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We introduce a method to reconstruct an element of a Hilbert space in termsof an arbitrary finite collection of linearly independent reconstructionvectors, given a finite number of its samples with respect to any Riesz basis.As we establish, provided the dimension of the reconstruction space is chosensuitably in relation to the number of samples, this procedure can benumerically implemented in a stable manner. Moreover, the accuracy of theresulting approximation is completely determined by the choice ofreconstruction basis, meaning that the reconstruction vectors can be tailoredto the particular problem at hand.  An important example of this approach is the accurate recovery of a piecewiseanalytic function from its first few Fourier coefficients. Whilst the standardFourier projection suffers from the Gibbs phenomenon, by reconstructing in apiecewise polynomial basis, we obtain an approximation with root exponentialaccuracy in terms of the number of Fourier samples and exponential accuracy interms of the degree of the reconstruction function. Numerical examplesillustrate the advantage of this approach over other existing methods.

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