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¨

• rmander FIOs

Gabor representations of Schr¨

• dinger-type propagators

Gabor Representations of Pseudodifferential Operators Almost diagonalization of Pseudodifferential Operators Sparsity of the Gabor matrix Applications to evolution equations

Gabor frames

g ∈ L2(Rd), Λ = αZd × βZd, α, β > 0: αβ ≤ 1, Mng(x) = e2πinxg(x) Tmg(x) = g(x − m), gm,n := MnTmg, (m, n) ∈ Λ. {gm,n} Gabor frame for L2(Rd) if there exist 0 < A ≤ B < ∞:

x η α 2α β 2β

Af 2

L2 ≤

• m,n∈Λ

|f , gm,n|2 ≤ Bf 2

L2

∀f ∈ L2(Rd).

This implies the reconstruction formula f =

• (m,n)∈Λ

f , gm,nγm,n =

• (m,n)∈Λ

f , γm,ngm,n (1) {γm,n}m,n dual Gabor frame

This implies the reconstruction formula f =

• (m,n)∈Λ

f , gm,nγm,n =

• (m,n)∈Λ

f , γm,ngm,n (1) {γm,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 =

• (m,n)∈Λ

f , gm,nγm,n =

• (m,n)∈Λ

f , γm,ngm,n (1) {γm,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(Rd) → S′(Rd) (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) =

• (m′,n′)∈Λ
• (m,n)∈Λ

Tgm,n, gm′,n′

• Tm′,n′,m,n

cm,nγm′,n′, cm,n = f , γm,n Tm′,n′,m,n : Gabor matrix of T.

T : S(Rd) → S′(Rd) (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) =

• (m′,n′)∈Λ
• (m,n)∈Λ

Tgm,n, gm′,n′

• Tm′,n′,m,n

cm,nγm′,n′, cm,n = f , γm,n Tm′,n′,m,n : Gabor matrix of T.

To obtain sparsity for this representation: estimate the decay properties of Tm′,n′,m,n, for large values of m′, n′, m, n.

Strictly hyperbolic problems: H¨

• rmander’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: Tf (x) =

• Rd e2πiΦ(x,η)σ(x, η)ˆ

f (η)dη.

• f H¨
• rmander’s type:
• Φ(x, η) is C∞(Rd × (Rd \ {0})), real-valued, with

Φ(x, λη) = λΦ(x, η), λ > 0 (positively homogeneous of degree 1 in η);

• σ(x, η) in the H¨
• rmander’s class S0

1,0(R2d), i.e., is C∞ and

|∂α

η ∂β x σ(x, η)| ≤ Cα,β(1 + |η|)−|α|;

• σ(x, η) is compactly supported in x.

Theorem

{ψµ}µ∈I : frame of curvelets (shearlets) in the plane (d = 2) T FIO as above. Then |Tψµ, ψµ′|µ,µ′∈I ≤ CN 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

{ψµ}µ∈I : frame of curvelets (shearlets) in the plane (d = 2) T FIO as above. Then |Tψµ, ψµ′|µ,µ′∈I ≤ CN 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¨

• dinger-type propagators

the FIO T Tf (x) =

• Rd e2πiΦ(x,η)σ(x, η)ˆ

f (η)dη.

• f Schr¨
• dinger-type:

◮ Φ(x, η) is C∞(R2d), real-valued, with

|∂α

x ∂β η Φ(x, η)| ≤ Cα,β, |α| + |β| ≥ 2, ◮ σ in the H¨

• rmander’s class S0

0,0(R2d):

|∂α

x ∂β η σ(x, η)| ≤ Cα,β .

Examples: Schr¨

• dinger Equations

Free particle.

• i∂tu + ∆u = 0,

u(0, x) = u0(x), (2) (t, x) ∈ R × Rd, d ≥ 1. The solution: u(t, x) =

• Rd e2πi(xη−2πt|η|2)

u0(η)dη, is the FIO: u(t, x) = Ttu0(x), for every fixed t ∈ R with Φ(x, η) = xη − 2πt|η|2, σ ≡ 1.

