Time-Frequency Methods for Pseudodifferential Calculus

Transcription

1 Time-Frequency Methods for Pseudodifferential Calculus Karlheinz Gröchenig European Center of Time-Frequency Analysis Faculty of Mathematics University of Vienna Harmonic Analysis and Partial Differential Equations, Nagoya Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 1 / 25

2 Outline 1 Time-Frequency Analysis and Pseudodifferential Operators 2 Function Spaces and Symbol Classes 3 Almost Diagonalization of Pseudodifferential Operators 4 Inversion of Pseudodifferential Operators 5 Fourier Multipliers and PDEs Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 2 / 25

3 Time-Frequency Analysis and Pseudodifferential Operators Motivation for Time-Frequency Representations I Weak definition σ(x, D)f(x) = σ(x, ξ)ˆf(ξ)e 2πix ξ dξ R d σ(x, D)f, g ) R d = R(g, f)(x, ξ) = g(x)ˆf(ξ)e 2πix ξ σ(x, ξ)ˆf(ξ)g(x) e 2πix ξ R d }{{} dξ = σ, R(g, f) R 2d R(g, f)(x, ξ)... Rihaczek distribution R(f, f)... joint time-frequency representation of f. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 3 / 25

4 Time-Frequency Analysis and Pseudodifferential Operators Motivation for Time-Frequency Representations I Weak definition σ(x, D)f(x) = σ(x, ξ)ˆf(ξ)e 2πix ξ dξ R d σ(x, D)f, g ) R d = R(g, f)(x, ξ) = g(x)ˆf(ξ)e 2πix ξ σ(x, ξ)ˆf(ξ)g(x) e 2πix ξ R d }{{} dξ = σ, R(g, f) R 2d R(g, f)(x, ξ)... Rihaczek distribution R(f, f)... joint time-frequency representation of f. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 3 / 25

5 Time-Frequency Analysis and Pseudodifferential Operators Motivation for Time-Frequency Representations I Weak definition σ(x, D)f(x) = σ(x, ξ)ˆf(ξ)e 2πix ξ dξ R d σ(x, D)f, g ) R d = R(g, f)(x, ξ) = g(x)ˆf(ξ)e 2πix ξ σ(x, ξ)ˆf(ξ)g(x) e 2πix ξ R d }{{} dξ = σ, R(g, f) R 2d R(g, f)(x, ξ)... Rihaczek distribution R(f, f)... joint time-frequency representation of f. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 3 / 25

6 Time-Frequency Analysis and Pseudodifferential Operators Motivation for Time-Frequency Representations II σ(x, D)f(x) = = = = σ(x, ξ)ˆf(ξ)e 2πix ξ dξ R d σ(x, ξ)f(y)e 2πi(x y) ξ dydξ R 2d ˆσ(η, y x)e 2πiη x f(y) dydη R 2d ˆσ(η, u) e 2πiη x f(u + x) R 2d }{{} dudη Time-Frequency shift : z = (x, ξ) R 2d, t R d π(z)f(t) = e 2πiξ t f(t x) σ(x, D)f = R 2d ˆσ(η, u)π( u, η) dudη Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 4 / 25

7 Time-Frequency Analysis and Pseudodifferential Operators Motivation for Time-Frequency Representations II σ(x, D)f(x) = = = = σ(x, ξ)ˆf(ξ)e 2πix ξ dξ R d σ(x, ξ)f(y)e 2πi(x y) ξ dydξ R 2d ˆσ(η, y x)e 2πiη x f(y) dydη R 2d ˆσ(η, u) e 2πiη x f(u + x) R 2d }{{} dudη Time-Frequency shift : z = (x, ξ) R 2d, t R d π(z)f(t) = e 2πiξ t f(t x) σ(x, D)f = R 2d ˆσ(η, u)π( u, η) dudη Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 4 / 25

