Representation and Approximation of Pseudodifferential Operators by Sums of Gabor Multipliers

We investigate a new representation of general operators by means of sums of shifted Gabor multipliers. These representations arise by studying the matrix of an operator with respect to a Gabor frame. Each shifted Gabor multiplier corresponds to a side-diagonal of this matrix. This representation is especially useful for operators whose associated matrix possesses some offdiagonal decay. In this case one can completely characterize the symbol class of the operator by the size of the symbols of the Gabor multipliers. As an application we derive approximation theorems for pseudodifferential operators in the Sjöstrand class.