On the approximate form of Kluvánek's theorem

An abstract form of the classical approximate sampling theorem is proved for functions on a locally compact abelian group that are continuous, square-integrable and have integrable Fourier transforms. An additional hypothesis that the samples of the function are square-summable is needed to ensure the convergence of the sampling series. As well as establishing the representation of the function as a sampling series plus a remainder term, an asymptotic formula is obtained under mild additional restrictions on the group. In conclusion a converse to Kluvánek's theorem is established.