On the approximate form of Kluvánek's theorem

• Published in: Journal of approximation theory Vol. 160; no. 1; pp. 281 - 303
• Main Authors: Beaty, M.G., Dodson, M.M., Eveson, S.P., Higgins, J.R.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-09-2009
• Subjects: Abstract harmonic analysis; An approximate Kluvánek theorem; Sampling theory; Signal theory; 94A05; An approximate Kluvánek theorem; Abstract harmonic analysis; Sampling theory; 43A99; Signal theory; 94A20
• ISSN: 0021-9045; 1096-0430
• DOI: 10.1016/j.jat.2009.02.013

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.