Gabor Representations of evolution operators Elena Cordero (joint - PowerPoint PPT Presentation

Gabor Representations of evolution operators Elena Cordero (joint work with Fabio Nicola and Luigi Rodino) Department of Mathematics University of Torino XXXIII Convegno Nazionale di Analisi Armonica Alba 17-20 Giugno 2013

Outline Gabor frames Represinting operators by Gabor frames Historical Backgrounds Curvelet (shearlet) representations of H¨ ormander FIOs Gabor representations of Schr¨ odinger-type propagators Gabor Representations of Pseudodifferential Operators Almost diagonalization of Pseudodifferential Operators Sparsity of the Gabor matrix Applications to evolution equations

Gabor frames g ∈ L 2 ( R d ), Λ = α Z d × β Z d , α, β > 0: αβ ≤ 1, M n g ( x ) = e 2 π inx g ( x ) T m g ( x ) = g ( x − m ) , g m , n := M n T m g , ( m , n ) ∈ Λ . { g m , n } Gabor frame for L 2 ( R d ) if there exist 0 < A ≤ B < ∞ : η � |� f , g m , n �| 2 ≤ B � f � 2 A � f � 2 L 2 ≤ L 2 2 β m , n ∈ Λ β ∀ f ∈ L 2 ( R d ). x α 2 α

This implies the reconstruction formula � � f = � f , g m , n � γ m , n = � f , γ m , n � g m , n (1) ( m , n ) ∈ Λ ( m , n ) ∈ Λ { γ m , n } m , n dual Gabor frame

This implies the reconstruction formula � � f = � f , g m , n � γ m , n = � f , γ m , n � g m , n (1) ( m , n ) ∈ Λ ( m , n ) ∈ Λ { γ m , n } m , n dual Gabor frame Aim of this work: representing by Gabor frames the solutions to Cauchy problems for a class of evolution operators T with constant coefficients, including hyperbolic and parabolic operators

This implies the reconstruction formula � � f = � f , g m , n � γ m , n = � f , γ m , n � g m , n (1) ( m , n ) ∈ Λ ( m , n ) ∈ Λ { γ m , n } m , n dual Gabor frame Aim of this work: representing by Gabor frames the solutions to Cauchy problems for a class of evolution operators T with constant coefficients, including hyperbolic and parabolic operators Results: ◮ Gabor frames provide a super-exponential decay away from the diagonal of the Gabor matrix of T ◮ The Gabor representation of T is sparse

T : S ( R d ) → S ′ ( R d ) (linear continuous), providing the solution of a well-posed problem for a PDE We expect T : ◮ a pseudodifferential operator (PSDO) ◮ a Fourier integral operator (FIO) Gabor decomposition of T : � � Tf ( x ) = � Tg m , n , g m ′ , n ′ � c m , n γ m ′ , n ′ , c m , n = � f , γ m , n � � �� � ( m ′ , n ′ ) ∈ Λ ( m , n ) ∈ Λ T m ′ , n ′ , m , n T m ′ , n ′ , m , n : Gabor matrix of T .

T : S ( R d ) → S ′ ( R d ) (linear continuous), providing the solution of a well-posed problem for a PDE We expect T : ◮ a pseudodifferential operator (PSDO) ◮ a Fourier integral operator (FIO) Gabor decomposition of T : � � Tf ( x ) = � Tg m , n , g m ′ , n ′ � c m , n γ m ′ , n ′ , c m , n = � f , γ m , n � � �� � ( m ′ , n ′ ) ∈ Λ ( m , n ) ∈ Λ T m ′ , n ′ , m , n T m ′ , n ′ , m , n : Gabor matrix of T . To obtain sparsity for this representation: estimate the decay properties of T m ′ , n ′ , m , n , for large values of m ′ , n ′ , m , n .

Strictly hyperbolic problems: H¨ ormander’s FIOs [Cordoba-Fefferman 1978, Smith 1998, Cand´ es-Demanet 2003-2007, Labate-Guo 2007] Solutions to strictly hyperbolic problems, represented by (type I) FIO T : � R d e 2 π i Φ( x ,η ) σ ( x , η )ˆ Tf ( x ) = f ( η ) d η. of H¨ ormander’s type: • Φ( x , η ) is C ∞ ( R d × ( R d \ { 0 } )), real-valued, with Φ( x , λη ) = λ Φ( x , η ) , λ > 0 (positively homogeneous of degree 1 in η ); 1 , 0 ( R 2 d ), i.e., is C ∞ and ormander’s class S 0 • σ ( x , η ) in the H¨ | ∂ α η ∂ β x σ ( x , η ) | ≤ C α,β (1 + | η | ) −| α | ; • σ ( x , η ) is compactly supported in x .

Theorem { ψ µ } µ ∈I : frame of curvelets (shearlets) in the plane (d = 2 ) T FIO as above. Then |� T ψ µ , ψ µ ′ �| µ,µ ′ ∈I ≤ C N ω ( µ, h ( µ ′ )) − N , ∀ N ∈ N ω pseudo-metric and h index mapping associated with the canonical transformation of the phase Φ of T. ◮ super-polynomial off diagonal decay ◮ sparsity of curvelet (shearlet) representation of T

Theorem { ψ µ } µ ∈I : frame of curvelets (shearlets) in the plane (d = 2 ) T FIO as above. Then |� T ψ µ , ψ µ ′ �| µ,µ ′ ∈I ≤ C N ω ( µ, h ( µ ′ )) − N , ∀ N ∈ N ω pseudo-metric and h index mapping associated with the canonical transformation of the phase Φ of T. ◮ super-polynomial off diagonal decay ◮ sparsity of curvelet (shearlet) representation of T curvelet and shearlet frames are effective in dealing with it!