Examples: Schr¨

• dinger Equations

Free particle.

• i∂tu + ∆u = 0,

u(0, x) = u0(x), (2) (t, x) ∈ R × Rd, d ≥ 1. The solution: u(t, x) =

• Rd e2πi(xη−2πt|η|2)

u0(η)dη, is the FIO: u(t, x) = Ttu0(x), for every fixed t ∈ R with Φ(x, η) = xη − 2πt|η|2, σ ≡ 1.

Further examples: replace ∆ in (2) by a Weyl quantization H

• f a quadratic form on R2d.

Gabor frames are effective for Schr¨

• dinger-type propagators T

[C., F. Nicola and L. Rodino, Sparsity of Gabor representation of Schr¨

• dinger

propagators, ACHA, 2009]

The Gabor matrix Tm′,n′,m,n of a FIO T as above satisfies: |Tm′,n′,m,n| ≤ CN(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¨

• dinger-type propagators T

[C., F. Nicola and L. Rodino, Sparsity of Gabor representation of Schr¨

• dinger

propagators, ACHA, 2009]

The Gabor matrix Tm′,n′,m,n of a FIO T as above satisfies: |Tm′,n′,m,n| ≤ CN(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¨

• dinger-type propagators T

[C., F. Nicola and L. Rodino, Sparsity of Gabor representation of Schr¨

• dinger

propagators, ACHA, 2009]

The Gabor matrix Tm′,n′,m,n of a FIO T as above satisfies: |Tm′,n′,m,n| ≤ CN(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 xu = 0,

u(0, x) = u0(x), ∂tu(0, x) = u1(x). with c(x) > 0, ∀x.

◮ Gabor frames do not provide an almost diagonal matrix

representation of the corresponding propagator Tt

◮ 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 xu = 0,

u(0, x) = u0(x), ∂tu(0, x) = u1(x). with c(x) > 0, ∀x.

◮ Gabor frames do not provide an almost diagonal matrix

representation of the corresponding propagator Tt

◮ 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

• rder 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 − ∆xu = 0,

u(0, x) = u0(x), ∂tu(0, x) = u1(x). The solution: u(t, x) = Ttu1(x) + ∂tTtu0(x), Tt is the Fourier multiplier Ttf (x) =

• e2πixησt(η)

f (η) dη, with symbol σt(η) = sin(2π|η|t) 2π|η| , η ∈ Rd.

{gm,n, (m, n) ∈ Λ} Gabor frame with g(x) = 2d/4e−π|x|2, Λ = Zd × (1/2)Zd. The Gabor matrix Tm′,n′,m,n of the Fourier multiplier Tt above satisfies |Tm′,n′,m,n| ≤ Cte−ǫt(|m′−m|2+|n′−n|2), (m, n), (m′, n′) ∈ Λ. for suitable Ct > 0, ǫt > 0.

With respect to the approach by H¨

• rmander FIOs:

◮ we study PSDO rather than FIO where the η-dependent phase

functions are here absorbed into the symbol σt(η).

◮ σt(η) is in the non-standard H¨

• rmander’s class S0

0,0

Gabor Representations of Pseudodifferential Operators

PSDO T = σw(x, D) in the Weyl form σw(x, D)f (x) =

• Rd σ

x + y 2 , η

• e2πi(x−y)ηf (y) dydη

with symbol σ ∈ C∞(R2d) satisfying, for s ≥ 0, z = (x, η) ∈ R2d, |∂ασ(z)| ≤ C |α|+1(α!)s (4) σ is ultra-analytic if 0 ≤ s < 1, analytic if s = 1, Gevrey if s > 1.

Estimates (4) satisfied by symbols of a large class of evolution

• perators with constant coefficients, with 0 ≤ s < 1. In

particular, s = 0 for the wave equation.

