Pseudospectral Fourier reconstruction with the modified inverse polynomial reconstruction method. (English) Zbl 1184.65123
The inverse polynomial reconstruction method (IPRM) was introduced by J.-H. Jung and B. D. Shizgal [J. Comput. Appl. Math. 172, No. 1, 131–151 (2004; Zbl 1053.65102)] in order to remedy the Gibbs phenomenon. Let \(f\) be a piecewise polynomial function defined on \([-1,\,1]\) and let \(m,\,n\in \mathbb N\) with \(m\geq n\) be given. In this paper, a modified IPRM is proposed that approximates \(f\) by a polynomial \(p(x) = \sum_{l=0}^{n-1} a_l\,P_l(x)\) with \(x\in [-1,\,1]\) such that
\[ \sum_{k=-\lfloor (m-1)/2\rfloor}^{\lfloor m/2\rfloor} |{\hat f}(k) - {\hat p}(k)|^2 \]
is minimal, where \(P_l\) are the normalized Legendre polynomials and \({\hat f}(k)\) are the Fourier coefficients of \(f\). The modified IPRM finds a truncated Legendre series of the given function \(f\) from its truncated Fourier series by solving a rectangular least squares problem. If \(m\geq n^2\), the authors show that the condition number of this least squares problem is small and that the convergence rate for an analytic function \(f\) is root exponential on \([-1,\,1]\). Numerical stability and accuracy of the proposed IPRM algorithm are validated experimentally.
\[ \sum_{k=-\lfloor (m-1)/2\rfloor}^{\lfloor m/2\rfloor} |{\hat f}(k) - {\hat p}(k)|^2 \]
is minimal, where \(P_l\) are the normalized Legendre polynomials and \({\hat f}(k)\) are the Fourier coefficients of \(f\). The modified IPRM finds a truncated Legendre series of the given function \(f\) from its truncated Fourier series by solving a rectangular least squares problem. If \(m\geq n^2\), the authors show that the condition number of this least squares problem is small and that the convergence rate for an analytic function \(f\) is root exponential on \([-1,\,1]\). Numerical stability and accuracy of the proposed IPRM algorithm are validated experimentally.
Reviewer: Manfred Tasche (Rostock)
MSC:
| 65T40 | Numerical methods for trigonometric approximation and interpolation |
| 65F20 | Numerical solutions to overdetermined systems, pseudoinverses |
| 65D10 | Numerical smoothing, curve fitting |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
Keywords:
inverse polynomial reconstruction method; Gibbs phenomenon; pseudospectral Fourier reconstruction; Legendre polynomial; condition number; convergence rate; rectangular least squares problem; numerical examples; numerical stabilityCitations:
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