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

Abstract

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.

Oversampled Collocation Approximation Method of Functions via Jacobi Frames

“…We now can conclude that the constants C 1 and C 2 both are bounded with the following assumptions proved in [6], i.e., linear oversampling is sufficient for C(m,n,γ,ε) to be bounded.…”

“…Recently, Adcock, Huybrechs and Shadrin proposed an approach termed polynomial frame approximation [2,6]. For some fixed γ > 1, polynomial frame approximation uses orthogonal polynomials on an extended interval [−γ,γ] to construct an approximation to a function over [−1,1].…”

“…For some fixed γ > 1, polynomial frame approximation uses orthogonal polynomials on an extended interval [−γ,γ] to construct an approximation to a function over [−1,1]. This method leads to an ill-conditioned least-squares problem, while Adcock and Shadrin showed that this problem can be computed accurately via a regularized singular value decomposition (SVD) at a set of m+1 linear oversampling equispaced nodes on [−1,1], and proved that the regularized frame approximation operator is well-conditioned [6]. Further, the two authors also showed that the exponential decay of the polynomial frame approximation error down to a finite user-determined tolerance ε>0 is indeed possible for functions that are analytic in a sufficiently large region.…”

“…Further, the two authors also showed that the exponential decay of the polynomial frame approximation error down to a finite user-determined tolerance ε>0 is indeed possible for functions that are analytic in a sufficiently large region. In other words, Adcock and Shadrin asserted the possibility of fast and stable approximation of analytic functions from equispaced samples [6].…”

“…When studying the theoretical analysis of polynomial frame approximation, Adcock and Shadrin adopted the Legendre polynomials for convenience [6]. To explore the generality of the polynomial frame approximation, we next consider the use of Jacobi polynomials as well as their special cases, including Chebyshev, Legendre and Gegenbauer polynomials, which are widely used in many branches of scientific computing such as approximation theory, Gauss-type quadrature and spectral methods for differential and integral equations (see, e.g., [8,14,17,18]).…”