Smoothing inversion of Fourier series using generalized cross-validation
Let \(f\) be a 1-periodic, absolutely continuous function with \(\int_I |f'(t)|^2 dt< \infty\), where \(I=\) \([-1/2, 1/2]\). Instead of the exact Fourier coefficients of \(f\), \(\widehat f_k:= \int_I f(t) e^{- 2\pi ikt} dt\), only a finite sequence of noisy values of \(\widehat f_k\), \(\widehat y_k= \widehat f_k+ \widehat\varepsilon_k\) \((k= 0, \dots, m)\) with \(m:= \lfloor n/2\rfloor\), \(n\in \mathbb{N}\) is known. The error terms \(\widehat\varepsilon_k\in \mathbb{C}\) are assumed to be normally distributed with \(E[\widehat\varepsilon_k]= 0\). Let \(n\in \mathbb{N}\) and \(m_1:= \lfloor(n- 1)/2\rfloor\). In the paper, the following reconstruction problem is considered: For given perturbed values \(\widehat y_k\) \((k= 0,\dots, m)\) and \(\widehat y_{- k}= \overline{\widehat y}_k\) \((k= 1,\dots, m_1)\), the unknown 1-periodic smooth function has to be reconstructed by a conveniently chosen trigonometric polynomial \(p\) of degree \(n\). Here the minimizing functional \(\widehat J_n\), \[ \widehat J_n(p):= \sum^m_{k= -m_1} |\widehat y_k- \widehat p_k|^2+ {\alpha\over (2\pi)^4} \int_I |p''(t)|^2 dt \] is used, where \(\alpha> 0\) is the smoothing parameter. The construction of an optimal parameter \(\alpha\) is based on \textit{G. Wahba's} method [Spline models for observational data, SIAM, Philadelphia, PA (1990; Zbl 0813.62001)] of generalized cross validation.
scientific article; zbMATH DE number 872711
scientific article