On the approximate form of Kluvánek's theorem

TY - JOUR

T1 - On the approximate form of Kluvánek's theorem

AU - Beaty, Michael

AU - Dodson, Maurice

AU - Eveson, Simon

AU - Higgins, John

PY - 2009/9

Y1 - 2009/9

N2 - 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.

AB - 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.

KW - Analysis,

KW - Pure Mathematics

UR - http://www.scopus.com/inward/record.url?scp=70349843886&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2009.02.013

DO - 10.1016/j.jat.2009.02.013

M3 - Article

SN - 0021-9045

VL - 160

SP - 281

EP - 303

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

IS - 1-2

ER -