Short-Time Fourier Transform and Modulation Spaces in Algebras of - - PowerPoint PPT Presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators

PhD within Initiativkolleg, the joint project of NuHAG and DIANA

Short-Time Fourier Transform and Modulation Spaces in Algebras of Generalized Functions

Šahbegović, Jasmin 28th August 2009

Šahbegović, Jasmin Short-Time Fourier Transform and Modulation Spaces in Algeb

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators

Contents

1 Introduction 2 Modulation spaces in the Colombeau algebra 3 Wiener-Amalgam spaces in the Colombeau framework.

The generalized modulation space Mp,q

G (Rd)

4 Generalized pseudo-differential operators Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Introduction

Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Introduction

Short-time Fourier transform - motivation and idea A single object containing both information on time and frequency? Consider the Fourier transform of a (suitably weighted) function on an arbitrary interval.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Introduction

Short-time Fourier transform - motivation and idea A single object containing both information on time and frequency? Consider the Fourier transform of a (suitably weighted) function on an arbitrary interval. Vgf (x, ω) := f , MωTxg, x, ω ∈ Rd, f ∈ S ′(Rd), g ∈ S (Rd) a nonzero function.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Introduction

Short-time Fourier transform - motivation and idea A single object containing both information on time and frequency? Consider the Fourier transform of a (suitably weighted) function on an arbitrary interval. Vgf (x, ω) := f , MωTxg, x, ω ∈ Rd, f ∈ S ′(Rd), g ∈ S (Rd) a nonzero function. STFT represents an object that simultaneously carries information both on time and frequency, yet not instantaneous - due to the uncertainty principle. Additional price to pay: doubled variables.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Introduction

Modulation spaces - motivation and idea A way to measure smoothness and decay properties of a function via T-F tools? Use Lp,q norms of STFT.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Introduction

Modulation spaces - motivation and idea A way to measure smoothness and decay properties of a function via T-F tools? Use Lp,q norms of STFT. f Mp,q := Vgf Lp,q =

• Vgf (x, ω)
• p

dx

• q

p dω

1

q < ∞ Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Introduction

Modulation spaces - motivation and idea A way to measure smoothness and decay properties of a function via T-F tools? Use Lp,q norms of STFT. f Mp,q

m

:= Vgf Lp,q

m

=

• Vgf (x, ω)m(x, ω)
• p

dx

• q

p dω

1

q < ∞. Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Introduction

Modulation spaces - motivation and idea A way to measure smoothness and decay properties of a function via T-F tools? Use Lp,q norms of STFT. f Mp,q

m

:= Vgf Lp,q

m

=

• Vgf (x, ω)m(x, ω)
• p

dx

• q

p dω

1

q < ∞.

Different windows yield equivalent norms Modulation spaces are Banach spaces Modulation spaces are invariant under time-frequency shifts

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Introduction

Colombeau algebra - motivation and idea Singular objects and nonlinearities in the framework of distribution theory? A differential algebra containing distributions as a subspace and a subset of smooth functions as a subalgebra.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Introduction

Colombeau algebra - motivation and idea Singular objects and nonlinearities in the framework of distribution theory? A differential algebra containing distributions as a subspace and a subset of smooth functions as a subalgebra. Algebra = moderate nets/negligible nets

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Introduction

Colombeau algebra - motivation and idea Singular objects and nonlinearities in the framework of distribution theory? A differential algebra containing distributions as a subspace and a subset of smooth functions as a subalgebra. Algebra = moderate nets/negligible nets Moderate nets - derivatives of a net bounded by some negative power of ε Negligible nets - derivatives of a net bounded by any non-negative power of ε

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Introduction

Colombeau algebras - advantages Singular objects well defined. Nonlinear operations enabled Product preserved on the level of (a subset of) C ∞-functions Existence of generalized solutions of PDE even when distributional solutions do not exist Notion of association - a path from nonlinear to linear

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras

Introduction

Colombeau algebras - advantages Singular objects well defined. Nonlinear operations enabled Product preserved on the level of (a subset of) C ∞-functions Existence of generalized solutions of PDE even when distributional solutions do not exist Notion of association - a path from nonlinear to linear Colombeau algebras - disadvantages Existence of zero divisors Association classes: algebraic properties not inherited Not every generalized element casts a distributional shadow

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Preparation for constructing a differential algebra The generalized modulation space GC ∞

Mp,q s,t

(Rd) Association in GC ∞

Mp,q s,t

(Rd) Regular Colombeau generalized functions

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

An embedding of Mp,q space into a differential algebra? Idea: smoothing via convolution with mollifier.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

An embedding of Mp,q space into a differential algebra? Idea: smoothing via convolution with mollifier.