8 Time-Frequency Analysis and Pseudodifferential Operators Motivation for Time-Frequency Representations II σ(x, D)f(x) = = = = σ(x, ξ)ˆf(ξ)e 2πix ξ dξ R d σ(x, ξ)f(y)e 2πi(x y) ξ dydξ R 2d ˆσ(η, y x)e 2πiη x f(y) dydη R 2d ˆσ(η, u) e 2πiη x f(u + x) R 2d }{{} dudη Time-Frequency shift : z = (x, ξ) R 2d, t R d π(z)f(t) = e 2πiξ t f(t x) σ(x, D)f = R 2d ˆσ(η, u)π( u, η) dudη Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 4 / 25

9 Time-Frequency Analysis and Pseudodifferential Operators Plan Signal transform Associated Function spaces Basis-like structure Operators Short-time Fourier transform Modulation spaces Gabor frames Pseudodifferential operators Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 5 / 25

10 Time-Frequency Analysis and Pseudodifferential Operators Plan Signal transform Associated Function spaces Basis-like structure Operators Short-time Fourier transform Modulation spaces Gabor frames Pseudodifferential operators Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 5 / 25

11 Time-Frequency Analysis and Pseudodifferential Operators Time-Frequency Analysis Short-time Fourier transform w.r.t. window g 0, g S, say: V g f(z) = f, π(z)g = f(t)g(t x)e 2πiξ t dt = ( f g(. x) ) (ξ) R d Represents distribution of f in phase-space (time-frequency plane ) Local version of Fourier transform Special case: for g(t) = e πt2, z = x + iξ C d V g f(x, ξ) = e πix ξ Bf(z)e π z 2 /2 Bf Bargmann transform of f, Bf is entire. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 6 / 25

12 Time-Frequency Analysis and Pseudodifferential Operators Time-Frequency Analysis Short-time Fourier transform w.r.t. window g 0, g S, say: V g f(z) = f, π(z)g = f(t)g(t x)e 2πiξ t dt = ( f g(. x) ) (ξ) R d Represents distribution of f in phase-space (time-frequency plane ) Local version of Fourier transform Special case: for g(t) = e πt2, z = x + iξ C d V g f(x, ξ) = e πix ξ Bf(z)e π z 2 /2 Bf Bargmann transform of f, Bf is entire. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 6 / 25

13 Time-Frequency Analysis and Pseudodifferential Operators Time-Frequency Analysis Short-time Fourier transform w.r.t. window g 0, g S, say: V g f(z) = f, π(z)g = f(t)g(t x)e 2πiξ t dt = ( f g(. x) ) (ξ) R d Represents distribution of f in phase-space (time-frequency plane ) Local version of Fourier transform Special case: for g(t) = e πt2, z = x + iξ C d V g f(x, ξ) = e πix ξ Bf(z)e π z 2 /2 Bf Bargmann transform of f, Bf is entire. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 6 / 25

14 Time-Frequency Analysis and Pseudodifferential Operators Time-Frequency Analysis Short-time Fourier transform w.r.t. window g 0, g S, say: V g f(z) = f, π(z)g = f(t)g(t x)e 2πiξ t dt = ( f g(. x) ) (ξ) R d Represents distribution of f in phase-space (time-frequency plane ) Local version of Fourier transform Special case: for g(t) = e πt2, z = x + iξ C d V g f(x, ξ) = e πix ξ Bf(z)e π z 2 /2 Bf Bargmann transform of f, Bf is entire. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 6 / 25

15 Function Spaces and Symbol Classes Modulation Spaces Fix non-zero test function g, 0 < p, q, weight m with norm f Mm p,q (R d ) V g f L p,q m f M p,q m = V gf L p,q m ( ( ) q/p ) 1/q = V g f(x, ξ) p m(x, ξ) p dx dξ R d R d Independence of g Many equivalent definitions Smoothness measured in time-frequency plane Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 7 / 25

