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Computing discrete convolutions with verified accuracy via Banach algebras and the FFT. (English) Zbl 1488.42047

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.

MSC:

42A85 Convolution, factorization for one variable harmonic analysis
42A05 Trigonometric polynomials, inequalities, extremal problems
65T50 Numerical methods for discrete and fast Fourier transforms
65G40 General methods in interval analysis
46B45 Banach sequence spaces
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
42B05 Fourier series and coefficients in several variables
35K35 Initial-boundary value problems for higher-order parabolic equations

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