Optimal sampling rates for approximating analytic functions from pointwise samples

Dr Alexei Shadrin, University of Cambridge

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Nov 23, 2017
from 02:00 PM to 03:00 PM


MAB 119

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We consider the problem of approximating an analytic function on a compact interval from its values at M + 1 distinct points. When the points are equispaced, a recent result (the so-called impossibility theorem) has shown that the best possible convergence rate of a stable method is root-exponential in M, and that any method with faster exponential convergence must also be exponentially ill-conditioned at a certain rate.

Here, we present an extension of the impossibility theorem valid for general nonequispaced points, and apply it to the case of points that are equidistributed with respect to modified Jacobi measures. This leads to a necessary sampling rate for stable approximation from such points. We prove that this rate is also sufficient, and therefore exactly quantify (up to constants) the precise sampling rate for approximating analytic functions from such node distributions with stable numerical methods.

In particular, we theoretically confirm the well-known heuristic that stable least-squares approximation using polynomials of degree N < M is possible only once M is sufficiently large for there to be a subset of N of the nodes that mimic the behaviour of the Nth Chebyshev nodes.

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