16 Function Spaces and Symbol Classes Modulation Spaces Fix non-zero test function g, 0 < p, q, weight m with norm f Mm p,q (R d ) V g f L p,q m f M p,q m = V gf L p,q m ( ( ) q/p ) 1/q = V g f(x, ξ) p m(x, ξ) p dx dξ R d R d Independence of g Many equivalent definitions Smoothness measured in time-frequency plane Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 7 / 25

17 Function Spaces and Symbol Classes Analogy to Besov spaces (smoothness measured with differences and derivatives) Frequency definition m(x, ξ) = (1 + ξ ) s f M p,q m ( κ Z d φ(d κ)f q p ) 1/q Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 8 / 25

18 Function Spaces and Symbol Classes Modulation Spaces II H. G. Feichtinger 1983 Hard Analysis Tachizawa 94, 98 Sjöstrand 94/95, 08 Boulkhemair 97 Lerner 06 Wang 06 Time-Frequency Analysis Heil 99 KG 99 Toft 01 Pilipovic, Teofanov 01 Torino group (Cordero, Rodino, etc.) 01 Strohmer 05 Sugimoto 07 Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 9 / 25

19 Function Spaces and Symbol Classes Modulation Spaces II H. G. Feichtinger 1983 Hard Analysis Tachizawa 94, 98 Sjöstrand 94/95, 08 Boulkhemair 97 Lerner 06 Wang 06 Time-Frequency Analysis Heil 99 KG 99 Toft 01 Pilipovic, Teofanov 01 Torino group (Cordero, Rodino, etc.) 01 Strohmer 05 Sugimoto 07 Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September 09 9 / 25

20 Function Spaces and Symbol Classes Sjöstrand s Class σ M,1 = R 2d sup z R 2d (σ Φ( z)) (ζ) }{{} dζ < V Φ σ(z, ζ) σ M,1 v = R 2d sup z R 2d V Φ σ(z, ζ) v(ζ) dζ < Embeddings: S 0 0,0 C2d+1 (R 2d ) M,1 (R 2d ) M,1 contains non-smooth symbols M,1 general symbol class for constant geometry. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

21 Function Spaces and Symbol Classes Sjöstrand s Class σ M,1 = R 2d sup z R 2d (σ Φ( z)) (ζ) }{{} dζ < V Φ σ(z, ζ) σ M,1 v = R 2d sup z R 2d V Φ σ(z, ζ) v(ζ) dζ < Embeddings: S 0 0,0 C2d+1 (R 2d ) M,1 (R 2d ) M,1 contains non-smooth symbols M,1 general symbol class for constant geometry. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

22 Function Spaces and Symbol Classes Gabor Frames Gabor frames are appropriate basis -like structure in time-frequency analysis. Discretization of continuous resolutions of the identity Fix: test function g S(R d ), g 0 lattice Λ = AZ 2d for 2d 2d-matrix A with det A 0. The set G(g, Λ) = {π(λ)g : λ Λ} is a Gabor frame, if there exist A, B > 0, such that A f 2 2 λ Λ f, π(λ)g 2 B f 2 2 f L 2 (R d ) Consequences: 1. Gabor expansions: there exists γ S(R d ), γ 0, such that f = λ Λ f, π(λ)g π(λ)γ f L 2 (R d ) 2. Characterization of modulation spaces Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

23 Function Spaces and Symbol Classes Gabor Frames Gabor frames are appropriate basis -like structure in time-frequency analysis. Discretization of continuous resolutions of the identity Fix: test function g S(R d ), g 0 lattice Λ = AZ 2d for 2d 2d-matrix A with det A 0. The set G(g, Λ) = {π(λ)g : λ Λ} is a Gabor frame, if there exist A, B > 0, such that A f 2 2 λ Λ f, π(λ)g 2 B f 2 2 f L 2 (R d ) Consequences: 1. Gabor expansions: there exists γ S(R d ), γ 0, such that f = λ Λ f, π(λ)g π(λ)γ f L 2 (R d ) 2. Characterization of modulation spaces Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

