[Paper Review] The Wiener Property for a Class of Fourier Integral Operators

Elena Cordero, Gröchenig, Karlheinz+2 more|arXiv (Cornell University)|Jan 19, 2012

Mathematical Analysis and Transform Methods30 references3 citations

TL;DR

This paper introduces a one-parameter family of algebras $FIO(\Xi,s)$, $s \geq 0$, for Fourier integral operators (FIOs) using Gabor matrix decay around the graph of a canonical transformation. It establishes boundedness, composition rules, and spectral invariance, proving that these algebras satisfy the Wiener property—i.e., invertibility in $L^2$ implies the inverse is also in the algebra—extending Sjöstrand's theory from pseudodifferential operators to a broad class of FIOs.

ABSTRACT

We construct a one-parameter family of algebras consisting of Fourier integral operators. We derive boundedness results, composition rules, and the spectral invariance of this class of operators. The operator algebra is defined by the decay properties of an associated Gabor matrix around the graph of the canonical transformation.

Motivation & Objective

• To construct a class of algebras $FIO(\Xi,s)$ of Fourier integral operators that generalize the Sjöstrand algebra to FIOs.
• To establish boundedness, composition rules, and spectral invariance for these algebras.
• To extend the Wiener property—i.e., inverse-closedness—to a class of FIOs beyond pseudodifferential operators.
• To characterize the algebras via decay properties of the Gabor matrix relative to the canonical transformation.
• To show that generalized metaplectic operators and their products with pseudodifferential operators lie within these algebras for $s > 2d$.

Proposed method

• Define the algebra $FIO(\Xi,s)$ using decay of the Gabor matrix of an operator outside the graph of a symplectic transformation $\Xi$.
• Use modulation spaces $M^{\infty,1}_{1 \otimes v_s}$ to characterize symbol classes $S^s_w$, with $v_s(z) = \langle z \rangle^s$.
• Apply the short-time Fourier transform (STFT) and Gabor frame theory to analyze the matrix representation of operators.
• Establish the continuous decay condition (25) on the Gabor matrix as a key criterion for membership in $FIO(\Xi,s)$.
• Leverage the covariance property of the STFT to relate the Gabor matrix of $\mu(\mathcal{A})$ to the action of $\mathcal{A}$ on time-frequency shifts.
• Use the metaplectic representation $\mu(\mathcal{A})$ for $\mathcal{A} \in Sp(d,\mathbb{R})$ and prove its membership in $\bigcap_{s \geq 0} FIO(\mathcal{A},s)$.

Experimental results

Research questions

• RQ1 Can a Wiener-type algebra of Fourier integral operators be constructed that is closed under inversion, even without a symbolic calculus?
• RQ2 How can the decay of the Gabor matrix relative to the canonical transformation be used to define a class of FIOs with desirable algebraic properties?
• RQ3 Does the spectral invariance (Wiener property) hold for FIOs when the symbol belongs to the generalized Sjöstrand class $S^s_w$ with $s > 2d$?
• RQ4 Can generalized metaplectic operators be factored into products of pseudodifferential operators and metaplectic operators within the algebra $FIO(\mathcal{A},s)$?
• RQ5 What is the relationship between the algebraic structure of FIOs of type I and II and the symplectic group action?

Key findings

• The algebra $FIO(\Xi,s)$ is spectrally invariant for all $s \geq 0$, meaning that if an operator in the algebra is invertible on $L^2$, its inverse is also in the algebra.
• For $s > 2d$, the algebra $FIO(\Xi,s)$ contains all generalized metaplectic operators $\mu(\mathcal{A})$ and is closed under composition with pseudodifferential operators.
• Every operator in $FIO(\Xi,s)$ with $s > 2d$ can be factored as a product of a pseudodifferential operator and a metaplectic operator, both with symbols in $S^s_w$.
• The Gabor matrix of $\mu(\mathcal{A})$ decays rapidly away from the graph of $\mathcal{A}$, satisfying the continuous decay condition (25) for all $s \geq 0$, which implies $\mu(\mathcal{A}) \in \bigcap_{s \geq 0} FIO(\mathcal{A},s)$.
• The class $FIO(\Xi,s)$ generalizes the Sjöstrand algebra $S_w = M^{\infty,1}(\mathbb{R}^{2d})$ to FIOs, preserving the Wiener property.
• Operators of type I and II do not form an algebra unless additional decay conditions are imposed, but the Gabor matrix approach ensures algebraic closure in $FIO(\Xi,s)$.