectors and wavelet inversion Definition ψ ∈ L 2 ( R d ) is called admissible if W ψ : L 2 ( R d ) ֒ → L 2 ( G ) isometrically. π is called discrete series representation if π is irreducible and has an admissible vector. Wavelet inversion If ψ is admissible, we obtain the wavelet inversion formula � f = W ψ f ( x , h ) π ( x , h ) ψ d ( x , h ) . G with weak-sense convergence. H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 11 / 31

Admissible vectors and wavelet inversion Definition ψ ∈ L 2 ( R d ) is called admissible if W ψ : L 2 ( R d ) ֒ → L 2 ( G ) isometrically. π is called discrete series representation if π is irreducible and has an admissible vector. Wavelet inversion If ψ is admissible, we obtain the wavelet inversion formula � f = W ψ f ( x , h ) π ( x , h ) ψ d ( x , h ) . G with weak-sense convergence. Furthermore: Right convolution with W ψ ψ is a reproducing kernel for the image space (important for frames and discretization). H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 11 / 31

Discrete-series representations and open dual orbits H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 12 / 31

Discrete-series representations and open dual orbits Theorem (HF, 2010) The quasiregular representation π is a discrete series representation iff there exists a single open orbit O under the dual action, with the additional property that, for some (equivalently: any) ξ 0 ∈ O , the associated dual stabilizer H ξ 0 = { h ∈ H ; h T ξ 0 = ξ 0 } ⊂ H is compact. H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 12 / 31

Discrete-series representations and open dual orbits Theorem (HF, 2010) The quasiregular representation π is a discrete series representation iff there exists a single open orbit O under the dual action, with the additional property that, for some (equivalently: any) ξ 0 ∈ O , the associated dual stabilizer H ξ 0 = { h ∈ H ; h T ξ 0 = ξ 0 } ⊂ H is compact. Remark O (if it exists) has full measure. H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 12 / 31

Discrete-series representations and open dual orbits Theorem (HF, 2010) The quasiregular representation π is a discrete series representation iff there exists a single open orbit O under the dual action, with the additional property that, for some (equivalently: any) ξ 0 ∈ O , the associated dual stabilizer H ξ 0 = { h ∈ H ; h T ξ 0 = ξ 0 } ⊂ H is compact. Remark O (if it exists) has full measure. Of particular interest will be the complement O c , the blind spot of the wavelet transform. H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 12 / 31

Overview Introduction: Nice wavelets in dimension one 1 Square-integrability over general dilation groups 2 Outline of coorbit theory: Analyzing vectors and frame atoms 3 Wavelet coorbit spaces over general dilation groups 4 Vanishing moment conditions and coorbit spaces 5 H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 13 / 31

H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 14 / 31

Construction of coorbit spaces Informal definition of coorbit spaces H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 14 / 31

Construction of coorbit spaces Informal definition of coorbit spaces Fix a Banach space Y of functions on G (solid, two-sided invariant). H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 14 / 31

Construction of coorbit spaces Informal definition of coorbit spaces Fix a Banach space Y of functions on G (solid, two-sided invariant). E.g., Y = L p ( G ) . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 14 / 31

Construction of coorbit spaces Informal definition of coorbit spaces Fix a Banach space Y of functions on G (solid, two-sided invariant). E.g., Y = L p ( G ) . Pick a suitable analyzing vector ψ ∈ L 2 ( R d ) H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 14 / 31

Construction of coorbit spaces Informal definition of coorbit spaces Fix a Banach space Y of functions on G (solid, two-sided invariant). E.g., Y = L p ( G ) . Pick a suitable analyzing vector ψ ∈ L 2 ( R d ) Coorbit space norm on L 2 ( R d ) : � f � CoY = �W ψ f � Y . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 14 / 31

Construction of coorbit spaces Informal definition of coorbit spaces Fix a Banach space Y of functions on G (solid, two-sided invariant). E.g., Y = L p ( G ) . Pick a suitable analyzing vector ψ ∈ L 2 ( R d ) Coorbit space norm on L 2 ( R d ) : � f � CoY = �W ψ f � Y . Define CoY as completion of { g ∈ L 2 ( R d ) : � g � CoY < ∞} . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 14 / 31