24 Function Spaces and Symbol Classes Gabor Frames Gabor frames are appropriate basis -like structure in time-frequency analysis. Discretization of continuous resolutions of the identity Fix: test function g S(R d ), g 0 lattice Λ = AZ 2d for 2d 2d-matrix A with det A 0. The set G(g, Λ) = {π(λ)g : λ Λ} is a Gabor frame, if there exist A, B > 0, such that A f 2 2 λ Λ f, π(λ)g 2 B f 2 2 f L 2 (R d ) Consequences: 1. Gabor expansions: there exists γ S(R d ), γ 0, such that f = λ Λ f, π(λ)g π(λ)γ f L 2 (R d ) 2. Characterization of modulation spaces Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

25 Function Spaces and Symbol Classes Gabor Frames Gabor frames are appropriate basis -like structure in time-frequency analysis. Discretization of continuous resolutions of the identity Fix: test function g S(R d ), g 0 lattice Λ = AZ 2d for 2d 2d-matrix A with det A 0. The set G(g, Λ) = {π(λ)g : λ Λ} is a Gabor frame, if there exist A, B > 0, such that A f 2 2 λ Λ f, π(λ)g 2 B f 2 2 f L 2 (R d ) Consequences: 1. Gabor expansions: there exists γ S(R d ), γ 0, such that f = λ Λ f, π(λ)g π(λ)γ f L 2 (R d ) 2. Characterization of modulation spaces Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

26 Almost Diagonalization of Pseudodifferential Operators Almost Diagonalization We will study σ(x, D) with respect to Gabor frame. Consider matrix M = M(σ) with entries M(σ) λ,µ = σ(x, D)(π(µ)g), π(λ)g λ, µ Λ. Theorem (KG, 04) Assume that g S(R d ) and G(g, Λ) is Gabor frame. Then σ M,1 if and only if there exists h l 1 (Λ) such that for all λ, µ Λ. M(σ) λ,µ = σ(x, D)π(µ)g, π(λ)g h(λ µ) Matrix with respect to Gabor frame is almost diagonal Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

27 Almost Diagonalization of Pseudodifferential Operators Almost Diagonalization We will study σ(x, D) with respect to Gabor frame. Consider matrix M = M(σ) with entries M(σ) λ,µ = σ(x, D)(π(µ)g), π(λ)g λ, µ Λ. Theorem (KG, 04) Assume that g S(R d ) and G(g, Λ) is Gabor frame. Then σ M,1 if and only if there exists h l 1 (Λ) such that for all λ, µ Λ. M(σ) λ,µ = σ(x, D)π(µ)g, π(λ)g h(λ µ) Matrix with respect to Gabor frame is almost diagonal Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

28 Almost Diagonalization of Pseudodifferential Operators Almost Diagonalization We will study σ(x, D) with respect to Gabor frame. Consider matrix M = M(σ) with entries M(σ) λ,µ = σ(x, D)(π(µ)g), π(λ)g λ, µ Λ. Theorem (KG, 04) Assume that g S(R d ) and G(g, Λ) is Gabor frame. Then σ M,1 if and only if there exists h l 1 (Λ) such that for all λ, µ Λ. M(σ) λ,µ = σ(x, D)π(µ)g, π(λ)g h(λ µ) Matrix with respect to Gabor frame is almost diagonal Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

29 Almost Diagonalization of Pseudodifferential Operators Almost Diagonalization Refinements We will study σ(x, D) with respect to Gabor frame. Consider matrix M = M(σ) with entries M(σ) λ,µ = σ(x, D)(π(µ)g), π(λ)g λ, µ Λ. Weights: v(z 1 + z 2 ) v(z 1 )v(z 2 ), z 1, z 2 R 2d, v( z) = v(z) and lim n v(nz) 1/n = 1 Theorem (KG) Assume that g M 1 v (R d ) and G(g, Λ) is Gabor frame. Then σ M,1 v if and only if there exists h l 1 v(λ) such that for all λ, µ Λ. M(σ) λ,µ = σ(x, D)π(µ)g, π(λ)g h(λ µ) Matrix with respect to Gabor frame is almost diagonal Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

