Uniform Bounds for Sampling Expansions Original Research Article

Letf∈B2σ, i.e.,f∈L2(R) and its Fourier transformF(s)=∫R f(t) e−2πist dtvanishes outside of [−σ, σ], then the Shannon sampling theorem says thatfcan be reconstructed by its infinitely many sampling points at {k/(2σ)},k∈Z, i.e.,f(t)=∑k=−∞∞ f k2σ sin π(2σt−k)π(2σt−k), ∀t∈R.But, in practice, only finitely many samples are available, so one would like to study the truncation errorTN(t)=f(t)−∑k=−NN f k2σ sinc(2σt−k), ∀f∈B2σ.The error bounds commonly seen in literature are not uniform. In this paper, the author gives uniform bounds for the truncation error forf∈B2σ, when its Fourier transform satisfies some smooth conditions.