# A Mapped Polynomial Method for High-Accuracy Approximations on Arbitrary Grids

@article{Adcock2016AMP, title={A Mapped Polynomial Method for High-Accuracy Approximations on Arbitrary Grids}, author={Ben Adcock and Rodrigo B. Platte}, journal={SIAM J. Numer. Anal.}, year={2016}, volume={54}, pages={2256-2281} }

The focus of this paper is the approximation of analytic functions on compact intervals from their pointwise values on arbitrary grids. We introduce a new method for this problem based on mapped polynomial approximation. By careful selection of the mapping parameter, we ensure both high accuracy of the approximation and an asymptotically optimal scaling of the polynomial degree with the grid spacing. As we explain, efficient implementation of this method can be achieved using nonuniform fast…

## 16 Citations

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## References

SHOWING 1-10 OF 44 REFERENCES

A note on fast Fourier transforms for nonequispaced grids

- Mathematics, Computer ScienceAdv. Comput. Math.
- 1998

A general efficient method for the fast evaluation of trigonometric polynomials at nonequispaced nodes based on the approximation of the poynomials by special linear combinations of translates of suitable functions ϕ is proposed.

A Windowed Fourier Method for Approximation of Non-periodic Functions on Equispaced Nodes

- Mathematics
- 2015

A windowed Fourier method is proposed for approximation of non-periodic functions on equispaced nodes. Spectral convergence is obtained in most of the domain, except near the boundaries, where…

Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2007

The approach is based on a very simple concept: use of Fourier analysis to continue smooth portions of a piecewise smooth function into new functions which, defined on larger domains, are both smooth and periodic.

Barycentric rational interpolation with no poles and high rates of approximation

- Mathematics, Computer ScienceNumerische Mathematik
- 2007

A family of barycentric rational interpolants that have no real poles and arbitrarily high approximation orders on any real interval, regardless of the distribution of the points are proposed and studied.

Nonperiodic Trigonometric Polynomial Approximation

- Mathematics, Computer ScienceJ. Sci. Comput.
- 2014

The non-optimallity of polynomial approximation is addressed and it is suggested to switch from powers of x to powers of sin (px) where p is a parameter which depends on the dimension of the approximating subspace.

Approximation properties of a mapped Chebyshev method

- Mathematics
- 2000

We analyze a mapped Chebyshev technique to approximate derivatives recently developed by Kosloff and Tal-Ezer. The technique is based on a one-parameter family of mappings. Earlier numerical…

New Quadrature Formulas from Conformal Maps

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2008

New nonpolynomial quadrature methods are proposed that avoid the usual ellipse of convergence to an infinite strip or another approximately straight-sided domain by conformally mapping the usual circle of convergence.

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

- Mathematics, Computer ScienceAppl. Math. Comput.
- 2009

The Runge Phenomenon can be completely defeated by interpolation on a ''mock-Chebyshev'' grid: a subset of (N+1) points from an equispaced grid with O(N^2) points chosen to mimic the non-uniform N+1-point ChebysheV-Lobatto grid.

On the Numerical Stability of Fourier Extensions

- Mathematics, Computer ScienceFound. Comput. Math.
- 2014

This paper shows that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic, and demonstrates that they are particularly well suited for this problem.

Nonuniform fast Fourier transforms using min-max interpolation

- Mathematics, Computer ScienceIEEE Trans. Signal Process.
- 2003

This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm and indicates that the proposed method easily generalizes to multidimensional signals.