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- W4300406710 abstract "The trigonometric interpolants to a periodic function $f$ in equispaced points converge if $f$ is Dini-continuous, and the associated quadrature formula, the trapezoidal rule, converges if $f$ is continuous. What if the points are perturbed? With equispaced grid spacing $h$, let each point be perturbed by an arbitrary amount $le alpha h$, where $alphain [kern .5pt 0,1/2)$ is a fixed constant. The Kadec 1/4 theorem of sampling theory suggests there may be be trouble for $alphage 1/4$. We show that convergence of both the interpolants and the quadrature estimates is guaranteed for all $alpha<1/2$ if $f$ is twice continuously differentiable, with the convergence rate depending on the smoothness of $f$. More precisely it is enough for $f$ to have $4alpha$ derivatives in a certain sense, and we conjecture that $2alpha$ derivatives is enough. Connections with the Fej'er--Kalm'ar theorem are discussed." @default.
- W4300406710 created "2022-10-03" @default.
- W4300406710 creator A5049730032 @default.
- W4300406710 creator A5065363123 @default.
- W4300406710 date "2016-12-12" @default.
- W4300406710 modified "2023-09-27" @default.
- W4300406710 title "Trigonometric Interpolation and Quadrature in Perturbed Points" @default.
- W4300406710 doi "https://doi.org/10.48550/arxiv.1612.04018" @default.
- W4300406710 hasPublicationYear "2016" @default.
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