30 Almost Diagonalization of Pseudodifferential Operators Almost Diagonalization Refinements We will study σ(x, D) with respect to Gabor frame. Consider matrix M = M(σ) with entries M(σ) λ,µ = σ(x, D)(π(µ)g), π(λ)g λ, µ Λ. Weights: v(z 1 + z 2 ) v(z 1 )v(z 2 ), z 1, z 2 R 2d, v( z) = v(z) and lim n v(nz) 1/n = 1 Theorem (KG) Assume that g M 1 v (R d ) and G(g, Λ) is Gabor frame. Then σ M,1 v if and only if there exists h l 1 v(λ) such that for all λ, µ Λ. M(σ) λ,µ = σ(x, D)π(µ)g, π(λ)g h(λ µ) Matrix with respect to Gabor frame is almost diagonal Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

31 Almost Diagonalization of Pseudodifferential Operators Refinements Decay Conditions σ M, v = sup z,ζ R 2d V Φ σ(z, ζ) v(ζ) Theorem (KG) Assume that g M 1 v (R d ) and G(g, Λ) is Gabor frame. Then σ M, v if and only if for all λ, µ Λ. M(σ) λ,µ = σ(x, D)π(µ)g, π(λ)g v(λ µ) 1 Example: v(ζ) = (1 + ζ ) s or v(ζ) = e a ζ b, a, b 0, b < 1. REMARK: appropriate symbol classes and basis imply off-diagonal decay (wavelets Meyer, Gabor frames Tachizawa, Rochberg, local Fourier bases Tachizawa) Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

32 Almost Diagonalization of Pseudodifferential Operators Refinements Decay Conditions σ M, v = sup z,ζ R 2d V Φ σ(z, ζ) v(ζ) Theorem (KG) Assume that g M 1 v (R d ) and G(g, Λ) is Gabor frame. Then σ M, v if and only if for all λ, µ Λ. M(σ) λ,µ = σ(x, D)π(µ)g, π(λ)g v(λ µ) 1 Example: v(ζ) = (1 + ζ ) s or v(ζ) = e a ζ b, a, b 0, b < 1. REMARK: appropriate symbol classes and basis imply off-diagonal decay (wavelets Meyer, Gabor frames Tachizawa, Rochberg, local Fourier bases Tachizawa) Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

33 Almost Diagonalization of Pseudodifferential Operators Refinements Decay Conditions σ M, v = sup z,ζ R 2d V Φ σ(z, ζ) v(ζ) Theorem (KG) Assume that g M 1 v (R d ) and G(g, Λ) is Gabor frame. Then σ M, v if and only if for all λ, µ Λ. M(σ) λ,µ = σ(x, D)π(µ)g, π(λ)g v(λ µ) 1 Example: v(ζ) = (1 + ζ ) s or v(ζ) = e a ζ b, a, b 0, b < 1. REMARK: appropriate symbol classes and basis imply off-diagonal decay (wavelets Meyer, Gabor frames Tachizawa, Rochberg, local Fourier bases Tachizawa) Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

34 Almost Diagonalization of Pseudodifferential Operators The Hörmander Class S 0 0,0 σ M, ζ s if and only if σ(x, D)π(µ)g, π(λ)g C (1 + λ µ ) s Corollary g S, g 0, G(g, Λ) Gabor frame. Then σ S0,0 0 if and only if ( σ(x, D)π(µ)g, π(λ)g = O (1 + λ µ ) s) s 0. Meyer, Tartaru Almost diagonalization w.r.t. time-frequency molecules and local Fourier bases (with Rzeszotnik) Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

35 Almost Diagonalization of Pseudodifferential Operators The Hörmander Class S 0 0,0 σ M, ζ s if and only if σ(x, D)π(µ)g, π(λ)g C (1 + λ µ ) s Corollary g S, g 0, G(g, Λ) Gabor frame. Then σ S0,0 0 if and only if ( σ(x, D)π(µ)g, π(λ)g = O (1 + λ µ ) s) s 0. Meyer, Tartaru Almost diagonalization w.r.t. time-frequency molecules and local Fourier bases (with Rzeszotnik) Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

