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Given $f \in C^{k}_{0}[a, b]\cap L^{2}(\mathbb{R})$, what can we say about the convergence rate of the cardinal series $$ s(t) = \sum_{j=0}^{n-1} f(a+jh) \mathrm{sinc}\left(\pi\left(\frac{t-a}{h} -j \right)\right), \quad hn = b -a $$ to $f$ as $h\to 0$?

(Moved in desperation from math.se.)

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  • $\begingroup$ Out of curiosity - is the term "cardinal series" well known for this expression? Where does it come from? $\endgroup$ – Amir Sagiv Jul 6 at 23:37
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    $\begingroup$ @AmirSagiv: This is the terminology from I.J. Schoenberg's 1973 book "Cardinal Spline Interpolation". It is also known as the Whittaker-Shannon interpolant. $\endgroup$ – user14717 Jul 7 at 1:22

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