1 Bound ∂α(u ∗ ρε)Mp,q by some (negative) power of ε? 2 Bound ∂α(uε · vε)Mp,q by some (negative) power of ε? 3 When does ∂α(u ∗ ρε − u)Mp,q → 0? Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

An embedding of Mp,q space into a differential algebra? Idea: smoothing via convolution with mollifier

1 Bound ∂α(u ∗ ρε)Mp,q by some (negative) power of ε? 2 Bound ∂α(uε · vε)Mp,q by some (negative) power of ε? 3 When does ∂α(u ∗ ρε − u)Mp,q → 0?

Classical multiplication and convolution properties u · vMp,q ≤ C · uMp,q · vM∞,1 u ∗ vMp,q ≤ C · uMp,q · vM1,∞

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Embedding theorems for modulation spaces Represent M∞,1-norm of a distribution f by Mp,q-norm of the derivatives of f ?

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Embedding theorems for modulation spaces Represent M∞,1-norm of a distribution f by Mp,q-norm of the derivatives of f ?

• Vgf (x, ω)
• ≤ C ·

|α|,|β|≤2k

• V∂βg∂αf (x, ω)
• ·

1 ω2k .

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Embedding theorems for modulation spaces Represent M∞,1-norm of a distribution f by Mp,q-norm of the derivatives of f ?

• Vgf (x, ω)
• ≤ C ·

|α|,|β|≤2k

• V∂βg∂αf (x, ω)
• ·

1 ω2k . sup

x

• Vgf (x, ω)
• ≤ C ·
• |α|,|β|≤2k

sup

x

• V∂βg∂αf (x, ω)
• ·

1 ω2k

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Embedding theorems for modulation spaces Represent M∞,1-norm of a distribution f by Mp,q-norm of the derivatives of f ?

• Vgf (x, ω)
• ≤ C ·

|α|,|β|≤2k

• V∂βg∂αf (x, ω)
• ·

1 ω2k . sup

x

• Vgf (x, ω)
• ≤ C ·
• |α|,|β|≤2k

∂αf M∞ · 1 ω2k .

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Embedding theorems for modulation spaces Represent M∞,1-norm of a distribution f by Mp,q-norm of the derivatives of f ?

• Vgf (x, ω)
• ≤ C ·

|α|,|β|≤2k

• V∂βg∂αf (x, ω)
• ·

1 ω2k . sup

x

• Vgf (x, ω)
• ≤ C ·
• |α|,|β|≤2k

∂αf M∞ · 1 ω2k . f M∞,1 ≤ C ·

• |α|,|β|≤2k

∂αf M∞ ·

• 1

ω2k dω

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Embedding theorems for modulation spaces Represent M∞,1-norm of a distribution f by Mp,q-norm of the derivatives of f ?

• Vgf (x, ω)
• ≤ C ·

|α|,|β|≤2k

• V∂βg∂αf (x, ω)
• ·

1 ω2k . sup

x

• Vgf (x, ω)
• ≤ C ·
• |α|,|β|≤2k

∂αf M∞ · 1 ω2k . f M∞,1 ≤ C ·

• |α|,|β|≤2k

∂αf M∞

−l,0 ·

• 1

ω2k−l dω, l ∈ N arbitrary, 2k > d + l.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Lemma For any k, l ∈ N (2k > d + l) there exists C ≥ 0 such that f M∞,1

m

≤ C ·

• |α|≤2k

∂αf Mp,q

m·n

holds for every f ∈ S (Rd), where n(x, ω) := ω−l.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Lemma For any k, l ∈ N (2k > d + l) there exists C ≥ 0 such that f M∞,1

m

≤ C ·

• |α|≤2k

∂αf Mp,q

m·n

holds for every f ∈ S (Rd), where n(x, ω) := ω−l. Corollary C ∞

Mp,q

s,t = C ∞

Mp,1

0,t Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Definition EC ∞

Mp,q s,t

(Rd) := {(uε)ε ∈ Hp

∞(Rd)(0,1]

