Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon

作者：Adcock Ben^*; Hansen Anders C

来源：Applied and Computational Harmonic Analysis, 2012, 32(3): 357-388.

DOI：10.1016/j.acha.2011.07.004

• Sampling theory
• Hilbert spaces
• Fourier series
• Gibbs phenomenon
• Stability

We introduce a simple and efficient method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we establish, provided the dimension of the reconstruction space is chosen suitably in relation to the number of samples, this procedure can be implemented in a completely numerically stable manner. Moreover, the accuracy of the resulting approximation is determined solely by the choice of reconstruction basis, meaning that reconstruction vectors can be readily tailored to the particular problem at hand. %26lt;br%26gt;An important example of this approach is the accurate recovery of a piecewise analytic function from its first few Fourier coefficients. Whilst the standard Fourier projection suffers from the Gibbs phenomenon, by reconstructing in a piecewise polynomial basis we obtain an approximation with root-exponential accuracy in terms of the number of Fourier samples and exponential accuracy in terms of the degree of the reconstruction. Numerical examples illustrate the advantage of this approach over other existing methods.

• 出版日期2012-5