Complex interpolation for Banach spaces
Computing discrete convolutions with verified accuracy via Banach algebras and the FFT
We introduce a method to compute rigorous component-wise enclosures of discrete convolutions using the fast Fourier transform, the properties of Banach algebras, and interval arithmetic. The purpose of this new approach is to improve the implementation and the applicability of computer-assisted proofs performed in weighed Banach algebras of Fourier/Chebyshev sequences, whose norms are known to be numerically unstable. We introduce some application examples, in particular a rigorous aposteriori...
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.