Construction of coorbit spaces Informal definition of coorbit spaces Fix a Banach space Y of functions on G (solid, two-sided invariant). E.g., Y = L p ( G ) . Pick a suitable analyzing vector ψ ∈ L 2 ( R d ) Coorbit space norm on L 2 ( R d ) : � f � CoY = �W ψ f � Y . Define CoY as completion of { g ∈ L 2 ( R d ) : � g � CoY < ∞} . If π is irreducible, CoY is independent of the choice of ψ � = 0, as long as W ψ ψ ∈ L 1 v 0 ( G ) . Here v 0 a (continuous, submultiplicative) control weight depending on Y . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 14 / 31

Construction of coorbit spaces Informal definition of coorbit spaces Fix a Banach space Y of functions on G (solid, two-sided invariant). E.g., Y = L p ( G ) . Pick a suitable analyzing vector ψ ∈ L 2 ( R d ) Coorbit space norm on L 2 ( R d ) : � f � CoY = �W ψ f � Y . Define CoY as completion of { g ∈ L 2 ( R d ) : � g � CoY < ∞} . If π is irreducible, CoY is independent of the choice of ψ � = 0, as long as W ψ ψ ∈ L 1 v 0 ( G ) . Here v 0 a (continuous, submultiplicative) control weight depending on Y . We define A v 0 as the set of all such ψ . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 14 / 31

Construction of coorbit spaces Informal definition of coorbit spaces Fix a Banach space Y of functions on G (solid, two-sided invariant). E.g., Y = L p ( G ) . Pick a suitable analyzing vector ψ ∈ L 2 ( R d ) Coorbit space norm on L 2 ( R d ) : � f � CoY = �W ψ f � Y . Define CoY as completion of { g ∈ L 2 ( R d ) : � g � CoY < ∞} . If π is irreducible, CoY is independent of the choice of ψ � = 0, as long as W ψ ψ ∈ L 1 v 0 ( G ) . Here v 0 a (continuous, submultiplicative) control weight depending on Y . We define A v 0 as the set of all such ψ . Key idea of coorbit theory: Use properties of the reproducing kernel W ψ ψ , and the fact that Y is a Banach convolution module over the algebra L 1 v 0 ( G ) . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 14 / 31

Discretization and Banach frames Discretization H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 15 / 31

Discretization and Banach frames Discretization Let Y be a Banach function space on G with well-defined CoY H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 15 / 31

Discretization and Banach frames Discretization Let Y be a Banach function space on G with well-defined CoY Pick a suitable frame atom ψ ∈ L 2 ( R d ) H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 15 / 31

Discretization and Banach frames Discretization Let Y be a Banach function space on G with well-defined CoY Pick a suitable frame atom ψ ∈ L 2 ( R d ) For all suitably dense uniformly discrete subsets Γ ⊂ G , the family ( π ( γ ) ψ ) γ ∈ Γ is a Banach frame of CoY . There exists a discrete coefficient norm � · � Y d such that ∀ f ∈ L 2 ( R d ) : � f � CoY ≍ �W ψ f | Γ � Y d . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 15 / 31

Discretization and Banach frames Discretization Let Y be a Banach function space on G with well-defined CoY Pick a suitable frame atom ψ ∈ L 2 ( R d ) For all suitably dense uniformly discrete subsets Γ ⊂ G , the family ( π ( γ ) ψ ) γ ∈ Γ is a Banach frame of CoY . There exists a discrete coefficient norm � · � Y d such that ∀ f ∈ L 2 ( R d ) : � f � CoY ≍ �W ψ f | Γ � Y d . Moreover, for all f ∈ CoY , there exist coefficients ( c γ ) γ ∈ Γ such that � f = c γ π ( γ ) ψ , � f � CoY ≍ � ( c γ ) γ ∈ Γ � Y d γ ∈ Γ H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 15 / 31

Discretization continued Examples, comments H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 16 / 31

Discretization continued Examples, comments For example, for Y = L p ( G ) , � f � CoY ≍ �W ψ f | Γ � ℓ p . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 16 / 31

