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