• ∀α ∈ Nd

0 ∃N ∈ N : ∂αuεMp,q

s,t = O(ε−N)},

NC ∞

Mp,q s,t

(Rd) := {(uε)ε ∈ Hp

∞(Rd)(0,1]

• ∀α ∈ Nd

0 ∀m ∈ N0 : ∂αuεMp,q

s,t = O(εm)}. Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Definition EC ∞

Mp,q s,t

(Rd) := {(uε)ε ∈ Hp

∞(Rd)(0,1]

• ∀α ∈ Nd

0 ∃N ∈ N : ∂αuεMp,q

s,t = O(ε−N)},

NC ∞

Mp,q s,t

(Rd) := {(uε)ε ∈ Hp

∞(Rd)(0,1]

• ∀α ∈ Nd

0 ∀m ∈ N0 : ∂αuεMp,q

s,t = O(εm)}.

Theorem EC ∞

Mp,q s,t

(Rd)is a (differential) algebra and NC ∞

Mp,q s,t

(Rd) is an ideal in EC ∞

Mp,q s,t

(Rd).

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Theorem Let ρ be a standard mollifier. The linear mapping ι(u) := (u ∗ ρε)ε + NC ∞

Mp,q s,t

(Rd)

• f Mp,q

s,t (Rd) into GC ∞

Mp,q s,t

(Rd) is an embedding.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Theorem Let ρ be a standard mollifier. The linear mapping ι(u) := (u ∗ ρε)ε + NC ∞

Mp,q s,t

(Rd)

• f Mp,q

s,t (Rd) into GC ∞

Mp,q s,t

(Rd) is an embedding. Theorem GC ∞

Mp,q s,t

(Rd) = GC ∞

Mp,1 0,t

(Rd), holds for all p, q ∈ [1, ∞), s ∈ R and t ≥ 0.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Theorem (Characterization of the Ideal) (uε)ε ∈ EC ∞

Mp,q s,t

(Rd) is C ∞

Mp,q

s,t (Rd)-negligible if and only if the

following condition is satisfied: ∀m ∈ N0 uεMp,q

s,t = O(εm). Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Theorem (Characterization of the Ideal) (uε)ε ∈ EC ∞

Mp,q s,t

(Rd) is C ∞

Mp,q

s,t (Rd)-negligible if and only if the

following condition is satisfied: ∀m ∈ N0 uεMp,q

s,t = O(εm).

Theorem Let σ : C ∞

Mp,q

s,t → GC ∞ Mp,q s,t

given by σ : f → [(f )ǫ]. Then ι|C∞

Mp,q s,t

= σ.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Theorem Let u ∈ C ∞

Mp1,q1

s1,t1

(Rd), v ∈ C ∞

Mp2,q2

s2,t2

(Rd), where p1, p2, q1, q2 ∈ [1, ∞] and s1, s2, t1, t2 ∈ R with t1, t2 ≥ 0. Then ι(u)ε · ι(v)ε − ι(uv)ε ∈ NC ∞

Mp1,q1 s1,t1

(Rd) ∩ NC ∞

Mp2,q2 s2,t2

(Rd), where ι(u)ε := (u ∗ ρε)ε.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Definition Let s ∈ R, t ≥ 0. An element u ∈ GC ∞

Mp,q s,t

(Rd) is called associated with 0 (denoted by u

Mp,q

s,t

≈ 0) if lim

ε→0 uεMp,q

s,t = 0. Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Definition Let s ∈ R, t ≥ 0. An element u ∈ GC ∞

Mp,q s,t

(Rd) is called associated with 0 (denoted by u

Mp,q

s,t

≈ 0) if lim

ε→0 uεMp,q

s,t = 0.

