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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.

##### MSC:
 65T40 Numerical methods for trigonometric approximation and interpolation
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