Discretization continued Examples, comments For example, for Y = L p ( G ) , � f � CoY ≍ �W ψ f | Γ � ℓ p . Criterion for frame atoms: W ψ ψ ∈ W R ( C 0 , L 1 v 0 ) , i.e., the function |W ψ ψ (( x , h )( y , g )) | ∈ R + G ∋ ( x , h ) �→ sup ( y , g ) ∈ U is in L 1 v 0 ( G ) , for some compact neighborhood U ⊂ G of the identity. H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 16 / 31

Discretization continued Examples, comments For example, for Y = L p ( G ) , � f � CoY ≍ �W ψ f | Γ � ℓ p . Criterion for frame atoms: W ψ ψ ∈ W R ( C 0 , L 1 v 0 ) , i.e., the function |W ψ ψ (( x , h )( y , g )) | ∈ R + G ∋ ( x , h ) �→ sup ( y , g ) ∈ U is in L 1 v 0 ( G ) , for some compact neighborhood U ⊂ G of the identity. Here v 0 is the control weight from above. H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 16 / 31

Discretization continued Examples, comments For example, for Y = L p ( G ) , � f � CoY ≍ �W ψ f | Γ � ℓ p . Criterion for frame atoms: W ψ ψ ∈ W R ( C 0 , L 1 v 0 ) , i.e., the function |W ψ ψ (( x , h )( y , g )) | ∈ R + G ∋ ( x , h ) �→ sup ( y , g ) ∈ U is in L 1 v 0 ( G ) , for some compact neighborhood U ⊂ G of the identity. Here v 0 is the control weight from above. We let B v 0 denote the set of all frame atoms associated to v 0 . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 16 / 31

Discretization continued Examples, comments For example, for Y = L p ( G ) , � f � CoY ≍ �W ψ f | Γ � ℓ p . Criterion for frame atoms: W ψ ψ ∈ W R ( C 0 , L 1 v 0 ) , i.e., the function |W ψ ψ (( x , h )( y , g )) | ∈ R + G ∋ ( x , h ) �→ sup ( y , g ) ∈ U is in L 1 v 0 ( G ) , for some compact neighborhood U ⊂ G of the identity. Here v 0 is the control weight from above. We let B v 0 denote the set of all frame atoms associated to v 0 . Note: One suitably chosen weight works for a whole scale of spaces � simultaneous Banach frames H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 16 / 31

Overview Introduction: Nice wavelets in dimension one 1 Square-integrability over general dilation groups 2 Outline of coorbit theory: Analyzing vectors and frame atoms 3 Wavelet coorbit spaces over general dilation groups 4 Vanishing moment conditions and coorbit spaces 5 H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 17 / 31

Further assumptions and notations From now on: H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 18 / 31

Further assumptions and notations From now on: π is assumed to be in the discrete series. H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 18 / 31

Further assumptions and notations From now on: π is assumed to be in the discrete series. The associated dual orbit is denoted by O = H T ξ , its complement by O c . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 18 / 31

Further assumptions and notations From now on: π is assumed to be in the discrete series. The associated dual orbit is denoted by O = H T ξ , its complement by O c . O c is a closed set of measure zero. H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 18 / 31

Further assumptions and notations From now on: π is assumed to be in the discrete series. The associated dual orbit is denoted by O = H T ξ , its complement by O c . O c is a closed set of measure zero. F − 1 ( C ∞ c ( O )) denotes the set of bandlimited Schwartz functions with Fourier support contained in O . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 18 / 31

Further assumptions and notations From now on: π is assumed to be in the discrete series. The associated dual orbit is denoted by O = H T ξ , its complement by O c . O c is a closed set of measure zero. F − 1 ( C ∞ c ( O )) denotes the set of bandlimited Schwartz functions with Fourier support contained in O . We fix a weight v : G → R + is of the form v ( x , h ) = ( 1 + | x | + � h � ) s w ( h ) with s ≥ 0, a matrix norm � · � , and w : H → R + an arbitrary weight. H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 18 / 31

