Abstract
Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the L^2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the L^2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.
Highlights
The Fourier series of a periodic function converges spectrally fast with respect to the number of terms in the series, that is, with an algebraic order that increases with the number of available derivatives and exponentially fast for analytic functions
Fourier extensions have been shown to be an effective means for the approximation of nonperiodic functions while avoiding the Gibbs phenomenon [1,4,7,17,23,24,26]
1 2 come from bounds on the Lebesgue function associated with the Fourier extension derived in Sect. 4, and the factor of N −k−α comes from a Jackson-type theorem proved for Fourier extensions derived in
Summary
The Fourier series of a periodic function converges spectrally fast with respect to the number of terms in the series, that is, with an algebraic order that increases with the number of available derivatives and exponentially fast for analytic functions. We prove a local pointwise convergence result, which states that if f ∈ L2(−1, 1), but f is uniformly Dini–Lipschitz in a subinterval [a, b], the Fourier extension converges uniformly in compact subintervals of (a, b) (see Theorem 3.5) This is done by generalizing a localization theorem of Freud on convergence of orthogonal polynomial expansions in [−1, 1] The appendix contains a derivation of asymptotics of Legendre polynomials on a circular arc, on the arc itself, from the Riemann–Hilbert analysis of Krasovsky [10,21,22]
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