Almost diagonalization of Pseudodifferential Operators

f ∈ S(Rd) in the Gelfand-Shilov space Ss

s (Rd): if ∃A, B > 0:

sup

x∈Rd |xα∂βf (x)| A|α|B|β|(α!)s(β!)s,

α, β ∈ Nd equivalently, if ∃h, k > 0: feh|x|1/sL∞ < ∞ and ˆ f ek|η|1/sL∞ < ∞. For z = (z1, z2) ∈ R2d, set π(z)g := Mz2Tz1g

Theorem

s ≥ 1/2, g ∈ Ss

s (Rd). Then TFAE for σ ∈ C∞(R2d):

(i) σ satisfies |∂ασ(z)| C |α|(α!)s, ∀ z ∈ R2d, ∀α ∈ N2d. (5) (ii) There exists ǫ > 0: |σw(x, D)π(z)g, π(w)g| e−ǫ|w−z|

1 s ,

∀ z, w ∈ R2d. (6)

Almost diagonalization of Pseudodifferential Operators

f ∈ S(Rd) in the Gelfand-Shilov space Ss

s (Rd): if ∃A, B > 0:

sup

x∈Rd |xα∂βf (x)| A|α|B|β|(α!)s(β!)s,

α, β ∈ Nd equivalently, if ∃h, k > 0: feh|x|1/sL∞ < ∞ and ˆ f ek|η|1/sL∞ < ∞. For z = (z1, z2) ∈ R2d, set π(z)g := Mz2Tz1g

Theorem

s ≥ 1/2, g ∈ Ss

s (Rd). Then TFAE for σ ∈ C∞(R2d):

(i) σ satisfies |∂ασ(z)| C |α|(α!)s, ∀ z ∈ R2d, ∀α ∈ N2d. (5) (ii) There exists ǫ > 0: |σw(x, D)π(z)g, π(w)g| e−ǫ|w−z|

1 s ,

∀ z, w ∈ R2d. (6) Gevrey case s > 1 already contained in [Gr¨

• chenig-Rzeszotnik. Ann. Inst.

Fourier, 2008]. s = 1 easily follows from that.

Sparsity of the Gabor matrix

σw(x, D) with symbol σ satisfying (5). {π(λ)g, λ ∈ Λ} Gabor frame for L2(Rd) with g ∈ Ss

s (Rd), s ≥ 1/2. Then

|σw(x, D)π(µ)g, π(λ)g| ≤ Ce−ǫ|λ−µ|

1 s ,

∀ λ, µ ∈ Λ, (7) for suitable C > 0, ǫ > 0.

Proposition

If (7) holds, then the Gabor matrix is sparse: let a be any column or raw of the matrix, and let |a|n be the n-largest entry of the sequence a. Then, |a|n satisfies |a|n ≤ Ce−ǫn

1 2ds ,

n ∈ N For comparison, the curvelet (shearlet) matrix gives: |a|n ≤ C n−M ∀M > 0.

Applications to evolution equations

P(∂t, Dx) = ∂m

t + m

• k=1

ak(Dx)∂m−k

t

, t ∈ R, x ∈ Rd ak(η), 1 ≤ k ≤ m, polynomials. Dxj =

1 2πi ∂xj, j = 1, . . . , d.

We study the forward C.P.:

• P(∂t, Dx)u = 0,

(t, x) ∈ R+ × Rd ∂k

t u(0, x) = uk(x),

0 ≤ k ≤ m − 1, (8) uk ∈ S(Rd), 0 ≤ k ≤ m − 1. C.P. (8) well-posed in the Schwartz class iff the forward Hadamard-Petrowsky condition is satisfied: ∃C > 0: (τ, η) ∈ C × Rd, P(iτ, η) = 0 = ⇒ Im τ ≥ −C. (9)

The solution: u(t, x) =

m−1

• k=0

∂k

t E(t, ·) ∗

• um−1−k +

m−k−1

• j=1

aj(Dx)um−k−1−j

• .

