On the Fourier Extension of Nonperiodic Functions

Abstract

We obtain exponentially accurate Fourier series for non-periodic functions on the interval [−1, 1] by extending these functions to periodic functions on a larger domain. The series may be evaluated, but not constructed, by means of the FFT. A complete convergence theory is given based on orthogonal polynomials that resemble Chebyshev polynomials of the first and second kinds. We analyze a previously proposed numerical method, which is unstable in theory but stable in practice. We propose a new numerical method that is stable both in theory and in practice.Keywords : Fourier series, orthogonal polynomials, frames, numerical integration AMS(MOS) Classification : Primary : 42A10, Secondary : 42C15, 65D32 On the Fourier extension of non-periodic functionsDaan Huybrechs * † AbstractWe obtain exponentially accurate Fourier series for non-periodic functions on the interval [−1, 1] by extending these functions to periodic functions on a larger domain. The series may be evaluated, but not constructed, by means of the FFT. A complete convergence theory is given based on orthogonal polynomials that resemble Chebyshev polynomials of the first and second kinds. We analyze a previously proposed numerical method, which is unstable in theory but stable in practice. We propose a new numerical method that is stable both in theory and in practice.

Impossibility of Fast Stable Approximation of Analytic Functions from Equispaced Samples

“…Related contributions are due to Gelb, Tadmor, and Tanner [23,43]. A Padé algorithm was proposed in [13], and further contributions in this area include [30] and [41]. For an excellent survey of much of this subject see [43].…”

Convergence of Linear Barycentric Rational Interpolation for Analytic Functions

“…Numerical tests indicate that this approach allows for a quite robust "black-box" construction of linear rational interpolants in the fewest possible nodes. We hope that our results enhance the understanding of the convergence behavior of linear barycentric interpolation and that they make it possible to compare such interpolation schemes with other available methods for interpolation in equispaced nodes, such as the approach proposed in [14] and in the references therein, that in [15], or the schemes reviewed and cited in [18].…”

“…, and let R > 0 be the smallest number such that f is analytic in the interior of C R defined in (14). Then the rational interpolants r n defined by (2), with limiting node measure μ and d(n)/n → C, satisfy lim sup In Figure 3 we illustrate the level lines C R for the parameter C = 0.2 with equispaced nodes on the left and with nodes distributed according to the density dμ/ dx = φ(x) = (4 + arctan(4x))/8 on the right.…”

Receding horizon pseudospectral optimal control for wave energy conversion

“…The orthogonal set of basis functions employed for the pseudospectral optimal control approach is based on half-range Chebyshev Fourier functions [10], defined from trigonometric polynomials called half-range Chebyshev polynomials T k and U k respectively of the first and second kind. Definition 1.…”