36 Almost Diagonalization of Pseudodifferential Operators The Hörmander Class S 0 0,0 σ M, ζ s if and only if σ(x, D)π(µ)g, π(λ)g C (1 + λ µ ) s Corollary g S, g 0, G(g, Λ) Gabor frame. Then σ S0,0 0 if and only if ( σ(x, D)π(µ)g, π(λ)g = O (1 + λ µ ) s) s 0. Meyer, Tartaru Almost diagonalization w.r.t. time-frequency molecules and local Fourier bases (with Rzeszotnik) Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

37 Almost Diagonalization of Pseudodifferential Operators Consequences of Almost Diagonalization Algebra property: if σ 1, σ 2 Mv,1, then σ 1 (x, D)σ 2 (x, D) = τ(x, D) for τ Mv,1. Boundedness property: If σ Mv,1, then σ(x, D) is bounded on the modulation space Mm p,q for 1 p, q and suitable weights m. Sjöstrand, Toft, KG, Rzesnotnik Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

38 Almost Diagonalization of Pseudodifferential Operators Consequences of Almost Diagonalization Algebra property: if σ 1, σ 2 Mv,1, then σ 1 (x, D)σ 2 (x, D) = τ(x, D) for τ Mv,1. Boundedness property: If σ Mv,1, then σ(x, D) is bounded on the modulation space Mm p,q for 1 p, q and suitable weights m. Sjöstrand, Toft, KG, Rzesnotnik Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

39 Almost Diagonalization of Pseudodifferential Operators Consequences of Almost Diagonalization Algebra property: if σ 1, σ 2 Mv,1, then σ 1 (x, D)σ 2 (x, D) = τ(x, D) for τ Mv,1. Boundedness property: If σ Mv,1, then σ(x, D) is bounded on the modulation space Mm p,q for 1 p, q and suitable weights m. Sjöstrand, Toft, KG, Rzesnotnik Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

40 Almost Diagonalization of Pseudodifferential Operators Wireless Communications Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

41 Almost Diagonalization of Pseudodifferential Operators Time-Varying Channels Continuous Case Time-delays caused by reflections time-shifts Doppler effect caused by mobile environment frequency shifts Received signal f = K σ f is a superposition of time-frequency shifts : K σ f(x) = σ(η, u) e 2πiη x f(x + u) dudη R 2d = σ(x, ξ)ˆf(ξ)e 2πix ξ dξ R d = σ(x, D)f(x) Pseudodifferential operator with (Kohn-Nirenberg) symbol σ Strohmer: M,1, 0 b 1, is suitable symbol class for wireless e a ζ b communications. Need to understand properties of inverse σ(x, D) 1. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

42 Almost Diagonalization of Pseudodifferential Operators Time-Varying Channels Continuous Case Time-delays caused by reflections time-shifts Doppler effect caused by mobile environment frequency shifts Received signal f = K σ f is a superposition of time-frequency shifts : K σ f(x) = σ(η, u) e 2πiη x f(x + u) dudη R 2d = σ(x, ξ)ˆf(ξ)e 2πix ξ dξ R d = σ(x, D)f(x) Pseudodifferential operator with (Kohn-Nirenberg) symbol σ Strohmer: M,1, 0 b 1, is suitable symbol class for wireless e a ζ b communications. Need to understand properties of inverse σ(x, D) 1. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

43 Almost Diagonalization of Pseudodifferential Operators Time-Varying Channels Continuous Case Time-delays caused by reflections time-shifts Doppler effect caused by mobile environment frequency shifts Received signal f = K σ f is a superposition of time-frequency shifts : K σ f(x) = σ(η, u) e 2πiη x f(x + u) dudη R 2d = σ(x, ξ)ˆf(ξ)e 2πix ξ dξ R d = σ(x, D)f(x) Pseudodifferential operator with (Kohn-Nirenberg) symbol σ Strohmer: M,1, 0 b 1, is suitable symbol class for wireless e a ζ b communications. Need to understand properties of inverse σ(x, D) 1. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

