Quasi-interpolation in the Fourier algebra

We derive new convergence results for the Schoenberg operator and more general quasi-interpolation operators. In particular, we prove that natural conditions on the generator function imply convergence of these operators in the Fourier algebra A(Rd)=FL1(Rd) and in S0(Rd), a function space developed by the first author and often used in time-frequency analysis. As a simple yet very useful consequence for applications in Gabor analysis we obtain that piecewise linear interpolation converges in A(R) as well as in S0(R). Generally, the results presented in this paper are motivated by discretization problems arising in time-frequency analysis and have important consequences in this field.