Further assumptions and notations From now on: π is assumed to be in the discrete series. The associated dual orbit is denoted by O = H T ξ , its complement by O c . O c is a closed set of measure zero. F − 1 ( C ∞ c ( O )) denotes the set of bandlimited Schwartz functions with Fourier support contained in O . We fix a weight v : G → R + is of the form v ( x , h ) = ( 1 + | x | + � h � ) s w ( h ) with s ≥ 0, a matrix norm � · � , and w : H → R + an arbitrary weight. For 1 ≤ p , q ≤ ∞ , let � � �� � q / p � dh L p , q R d | F ( x , h ) | p v ( x , h ) p dx v ( G ) = F : G → C : | det ( h ) | < ∞ H with obvious modifications for p = ∞ and/or q = ∞ . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 18 / 31

Further assumptions and notations From now on: π is assumed to be in the discrete series. The associated dual orbit is denoted by O = H T ξ , its complement by O c . O c is a closed set of measure zero. F − 1 ( C ∞ c ( O )) denotes the set of bandlimited Schwartz functions with Fourier support contained in O . We fix a weight v : G → R + is of the form v ( x , h ) = ( 1 + | x | + � h � ) s w ( h ) with s ≥ 0, a matrix norm � · � , and w : H → R + an arbitrary weight. For 1 ≤ p , q ≤ ∞ , let � � �� � q / p � dh L p , q R d | F ( x , h ) | p v ( x , h ) p dx v ( G ) = F : G → C : | det ( h ) | < ∞ H with obvious modifications for p = ∞ and/or q = ∞ . Note: There is a control weight v 0 for L p , q v ( G ) of the same type as v H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 18 / 31

Wavelet coorbit spaces Theorem (Kaniuth/Taylor ’96,HF ’12) The quasiregular representation is v 0 -integrable: If ψ ∈ F − 1 C ∞ c ( O ) , then W ψ ψ ∈ L 1 v 0 ( G ) . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 19 / 31

Wavelet coorbit spaces Theorem (Kaniuth/Taylor ’96,HF ’12) The quasiregular representation is v 0 -integrable: If ψ ∈ F − 1 C ∞ c ( O ) , then W ψ ψ ∈ L 1 v 0 ( G ) . Corollary c ( O ) ⊂ Co ( L p , q F − 1 C ∞ v ( G )) . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 19 / 31

Wavelet coorbit spaces Theorem (Kaniuth/Taylor ’96,HF ’12) The quasiregular representation is v 0 -integrable: If ψ ∈ F − 1 C ∞ c ( O ) , then W ψ ψ ∈ L 1 v 0 ( G ) . Corollary c ( O ) ⊂ Co ( L p , q F − 1 C ∞ v ( G )) . Theorem (HF, ’12) For all control weights v 0 satisfying v 0 ( x , h ) ≤ ( 1 + | x | ) t w 0 ( h ) , with suitable t > 0 and continuous weights w 0 on H , we have F − 1 C ∞ c ( O ) ⊂ B v 0 . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 19 / 31

Overview Introduction: Nice wavelets in dimension one 1 Square-integrability over general dilation groups 2 Outline of coorbit theory: Analyzing vectors and frame atoms 3 Wavelet coorbit spaces over general dilation groups 4 Vanishing moment conditions and coorbit spaces 5 H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 20 / 31

Chief problem: Measuring and controlling overlap Recall: Wavelet coefficient decay is related to overlap on the Fourier transform side H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 21 / 31

Chief problem: Measuring and controlling overlap Recall: Wavelet coefficient decay is related to overlap on the Fourier transform side Main questions H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 21 / 31

Chief problem: Measuring and controlling overlap Recall: Wavelet coefficient decay is related to overlap on the Fourier transform side Main questions Which vanishing moment conditions do we need to impose? H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 21 / 31

Chief problem: Measuring and controlling overlap Recall: Wavelet coefficient decay is related to overlap on the Fourier transform side Main questions Which vanishing moment conditions do we need to impose? (Answer: � ψ needs to vanish on O c ) H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 21 / 31

Chief problem: Measuring and controlling overlap Recall: Wavelet coefficient decay is related to overlap on the Fourier transform side Main questions Which vanishing moment conditions do we need to impose? (Answer: � ψ needs to vanish on O c ) How do we control overlap from vanishing moment conditions and smoothness? H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 21 / 31