Theorem Let u ∈ Mp,q

s,t , p, q ∈ [1, ∞). If ι(u) Mp,q

s,t

≈ 0, then u = 0.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Theorem

1 Let u, v ∈ Mp,q

s+s0,t, s0 ∈ N0. Then

u

Mp,q

s+s0,t

≈ v ⇔ ∂αu

Mp,q

s,t

≈ ∂αv for all |α| ≤ s0.

2 Let u, v ∈ GC ∞ Mp1,q1 s1,t1

(Rd), u

Mp1,q1

s1,t1

≈ v, f ∈ C ∞

Mp2,q2

s2,t2

, t2 ≥ 0 for any p2, q2, s2 and t2). Then f · u

Mp1,q1

s1,t1

≈ f · v.

3 Let u ∈ Mp1,q1

s1,t1 and v ∈ Mp0,1. Then

ι(u) · ι(v)

Mp1,q1

s1,t1

≈ ι(u · v).

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Definition G ∞

C ∞

Mp,q s,t

(Rd) := E ∞

C ∞

Mp,q s,t

(Rd)/NC ∞

Mp,q s,t

(Rd), where E ∞

C ∞

Mp,q s,t

(Rd) := {(uε)ε ∈ Hp

∞

• ∃N ∈ N : ∀α ∈ Nd

0 ∂αuεMp,q = O(ε−N)}.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Preparation for constructing a differential algebra The generalized modulation space Association in generalized modulation spaces Regular Colombeau generalized functions

Modulation spaces in the Colombeau algebra

Definition G ∞

C ∞

Mp,q s,t

(Rd) := E ∞

C ∞

Mp,q s,t

(Rd)/NC ∞

Mp,q s,t

(Rd), where E ∞

C ∞

Mp,q s,t

(Rd) := {(uε)ε ∈ Hp

∞

• ∃N ∈ N : ∀α ∈ Nd

0 ∂αuεMp,q = O(ε−N)}.

Conjecture G ∞

C ∞

Mp,q s,0

(Rd) ∩ S ′(Rd) = C ∞

Mp,q(Rd) = Hp ∞(Rd).

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Why generalized Wiener-Amalgam space? The generalized modulation space Mp,q

G

Wiener-Amalgam spaces in the Colombeau framework

Why generalized Wiener-Amalgam space? The generalized modulation space Mp,q

G (Rd)

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Why generalized Wiener-Amalgam space? The generalized modulation space Mp,q

G

Wiener-Amalgam spaces in the Colombeau framework

The loss of the ”Lq-information“ due to the smoothness assumption Recall: GC ∞

Mp,q s,t

(Rd) = GC ∞

Mp,1 0,t

(Rd).

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Why generalized Wiener-Amalgam space? The generalized modulation space Mp,q

G

Wiener-Amalgam spaces in the Colombeau framework

The loss of the ”Lq-information“ due to the smoothness assumption Recall: GC ∞

Mp,q s,t

(Rd) = GC ∞

Mp,1 0,t

(Rd). A remedy? Retrieve the information by involving the Fourier transform side of the STFT - i.e. consider Wiener-Amalgam spaces.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Why generalized Wiener-Amalgam space? The generalized modulation space Mp,q

G

Wiener-Amalgam spaces in the Colombeau framework

The loss of the ”Lq-information“ due to the smoothness assumption Recall: GC ∞

Mp,q s,t

(Rd) = GC ∞

Mp,1 0,t

(Rd). A remedy? Retrieve the information by involving the Fourier transform side of the STFT - i.e. consider Wiener-Amalgam spaces. The generalized modulation space Mp,q

G

The algebra of generalized functions composed of classes of moderate nets corresponding to modulation spaces and classes of negligible nets corresponding to Wiener-Amalgam spaces.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Why generalized Wiener-Amalgam space? The generalized modulation space Mp,q

G

Wiener-Amalgam spaces in the Colombeau framework

Mp,q

G (Rd) = EC ∞

Mp,q (Rd)/FMp,q

N (Rd)

FMp,q

N (Rd) := {(uε)ε ∈ EC ∞

Mp,q (Rd)

• ∀α ∈ Nd

0 ∀m ∈ N0 : ∂αuεFMp,q = O(εm)}.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Why generalized Wiener-Amalgam space? The generalized modulation space Mp,q

G

Wiener-Amalgam spaces in the Colombeau framework

Mp,q

G (Rd) = EC ∞

Mp,q (Rd)/FMp,q

N (Rd)

FMp,q

N (Rd) := {(uε)ε ∈ EC ∞

Mp,q (Rd)

• ∀α ∈ Nd

0 ∀m ∈ N0 : ∂αuεFMp,q = O(εm)}.

