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From high oscillation to rapid approximation. I: Modified Fourier expansions. (English) Zbl 1221.65348
Let \(f\) be an analytic function in an open set containing the interval \([-1,1]\) and periodic with period 2. The following modified Fourier expansion is investigated: \[ \tfrac 12\widehat f_0^C+ \sum^\infty_{n=1}\left[\widehat f_n^C \cos\pi nx+\widehat f_n^S\sin\pi \left(n-\tfrac 12\right)x\right], \] where \[ \widehat f_n^C:=\int^1_{-1} f(x)\cos\pi n\,dx,\quad\widehat f_n^S:=\int^1_{-1}f(x)\sin\pi\left(n-\tfrac 12\right)x\,dx. \] The authors prove that the system \[ \left\{\cos \pi nx:n\in\mathbb{Z}_+\quad\text{and}\qquad\sin\pi\left(n-\tfrac 12\right)x:n \in\mathbb{N}\right\} \] is orthogonal and dense in \(L_2[-1,1]\). Suitably amended, the classical Fejér and de la Vallée Poussin theorems remain valid in this setting. Even the modified Fourier expansion has a number of advantages in the approximation of analytic, nonperiodic functions. In particular, expansion coefficients decay like \(O(n^{-2})\), rather than like \(O(n^{-1})\) which is the case in the classical setting. Furthermore, instead of approximating expansion coefficients by discrete Fourier transform, the authors expand them into asymptotic series and present algorithms which require \(O(m)\) operations in the computation of \(\widehat f^C_n\) and \(\widehat f_n^C\) to suitably high precision for \(n\leq m\). They also employ techniques for the computation of highly oscillatory integrals based on Filon type quadrature.

65T40 Numerical methods for trigonometric approximation and interpolation
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