The Fourier extension method and discrete orthogonal polynomials on an arc of the circle

Geronimo, Jeffrey S.; Liechty, Karl

doi:10.1016/j.aim.2020.107064

Cited by 4 publications

(2 citation statements)

References 28 publications

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“…Proving a similar possibility theorem for this scheme is an open problem. Note that Fourier extension is equivalent to a polynomial approximation problem on an arc of the complex unit circle [23,40]. Fourier extension schemes have several advantages over the polynomial extension scheme studied herein.…”

Section: Discussionmentioning

confidence: 99%

Fast and Stable Approximation of Analytic Functions from Equispaced Samples via Polynomial Frames

Adcock

Shadrin

2022

Constr Approx

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We consider approximating analytic functions on the interval $$[-1,1]$$ [ - 1 , 1 ] from their values at a set of $$m+1$$ m + 1 equispaced nodes. A result of Platte, Trefethen & Kuijlaars states that fast and stable approximation from equispaced samples is generally impossible. In particular, any method that converges exponentially fast must also be exponentially ill-conditioned. We prove a positive counterpart to this ‘impossibility’ theorem. Our ‘possibility’ theorem shows that there is a well-conditioned method that provides exponential decay of the error down to a finite, but user-controlled tolerance $$\epsilon > 0$$ ϵ > 0 , which in practice can be chosen close to machine epsilon. The method is known as polynomial frame approximation or polynomial extensions. It uses algebraic polynomials of degree n on an extended interval $$[-\gamma ,\gamma ]$$ [ - γ , γ ] , $$\gamma > 1$$ γ > 1 , to construct an approximation on $$[-1,1]$$ [ - 1 , 1 ] via a SVD-regularized least-squares fit. A key step in the proof of our main theorem is a new result on the maximal behaviour of a polynomial of degree n on $$[-1,1]$$ [ - 1 , 1 ] that is simultaneously bounded by one at a set of $$m+1$$ m + 1 equispaced nodes in $$[-1,1]$$ [ - 1 , 1 ] and $$1/\epsilon $$ 1 / ϵ on the extended interval $$[-\gamma ,\gamma ]$$ [ - γ , γ ] . We show that linear oversampling, i.e. $$m = c n \log (1/\epsilon ) / \sqrt{\gamma ^2-1}$$ m = c n log ( 1 / ϵ ) / γ 2 - 1 , is sufficient for uniform boundedness of any such polynomial on $$[-1,1]$$ [ - 1 , 1 ] . This result aside, we also prove an extended impossibility theorem, which shows that such a possibility theorem (and consequently the method of polynomial frame approximation) is essentially optimal.

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Section: Discussionmentioning

confidence: 99%

Fast and Stable Approximation of Analytic Functions from Equispaced Samples via Polynomial Frames

Adcock

Shadrin

2022

Constr Approx

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show abstract

“…Proving a similar possibility theorem for this scheme is an open problem. Note that Fourier extension is equivalent to polynomial approximation problem on an arc of the complex unit circle [20,34].…”

Section: Discussionmentioning

confidence: 99%

Fast and stable approximation of analytic functions from equispaced samples via polynomial frames

Adcock¹,

Shadrin²

2021

Preprint

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We consider approximating analytic functions on the interval [−1, 1] from their values at a set of m+1 equispaced nodes. A result of Platte, Trefethen & Kuijlaars states that fast and stable approximation from equispaced samples is generally impossible. In particular, any method that converges exponentially fast must also be exponentially ill-conditioned. We prove a positive counterpart to this 'impossibility' theorem. Our 'possibility' theorem shows that there is a well-conditioned method that provides exponential decay of the error down to a finite, but user-controlled tolerance ǫ > 0, which in practice can be chosen close to machine epsilon. The method is known as polynomial frame approximation or polynomial extensions. It uses algebraic polynomials of degree n on an extended interval [−γ, γ], γ > 1, to construct an approximation on [−1, 1] via a SVD-regularized leastsquares fit. A key step in the proof of our possibility theorem is a new result on the maximal behaviour of a polynomial of degree n on [−1, 1] that is simultaneously bounded by one at a set of m + 1 equispaced nodes in [−1, 1] and 1/ǫ on the extended interval [−γ, γ]. We show that linear oversampling, i.e., m = cn log(1/ǫ)/ γ 2 − 1, is sufficient for uniform boundedness of any such polynomial on [−1, 1]. This result aside, we also prove an extended impossibility theorem, which shows that the possibility theorem (and consequently the method of polynomial frame approximation) is essentially optimal.

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On the adaptive spectral approximation of functions using redundant sets and frames

Coppé

Huybrechs

2021

IMA Journal of Numerical Analysis

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The approximation of smooth functions with a spectral basis typically leads to rapidly decaying coefficients, where the rate of decay depends on the smoothness of the function and vice versa. The optimal number of degrees of freedom in the approximation can be determined with relative ease by truncating the coefficients once a threshold is reached. Recent approximation schemes based on redundant sets and frames extend the applicability of spectral approximations to functions defined on irregular geometries and to certain nonsmooth functions. However, due to their inherent redundancy, the expansion coefficients in frame approximations do not necessarily decay even for very smooth functions. In this paper, we highlight this lack of equivalence between smoothness and coefficient decay, and we explore approaches to determine an optimal number of degrees of freedom for such redundant approximations.

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The Fourier extension method and discrete orthogonal polynomials on an arc of the circle

Cited by 4 publications

References 28 publications

Fast and Stable Approximation of Analytic Functions from Equispaced Samples via Polynomial Frames

Fast and Stable Approximation of Analytic Functions from Equispaced Samples via Polynomial Frames

Fast and stable approximation of analytic functions from equispaced samples via polynomial frames

On the adaptive spectral approximation of functions using redundant sets and frames

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