Embedding map? Is the suggested algebra of negligible nets really an ideal in the algebra of moderate nets? Is it possible to characterize the ideal in the usual way?

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Why generalized Wiener-Amalgam space? The generalized modulation space Mp,q

G

Wiener-Amalgam spaces in the Colombeau framework

Theorem Let ρ be a standard mollifier. The linear mapping

• ι(u) := (u ∗ ρε)ε + FMp,q

N (Rd)

• f Mp,q(Rd) into Mp,q

G (Rd) is an embedding.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Why generalized Wiener-Amalgam space? The generalized modulation space Mp,q

G

Wiener-Amalgam spaces in the Colombeau framework

Theorem Let ρ be a standard mollifier. The linear mapping

• ι(u) := (u ∗ ρε)ε + FMp,q

N (Rd)

• f Mp,q(Rd) into Mp,q

G (Rd) is an embedding.

Theorem FMp,q

N (Rd) is an ideal in EC ∞

Mp,q (Rd). Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Why generalized Wiener-Amalgam space? The generalized modulation space Mp,q

G

Wiener-Amalgam spaces in the Colombeau framework

Characterization of the Ideal Let p ≤ q. (uε)ε ∈ EC ∞

Mp,q (Rd) is Mp,q

G (Rd)-negligible if and only if

the following condition is satisfied: ∀m ∈ N0 uεFMp,q = O(εm).

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Generalized pseudo-differential operators

Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Generalized pseudo-differential operators

Partial differential operator with non-constant coefficients P = P(x, D) :=

|α|≤N aα(x)Dα x .

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Generalized pseudo-differential operators

Partial differential operator with non-constant coefficients P = P(x, D) :=

|α|≤N aα(x)Dα x .

Action of a PDO on a function P(x, D)f (x) =

• P(x, ξ) · ˆ

f (ξ) · e2πiξx dξ.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Generalized pseudo-differential operators

Partial differential operator with non-constant coefficients P = P(x, D) :=

|α|≤N aα(x)Dα x .

Action of a PDO on a function P(x, D)f (x) =

• P(x, ξ) · ˆ

f (ξ) · e2πiξx dξ. Definition: ΨDO in time-frequency context σ ∈ S ′(R2d) and f ∈ S (Rd). We define the ΨDO σKN by: σKNf (x) = σ(x, D)f (x) := σ(x, ξ), ¯ ˆ f (ξ) · e−2πixξξ.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Generalized pseudo-differential operators

σKNf , g = σ(x, ξ), ¯ ˆ f (ξ) · g(x) · e−2πixξx,ξ,

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Generalized pseudo-differential operators

σKNf , g = σ(x, ξ), ¯ ˆ f (ξ) · g(x) · e−2πixξx,ξ, Definition f , g ∈ S (Rd). The (cross) Rihaczek distribution of f and g is defined as the frequency-shifted tensor product R(f , g)(x, ξ) := f (x)ˆ g(ξ)e−2πixξ.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Generalized pseudo-differential operators

σKNf , g = σ(x, ξ), ¯ ˆ f (ξ) · g(x) · e−2πixξx,ξ, Definition f , g ∈ S (Rd). The (cross) Rihaczek distribution of f and g is defined as the frequency-shifted tensor product R(f , g)(x, ξ) := f (x)ˆ g(ξ)e−2πixξ. Lemma: Let σ ∈ S ′(R2d). The sesquilinear form σKNf , g := σ, R(g, f ). defines a continuous operator σKN from S (Rd) to S ′(Rd).

