Quasi-interpolation in the Fourier algebra

Abstract

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 ( R d ) = F L 1 ( R d ) and in S 0 ( R d ) , 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 S 0 ( 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.