# On the Numerical Stability of Fourier Extensions

@article{Adcock2014OnTN, title={On the Numerical Stability of Fourier Extensions}, author={B. Adcock and D. Huybrechs and J. Mart{\'i}n-Vaquero}, journal={Foundations of Computational Mathematics}, year={2014}, volume={14}, pages={635-687} }

An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically fast in the truncation parameter. Unfortunately, computing a Fourier extension requires solving an ill-conditioned linear system, and hence one might expect such rapid convergence to be destroyed when carrying out computations in finite precision. The purpose… CONTINUE READING

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#### References

##### Publications referenced by this paper.

SHOWING 1-10 OF 44 REFERENCES

On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function

- Mathematics
- 1992

216

On the resolution power of Fourier extensions for oscillatory functions

- Computer Science, Mathematics
- 2014

29

A Comparison of Numerical Algorithms for Fourier Extension of the First, Second, and Third Kinds

- Mathematics
- 2002

113

Exponentially-Convergent Strategies for Defeating the Runge Phenomenon for the Approximation of Non-PeriodicFunctions, PartI:Single-IntervalSchemes

- Mathematics
- 2009

53

Divergence (Runge Phenomenon) for least-squares polynomial approximation on an equispaced grid and Mock-Chebyshev subset interpolation

- Mathematics, Computer Science
- 2009

53