Schr¨ odinger-type propagators the FIO T � R d e 2 π i Φ( x ,η ) σ ( x , η )ˆ Tf ( x ) = f ( η ) d η. of Schr¨ odinger-type: ◮ Φ( x , η ) is C ∞ ( R 2 d ), real-valued, with | ∂ α x ∂ β η Φ( x , η ) | ≤ C α,β , | α | + | β | ≥ 2 , ◮ σ in the H¨ ormander’s class S 0 0 , 0 ( R 2 d ): | ∂ α x ∂ β η σ ( x , η ) | ≤ C α,β .

Examples: Schr¨ odinger Equations Free particle. � i ∂ t u + ∆ u = 0 , (2) u (0 , x ) = u 0 ( x ) , ( t , x ) ∈ R × R d , d ≥ 1. The solution: � R d e 2 π i ( x η − 2 π t | η | 2 ) � u ( t , x ) = u 0 ( η ) d η, is the FIO: u ( t , x ) = T t u 0 ( x ) , for every fixed t ∈ R with Φ( x , η ) = x η − 2 π t | η | 2 , σ ≡ 1.

Examples: Schr¨ odinger Equations Free particle. � i ∂ t u + ∆ u = 0 , (2) u (0 , x ) = u 0 ( x ) , ( t , x ) ∈ R × R d , d ≥ 1. The solution: � R d e 2 π i ( x η − 2 π t | η | 2 ) � u ( t , x ) = u 0 ( η ) d η, is the FIO: u ( t , x ) = T t u 0 ( x ) , for every fixed t ∈ R with Φ( x , η ) = x η − 2 π t | η | 2 , σ ≡ 1. Further examples: replace ∆ in (2) by a Weyl quantization H of a quadratic form on R 2 d .

Gabor frames are effective for Schr¨ odinger-type propagators T [C., F. Nicola and L. Rodino, Sparsity of Gabor representation of Schr¨ odinger propagators, ACHA, 2009] The Gabor matrix T m ′ , n ′ , m , n of a FIO T as above satisfies: | T m ′ , n ′ , m , n | ≤ C N (1 + | χ ( m , n ) − ( m ′ , n ′ ) | 2 ) − N , ∀ N ∈ N (3) where χ is the canonical transformation generated by Φ.

Gabor frames are effective for Schr¨ odinger-type propagators T [C., F. Nicola and L. Rodino, Sparsity of Gabor representation of Schr¨ odinger propagators, ACHA, 2009] The Gabor matrix T m ′ , n ′ , m , n of a FIO T as above satisfies: | T m ′ , n ′ , m , n | ≤ C N (1 + | χ ( m , n ) − ( m ′ , n ′ ) | 2 ) − N , ∀ N ∈ N (3) where χ is the canonical transformation generated by Φ. ◮ Gabor frames provide an almost diagonal matrix representation of T ◮ The Gabor representation of T is sparse

Gabor frames are effective for Schr¨ odinger-type propagators T [C., F. Nicola and L. Rodino, Sparsity of Gabor representation of Schr¨ odinger propagators, ACHA, 2009] The Gabor matrix T m ′ , n ′ , m , n of a FIO T as above satisfies: | T m ′ , n ′ , m , n | ≤ C N (1 + | χ ( m , n ) − ( m ′ , n ′ ) | 2 ) − N , ∀ N ∈ N (3) where χ is the canonical transformation generated by Φ. ◮ Gabor frames provide an almost diagonal matrix representation of T ◮ The Gabor representation of T is sparse ...what about representing hyperbolic PDEs by Gabor frames?

Drawbacks: Gabor frames are not fit for every hyperbolic problem Example: Variable coefficient wave equation in d = 1: � ∂ 2 t u − c ( x ) ∂ 2 x u = 0 , u (0 , x ) = u 0 ( x ) , ∂ t u (0 , x ) = u 1 ( x ) . with c ( x ) > 0, ∀ x . ◮ Gabor frames do not provide an almost diagonal matrix representation of the corresponding propagator T t ◮ Curvelets/shearlets do the job! ⇒ limit the study to Gabor representations of solutions to PDEs with constant coefficients

Drawbacks: Gabor frames are not fit for every hyperbolic problem Example: Variable coefficient wave equation in d = 1: � ∂ 2 t u − c ( x ) ∂ 2 x u = 0 , u (0 , x ) = u 0 ( x ) , ∂ t u (0 , x ) = u 1 ( x ) . with c ( x ) > 0, ∀ x . ◮ Gabor frames do not provide an almost diagonal matrix representation of the corresponding propagator T t ◮ Curvelets/shearlets do the job! ⇒ limit the study to Gabor representations of solutions to PDEs with constant coefficients Advantages with respect to curvelet/shearlet frames: ◮ treat hyperbolic equations, not necessarily strictly hyperbolic, of any order and dimension ◮ our class of equations includes parabolic equations ◮ the Gabor matrix off diagonal decay of super-exponential type

Example: constant coefficient wave equation � ∂ 2 t u − ∆ x u = 0 , u (0 , x ) = u 0 ( x ) , ∂ t u (0 , x ) = u 1 ( x ) . The solution: u ( t , x ) = T t u 1 ( x ) + ∂ t T t u 0 ( x ) , T t is the Fourier multiplier � e 2 π ix η σ t ( η ) � T t f ( x ) = f ( η ) d η, with symbol σ t ( η ) = sin(2 π | η | t ) η ∈ R d . , 2 π | η |