Computing discrete convolutions with verified accuracy via Banach algebras and the FFT. (Q1785642)

Let \(u_j\), \(j=1,\ldots,p\) be given \((2\pi/L)\)-periodic trigonometric polynomials of order \(M-1\) with corresponding sequences of Fourier coefficients \(a^{(j)}\in\mathbb{C}^{2M-1}\). Then the Fourier coefficients of the product \(u_1\,u_2\,\ldots\,u_p\) can be computed by discrete convolutions \(a^{(1)}\ast a^{(2)}\ast\,\ldots\,\ast a^{(p)}\). Using fast Fourier transforms, properties of Banach sequence spaces, and interval arithmetic, the author determines rigorous enclosures of these discrete convolutions. The method is applied to the numerical solution of the nonlinear Swift-Hohenberg partial differential equation \[u_t=(\lambda -1)\,u-2u_{xx}-u_{xxxx}+\mu u^3-u^5\] with periodic boundary conditions.