Chief problem: Measuring and controlling overlap Recall: Wavelet coefficient decay is related to overlap on the Fourier transform side Main questions Which vanishing moment conditions do we need to impose? (Answer: � ψ needs to vanish on O c ) How do we control overlap from vanishing moment conditions and smoothness? (Answer: Fourier envelopes, see next slide) H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 21 / 31

Controlling overlap: Fourier envelopes Definition | · | : R d → R + 0 denotes the euclidean norm. For r , m ≥ 0 and f : R d → C , let ( 1 + | x | ) m | ∂ α f ( x ) | . | f | r , m = sup x ∈ R d , | α |≤ r H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 22 / 31

Controlling overlap: Fourier envelopes Definition | · | : R d → R + 0 denotes the euclidean norm. For r , m ≥ 0 and f : R d → C , let ( 1 + | x | ) m | ∂ α f ( x ) | . | f | r , m = sup x ∈ R d , | α |≤ r Definition (Fourier envelope function) Let O ⊂ R d denote the dual orbit. Given ξ ∈ O , let dist ( ξ, O c ) denote the euclidean distance of ξ to O c . Let � � dist ( ξ, O c ) 1 A ( ξ ) = min � | ξ | 2 − dist ( ξ, O c ) 2 , . 1 + | ξ | 1 + H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 22 / 31

Vanishing moment conditions and wavelet coefficient decay Definition Let r ∈ N be given. f ∈ L 1 ( R d ) has vanishing moments in O c of order r if all distributional derivatives ∂ α � f with | α | < r are continuous functions, identically vanishing on O c . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 23 / 31

Vanishing moment conditions and wavelet coefficient decay Definition Let r ∈ N be given. f ∈ L 1 ( R d ) has vanishing moments in O c of order r if all distributional derivatives ∂ α � f with | α | < r are continuous functions, identically vanishing on O c . Lemma Let α be a multiindex with | α | < r . Assume that f , ψ ∈ L 1 ( R d ) have vanishing moments of order r in O c , and fulfill | � f | r , r −| α | < ∞ , | � ψ | r , r −| α | < ∞ . Then there exists a constant C > 0 , independent of f and ψ , such that | ∂ α ( � f · D h � ψ )( ξ ) | C | � f | r , r −| α | | � ψ | r , r −| α | | det ( h ) | 1 / 2 ( 1 + � h � ) | α | A ( ξ ) r −| α | A ( h T ξ ) r −| α | ≤ H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 23 / 31

Quantifying overlap of Fourier envelopes H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 24 / 31

Quantifying overlap of Fourier envelopes Definition Let Φ ℓ : H → R + ∪ {∞} via � R d A ( ξ ) ℓ A ( h T ξ ) ℓ d ξ Φ ℓ ( h ) = H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 24 / 31

Quantifying overlap of Fourier envelopes Definition Let Φ ℓ : H → R + ∪ {∞} via � R d A ( ξ ) ℓ A ( h T ξ ) ℓ d ξ Φ ℓ ( h ) = Lemma (Wavelet coefficient decay) Let 0 < m < r , and let ψ ∈ L 1 ( R d ) denote a function with vanishing moments of order r in O c and | � ψ | r , r < ∞ . Then |W ψ ψ ( x , h ) | ≺ | � r , r ( 1 + | x | ) − m | det ( h ) | 1 / 2 ( 1 + � h � ∞ ) m Φ r − m ( h ) . ψ | 2 H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 24 / 31

Main result: Vanishing moment criteria for atoms H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 25 / 31

Main result: Vanishing moment criteria for atoms Definition Let w 0 : H → R + denote a weight, s ≥ 0. We call O strongly ( s , w 0 ) -temperately embedded (with index ℓ ∈ N ) if Φ ℓ ∈ W ( C 0 , L 1 m ) , where the weight m : H → R + is defined by m ( h ) = w 0 ( h ) | det ( h ) | − 1 / 2 ( 1 + � h � ) 2 ( s + d + 1 ) . H. Führ (RWTH Aachen) Wavelet coorbit spaces AHA Granada, 2013 25 / 31