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Generalized pseudo-differential operators

Theorem (Complete picture of L2-boundedness) Let σ ∈ Mp,q, p, q ∈ [1, ∞]. If q ≤ 2 and either 1 ≤ p ≤ 2 or p ≤ q′, then σKN is a bounded operator on L2(Rd). If q > 2 or if p ≥ 2 and p > q′, then there exist σ ∈ Mp,q such that σKN is unbounded on L2(Rd).

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Generalized pseudo-differential operators

Theorem σ ∈ GC ∞

Mp,q (R2d) and f ∈ GC ∞ Mr1,s1 (Rd). Then for 1 ≤ r1, s1, r2,

s2 < ∞ and every multi-index α there exists a constant Cα such that ∂α(σKN

ε

fε)Mr2,s2 ≤ Cα ·

• β≤α

∂α−β

x

σεMp,q · ∂βfεMr1,s1, σKN

ε

is a ΨDO corresponding to the symbol σε. In particular, σKN : GC ∞

Mr1,s1 (Rd) → GC ∞ Mr2,s2 (Rd),

where σKNf := [(σKN

ε

fε)ε], for f ∈ GC ∞

Mr1,s1 (Rd). Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Generalized pseudo-differential operators

Definition (σ1

ε)ε ∈ EC ∞

Mp1,q1 (Rd) and (σ2

ε)ε ∈ EC ∞

Mp2,q2 (Rd). The net

R(σ1

ε, σ2 ε)ε is called the generalized Rihaczek distribution and is

given by R(σ1

ε, σ2 ε)(x, ξ) := σ1 ε(x) ·

σ2

ε(ξ) · e−2πixξ.

Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Generalized pseudo-differential operators

Definition (σ1

ε)ε ∈ EC ∞

Mp1,q1 (Rd) and (σ2

ε)ε ∈ EC ∞

Mp2,q2 (Rd). The net

R(σ1

ε, σ2 ε)ε is called the generalized Rihaczek distribution and is

given by R(σ1

ε, σ2 ε)(x, ξ) := σ1 ε(x) ·

σ2

ε(ξ) · e−2πixξ.

Theorem Let R(σ1

ε, σ2 ε)ε ∈ R(EC ∞

Mp,q (Rd), EC ∞ Mp′,q′ (Rd)). Then

(R(σ1

ε, σ2 ε)ε)KN : Mp,q(Rd) → EC ∞

Mp,q (Rd). Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Generalized pseudo-differential operators

Definition (σ1

ε)ε ∈ EC ∞

Mp1,q1 (Rd) and (σ2

ε)ε ∈ EC ∞

Mp2,q2 (Rd). The net

R(σ1

ε, σ2 ε)ε is called the generalized Rihaczek distribution and is

given by R(σ1

ε, σ2 ε)(x, ξ) := σ1 ε(x) ·

σ2

ε(ξ) · e−2πixξ.

Theorem Let R(σ1

ε, σ2)ε ∈ R(EC ∞

Mp,q (Rd), Mp′,q′(Rd)). Then

(R(σ1

ε, σ2)ε)KN : Mp,q(Rd) → EC ∞

Mp,q (Rd). Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators Basic facts and definitions Boundedness of ΨDO in the framework of modulation spaces Generalized modulation spaces as symbol classes for ΨDO

Generalized pseudo-differential operators

Theorem (σε)ε = (σ1

ε ⊗ ˆ

σ2

ε)ε ∈ EC ∞

Mp,q (Rd) × FEC ∞ FMp,q (Rd) and f ∈ Mp,q.

For 1 ≤ r1, s1, r2, s2 < ∞ and every multi-index α there exists a constant Cα such that ∂α(σKN

ε

f )Mr2,s2 ≤ Cα·

• β≤α

∂α−βσ1

εMp,q·∂βσ2 εFMp,q·f Mr1,s1.

In particular, σKN : Mr1,s1(Rd) → EC ∞

Mr2,s2 (Rd). Šahbegović, Jasmin Dissertation presentation

Introduction Modulation spaces in the Colombeau algebra Wiener-Amalgam spaces in the Colombeau framework Generalized pseudo-differential operators

Thank you very much for your attention.

Šahbegović, Jasmin Dissertation presentation