E(t, x) = F−1

η→xσ(t, η), where σ(t, η) is the unique solution to

• ∂m

t + m

• k=1

ak(η)∂m−k

t

• σ(t, η) = δ(t)

supported in [0, +∞) × Rd. E(t, x) : the fundamental solution of P supported in [0, +∞) × Rd. ⇒ Reduce the study of the C.P. (8) to that of the Fourier multiplier σw(t, Dx) = σ(t, Dx)f = F−1σ(t, ·)Ff = E(t, ·) ∗ f . (10) (For Fourier multipliers Weyl and Kohn-Nirenberg quantizations coincide).

Examples:

Wave operator ∂2

t − ∆:

σ(t, η) = sin(2π|η|t) 2π|η| Klein-Gordon operator ∂2

t − ∆ + m2, m > 0:

σ(t, η) = sin(t

• 4π2|η|2 + m2)
• 4π2|η|2 + m2

Heat operator ∂t − ∆: σ(t, η) = e−4π2|η|2t

Examples:

Wave operator ∂2

t − ∆:

σ(t, η) = sin(2π|η|t) 2π|η| Klein-Gordon operator ∂2

t − ∆ + m2, m > 0:

σ(t, η) = sin(t

• 4π2|η|2 + m2)
• 4π2|η|2 + m2

Heat operator ∂t − ∆: σ(t, η) = e−4π2|η|2t

Assume: ∃ C > 0, r ≥ 1: (τ, ζ) ∈ C × Cd, P(iτ, ζ) = 0 = ⇒ Im τ ≥ −C(1 + |Im ζ|)r. (11) (condition (11) stronger than the forward Hadamard-Petrowsky condition).

Theorem

Assume P satisfies (11) for some C > 0, r ≥ 1. Then the symbol σ(t, η)

• f the corresponding propagator σ(t, Dx) satisfies:

|∂α

η σ(t, η)| ≤ C (t+1)|α|+t(α!)µ,

η ∈ Rd, t ≥ 0, α ∈ Nd, with µ = 1 − 1/r. (assumption r ≥ 1 ⇒ 0 ≤ µ < 1). Combining this theorem with the almost diagonalization result:

Theorem

Assume P satisfies (11) for some C > 0, r ≥ 1, and set s = max{1/2, 1 − 1/r}. If g ∈ Ss

s (Rd) then σ(t, Dx) satisfies:

|σ(t, Dx)π(z)g, π(w)g| ≤ Ce−ǫ|w−z|

1 s ,

∀ z, w ∈ R2d, (12) for some ǫ > 0 and C > 0. The constants ǫ and C are uniform when t lies in bounded subsets of [0, +∞). Notice that 1/2 ≤ s < 1, ⇒ super-exponential decay of the Gabor matrix.

Examples of operators satisfying the previous theorem

Proposition

Assume P(∂t, Dx) is hyperbolic with respect to t. Then the condition (11) is satisfied with r = 1 for some C > 0. Hence |σ(t, Dx)π(z)g, π(w)g| ≤ Ce−ǫ|w−z|2, ∀ z, w ∈ R2d, (13) if g ∈ S1/2

1/2(Rd), for some ǫ > 0 and C > 0.

Proposition

Consider the operator P(∂t, Dx) = ∂t + (−∆)k, with k ∈ N+ (k = 1 ⇒ heat operator). Then P satisfies (11) with r = 2k. Hence for s = 1 − 1/(2k) |σ(t, Dx)π(z)g, π(w)g| ≤ Ce−ǫ|w−z|

1 s ,

∀ z, w ∈ R2d, (14) if g ∈ Ss

s (Rd), for some ǫ > 0 and for a new constant C > 0.

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Schr¨

• dinger propagators, Appl. Comput. Harmon. Anal., 26:357-370,

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