Pointwise and Uniform Convergence of Fourier Extensions

Webb, Marcus; Coppé, Vincent; Huybrechs, Daan

doi:10.1007/s00365-019-09486-x

“…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

¹

,

Shadrin

²

2022

Constr Approx

4

0

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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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“…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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“…These infinite series composed of sine and cosine functions are called Fourier series. Simultaneously, for nonperiodic functions with finite intervals, it is also possible to make them decomposable using Fourier series through period extensions 62 . Fourier series is widely used as an essential mathematical tool in signal processing and mathematical analysis 63 .…”

Section: Symmetric Projection Optimizermentioning

confidence: 99%

Symmetric projection optimizer: concise and efficient solving engineering problems using the fundamental wave of the Fourier series

Su,

Dong,

Liu

et al. 2024

Sci Rep

1

0

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The fitness function value is a kind of important information in the search process, which can be more targeted according to the guidance of the fitness function value. Most existing meta-heuristic algorithms only use the fitness function value as an indicator to compare the current variables as good or bad but do not use the fitness function value in the search process. To address this problem, the mathematical idea of the fitting is introduced into the meta-heuristic algorithm, and a symmetric projection optimizer (SPO) is proposed to solve numerical optimization and engineering problems more efficiently. The SPO algorithm mainly utilizes a new search mechanism, the symmetric projection search (SP) method. The SP method quickly completes the fitting of the projection plane, which is located through the symmetry of the two points and finds the minima in the projection plane according to the fitting result. Fitting by using the fitness function values allows the SP to find regions where extreme values may exist more quickly. Based on the SP method, exploration and exploitation strategies are constructed, respectively. The exploration strategy is used to find better regions, and the exploitation strategy is used to optimize the discovered regions continuously. The timing of the use of the two strategies is designed so that the SPO algorithm can converge faster while avoiding falling into local optima. The effectiveness of the SPO algorithm is extensively evaluated using seven test suites, including CEC2017, CEC2019, CEC2020, and CEC2022. It is also compared with two sets of 19 recent competitive algorithms. Statistical analyses are performed using five metrics such as the Wilcoxon test, the Friedman test, and variance. Finally, the practicality of the SPO algorithm is verified by four typical engineering problems and a real spacecraft trajectory optimization problem. The results show that the SPO algorithm can find superior results in 94.6% of the comparison tests and is a promising alternative for solving real-world problems.

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Pointwise and Uniform Convergence of Fourier Extensions

Cited by 10 publications

References 32 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

Symmetric projection optimizer: concise and efficient solving engineering problems using the fundamental wave of the Fourier series

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