44 Inversion of Pseudodifferential Operators Wiener s Lemma for M,1 v Theorem (Sjöstrand) If σ M,1 (R 2d ) and K σ is invertible on L 2 (R d ), then Kσ 1 some τ M,1. = K τ for Theorem (KG) Assume that v is submultiplicative and GRS lim n v(nz) 1/n = 1, z R 2d. If σ Mv,1 (R 2d ) and K σ is invertible on L 2 (R d ), then Kσ 1 = K τ for some τ M,1 v. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

45 Inversion of Pseudodifferential Operators Wiener s Lemma for M,1 v Theorem (Sjöstrand) If σ M,1 (R 2d ) and K σ is invertible on L 2 (R d ), then Kσ 1 some τ M,1. = K τ for Theorem (KG) Assume that v is submultiplicative and GRS lim n v(nz) 1/n = 1, z R 2d. If σ Mv,1 (R 2d ) and K σ is invertible on L 2 (R d ), then Kσ 1 = K τ for some τ M,1 v. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

46 Inversion of Pseudodifferential Operators Hörmander s Class Hörmander class: σ S 0 0,0 if and only if α σ L (R 2d ), α 0. Observation: If v s (ζ) = (1 + ζ ) s, then S 0 0,0 = s 0 M,1 v s Corollary (Beals 75) If σ S0,0 0 and K σ is invertible on L 2 (R d ), then Kσ 1 τ S0,0 0. = K τ for some Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

47 Inversion of Pseudodifferential Operators Hörmander s Class Hörmander class: σ S 0 0,0 if and only if α σ L (R 2d ), α 0. Observation: If v s (ζ) = (1 + ζ ) s, then S 0 0,0 = s 0 M,1 v s Corollary (Beals 75) If σ S0,0 0 and K σ is invertible on L 2 (R d ), then Kσ 1 τ S0,0 0. = K τ for some Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

48 Inversion of Pseudodifferential Operators Hörmander s Class Hörmander class: σ S 0 0,0 if and only if α σ L (R 2d ), α 0. Observation: If v s (ζ) = (1 + ζ ) s, then S 0 0,0 = s 0 M,1 v s Corollary (Beals 75) If σ S0,0 0 and K σ is invertible on L 2 (R d ), then Kσ 1 τ S0,0 0. = K τ for some Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

49 Inversion of Pseudodifferential Operators Regularity Assume that σ Mv,1 (R 2d ) and σ(x, D) is invertible on L 2 (R d ) (order 0) σ(x, D)f = u If u M p,q m, then f M p,q m [p, q [1, ] and suitable weight m ( m must satisfy m(z 1 + z 2 ) Cv(z 1 )m(z 2 ), z 1, z 2 R 2d )XS] Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

50 Inversion of Pseudodifferential Operators Regularity Assume that σ Mv,1 (R 2d ) and σ(x, D) is invertible on L 2 (R d ) (order 0) σ(x, D)f = u If u M p,q m, then f M p,q m [p, q [1, ] and suitable weight m ( m must satisfy m(z 1 + z 2 ) Cv(z 1 )m(z 2 ), z 1, z 2 R 2d )XS] Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

51 Inversion of Pseudodifferential Operators Regularity Assume that σ Mv,1 (R 2d ) and σ(x, D) is invertible on L 2 (R d ) (order 0) σ(x, D)f = u If u M p,q m, then f M p,q m [p, q [1, ] and suitable weight m ( m must satisfy m(z 1 + z 2 ) Cv(z 1 )m(z 2 ), z 1, z 2 R 2d )XS] Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

