# Convergence rate of cardinal series (Whittaker-Shannon interpolant)

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.)

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