Characterization of L...(R...) Using Gabor Frames.
Convergence of series of dilated functions and spectral norms of GCD matrices
We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.