52 Inversion of Pseudodifferential Operators Lifting Properties and the Isomorphism Theorem µ... moderate function of polynomial growth Φ(z) = e πz z, z R 2d Theorem (KG, Toft) Set σ = µ Φ. Then the operator σ(x, D) is a isomorphism from Mm p,q onto M p,q m/µ for 1 p, q and every polynomially moderate weight m. Conclusion: Often it suffices to restrict to the unweighted case or to operators of order 0. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

53 Fourier Multipliers and PDEs Free Schrödinger Equation i u t (x, t) = xu(x, t) u(x, 0) = f(x), x R d, t > 0 Theorem Time evolution preserves M p,q (constant weight m 1. u(, t) M p,q C(1 + t 2 ) d/4 f M p,q Initial conditions in M p,q are preserved, but not in L p. Time-frequency distribution is preserved by Schrödinger equation. Wang etc. (2006), Benyi, KG, Okoudjou, Rogers (2007) with a multiplier theorem; Cordero, etc., de Gosson (2008) Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

54 Fourier Multipliers and PDEs Free Schrödinger Equation i u t (x, t) = xu(x, t) u(x, 0) = f(x), x R d, t > 0 Theorem Time evolution preserves M p,q (constant weight m 1. u(, t) M p,q C(1 + t 2 ) d/4 f M p,q Initial conditions in M p,q are preserved, but not in L p. Time-frequency distribution is preserved by Schrödinger equation. Wang etc. (2006), Benyi, KG, Okoudjou, Rogers (2007) with a multiplier theorem; Cordero, etc., de Gosson (2008) Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

55 Fourier Multipliers and PDEs Free Schrödinger Equation i u t (x, t) = xu(x, t) u(x, 0) = f(x), x R d, t > 0 Theorem Time evolution preserves M p,q (constant weight m 1. u(, t) M p,q C(1 + t 2 ) d/4 f M p,q Initial conditions in M p,q are preserved, but not in L p. Time-frequency distribution is preserved by Schrödinger equation. Wang etc. (2006), Benyi, KG, Okoudjou, Rogers (2007) with a multiplier theorem; Cordero, etc., de Gosson (2008) Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

56 Fourier Multipliers and PDEs Wave Equation i 2 u t 2 (x, t) = xu(x, t) u(x, 0) = f(x), x R d, t > 0 u (x, 0) = g(x) t Theorem Time evolution preserve M p,q (constant weight m 1). u(, t) M p,q C(t) ( f M p,q + g M p,q). Initial conditions in M p,q are preserved, but not in L p. Nonlinear Schrödinger equation on modulation spaces Wang, etc. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

57 Fourier Multipliers and PDEs Wave Equation i 2 u t 2 (x, t) = xu(x, t) u(x, 0) = f(x), x R d, t > 0 u (x, 0) = g(x) t Theorem Time evolution preserve M p,q (constant weight m 1). u(, t) M p,q C(t) ( f M p,q + g M p,q). Initial conditions in M p,q are preserved, but not in L p. Nonlinear Schrödinger equation on modulation spaces Wang, etc. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

58 Fourier Multipliers and PDEs Wave Equation i 2 u t 2 (x, t) = xu(x, t) u(x, 0) = f(x), x R d, t > 0 u (x, 0) = g(x) t Theorem Time evolution preserve M p,q (constant weight m 1). u(, t) M p,q C(t) ( f M p,q + g M p,q). Initial conditions in M p,q are preserved, but not in L p. Nonlinear Schrödinger equation on modulation spaces Wang, etc. Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25

59 Fourier Multipliers and PDEs Summary Ingredients of time-frequency analysis (phase-space analysis): Use short-time Fourier transform as basic object and base analysis on STFT Modulation spaces (quantify time-frequency content/phase space content) physics Gabor frames characterizations of modulation spaces, sparsity (compare wavelet bases and Besov-Triebel-Lizorkin spaces) Pseudodifferential operators Karlheinz Gröchenig (EUCETIFA) Time-Frequency Methods September / 25