Kernel representations of operators commuting with Gabor systems

Abstract Gabor systems consist of translations and modulations of a single function and are an important tool for signal analysis. We want to analyze and characterize operators which preserve Gabor structure in the sense that they commute with the translation operator $$T^a$$ T a and the modulation operator $$M^b$$ M b , i.e. we analyze the commutant $$\{T^a, M^b\}'$$ { T a , M b } ′ . In the case of $$ab=1$$ a b = 1 , the commutant turns out to be a maximal abelian von Neumann algebra. This means that operators in the commutant are unitarily equivalent via the Zak transform to multiplication operators, which we call a kernel representation. One important example of an operator in the commutant is the mixed frame operator of two Gabor frames. This leads to another representation of invertible operators in the commutant via the Zak transform with generators of Gabor Riesz bases. In the general case of $$a,b>0$$ a , b > 0 , operators in the commutant can be studied via distributional kernel representations of pseudo differential operators. These kernels admit a series representation of operators via generalized Fourier series. In particular in the case of $$ab=1$$ a b = 1 , the relation of all three approaches yields a commutative diagram.