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 M p , q G ( R d ) 4 Generalized pseudo-differential operators Šahbegović, Jasmin Dissertation presentation
Introduction Short-time Fourier transform (STFT) Modulation spaces in the Colombeau algebra Modulation spaces Wiener-Amalgam spaces in the Colombeau framework Colombeau algebras Generalized pseudo-differential operators Introduction Short-time Fourier transform (STFT) Modulation spaces Colombeau algebras Šahbegović, Jasmin Dissertation presentation
Introduction Short-time Fourier transform (STFT) Modulation spaces in the Colombeau algebra Modulation spaces Wiener-Amalgam spaces in the Colombeau framework Colombeau algebras Generalized pseudo-differential operators 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 Short-time Fourier transform (STFT) Modulation spaces in the Colombeau algebra Modulation spaces Wiener-Amalgam spaces in the Colombeau framework Colombeau algebras Generalized pseudo-differential operators 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. V g f ( x , ω ) := � f , M ω T x g � , x , ω ∈ R d , f ∈ S ′ ( R d ) , g ∈ S ( R d ) a nonzero function. Šahbegović, Jasmin Dissertation presentation
Introduction Short-time Fourier transform (STFT) Modulation spaces in the Colombeau algebra Modulation spaces Wiener-Amalgam spaces in the Colombeau framework Colombeau algebras Generalized pseudo-differential operators 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. V g f ( x , ω ) := � f , M ω T x g � , x , ω ∈ R d , f ∈ S ′ ( R d ) , g ∈ S ( R d ) 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 Short-time Fourier transform (STFT) Modulation spaces in the Colombeau algebra Modulation spaces Wiener-Amalgam spaces in the Colombeau framework Colombeau algebras Generalized pseudo-differential operators Introduction Modulation spaces - motivation and idea A way to measure smoothness and decay properties of a function via T-F tools? Use L p , q norms of STFT. Šahbegović, Jasmin Dissertation presentation
Introduction Short-time Fourier transform (STFT) Modulation spaces in the Colombeau algebra Modulation spaces Wiener-Amalgam spaces in the Colombeau framework Colombeau algebras Generalized pseudo-differential operators Introduction Modulation spaces - motivation and idea A way to measure smoothness and decay properties of a function via T-F tools? Use L p , q norms of STFT. � f � M p , q := � V g f � L p , q � � � � � � � � 1 q p p d ω q < ∞ � � � � = � � V g f ( x , ω ) � dx � Šahbegović, Jasmin Dissertation presentation
Introduction Short-time Fourier transform (STFT) Modulation spaces in the Colombeau algebra Modulation spaces Wiener-Amalgam spaces in the Colombeau framework Colombeau algebras Generalized pseudo-differential operators Introduction Modulation spaces - motivation and idea A way to measure smoothness and decay properties of a function via T-F tools? Use L p , q norms of STFT. � f � M p , q := � V g f � L p , q m m � � � � � � � � 1 q p p d ω q < ∞ . � � � � = � � V g f ( x , ω ) m ( x , ω ) � dx � Šahbegović, Jasmin Dissertation presentation
Introduction Short-time Fourier transform (STFT) Modulation spaces in the Colombeau algebra Modulation spaces Wiener-Amalgam spaces in the Colombeau framework Colombeau algebras Generalized pseudo-differential operators Introduction Modulation spaces - motivation and idea A way to measure smoothness and decay properties of a function via T-F tools? Use L p , q norms of STFT. � f � M p , q := � V g f � L p , q m m � � � � � � � � 1 q p p d ω q < ∞ . � � � � = � � V g f ( x , ω ) m ( x , ω ) � dx � Different windows yield equivalent norms Modulation spaces are Banach spaces Modulation spaces are invariant under time-frequency shifts Šahbegović, Jasmin Dissertation presentation
Introduction Short-time Fourier transform (STFT) Modulation spaces in the Colombeau algebra Modulation spaces Wiener-Amalgam spaces in the Colombeau framework Colombeau algebras Generalized pseudo-differential operators 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 Short-time Fourier transform (STFT) Modulation spaces in the Colombeau algebra Modulation spaces Wiener-Amalgam spaces in the Colombeau framework Colombeau algebras Generalized pseudo-differential operators 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 Short-time Fourier transform (STFT) Modulation spaces in the Colombeau algebra Modulation spaces Wiener-Amalgam spaces in the Colombeau framework Colombeau algebras Generalized pseudo-differential operators 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 Short-time Fourier transform (STFT) Modulation spaces in the Colombeau algebra Modulation spaces Wiener-Amalgam spaces in the Colombeau framework Colombeau algebras Generalized pseudo-differential operators 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 Short-time Fourier transform (STFT) Modulation spaces in the Colombeau algebra Modulation spaces Wiener-Amalgam spaces in the Colombeau framework Colombeau algebras Generalized pseudo-differential operators 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 Preparation for constructing a differential algebra Modulation spaces in the Colombeau algebra The generalized modulation space Wiener-Amalgam spaces in the Colombeau framework Association in generalized modulation spaces Generalized pseudo-differential operators Regular Colombeau generalized functions Modulation spaces in the Colombeau algebra Preparation for constructing a differential algebra The generalized modulation space G C ∞ ( R d ) Mp , q s , t Association in G C ∞ ( R d ) Mp , q s , t Regular Colombeau generalized functions Šahbegović, Jasmin Dissertation presentation
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