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On the stability and accuracy of least s...
Cohen, Albert...

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On the stability and accuracy of least squares approximations by Cohen, Albert ( Author )
Lemonjar Open Access
Australian National University
13-07-2023
We consider the problem of reconstructing an unknown function $f$ on a domain $X$ from samples of $f$ at $n$ randomly chosen points with respect to a given measure $\rho_X$. Given a sequence of linear spaces $(V_m)_{m>0}$ with ${\rm dim}(V_m)=m\leq n$, we study the least squares approximations from the spaces $V_m$. It is well known that such approximations can be inaccurate when $m$ is too close to $n$, even when the samples are noiseless. Our main result provides a criterion on $m$ that describes the needed amount of regularization to ensure that the least squares method is stable and that its accuracy, measured in $L^2(X,\rho_X)$, is comparable to the best approximation error of $f$ by elements from $V_m$. We illustrate this criterion for various approximation schemes, such as trigonometric polynomials, with $\rho_X$ being the uniform measure, and algebraic polynomials, with $\rho_X$ being either the uniform or Chebyshev measure. For such examples we also prove similar stability results using deterministic samples that are equispaced with respect to these measures.
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Article
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29.34 KB
English
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MYR 0.01
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http://arxiv.org/abs/1111.4422
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