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Positive linear operators and the approximation of continuous functions on locally compact abelian groups

Part of: Abstract harmonic analysis Approximations and expansions

Published online by Cambridge University Press:  09 April 2009

Walter R. Bloom  and
Joseph F. Sussich
Walter R. Bloom
Affiliation:
School of Mathematical and Physical Sciences, Murdoch University, Murdoch, Western Australia 6153, Australia
Joseph F. Sussich
Affiliation:
School of Mathematical and Physical Sciences, Murdoch University, Murdoch, Western Australia 6153, Australia
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Abstract

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In 1953 P. P. Korovkin proved that if (Tn) is a sequence of positive linear operators defined on the space C of continuous real 2π-periodic functions and limn→rTnf = f uniformly for f = 1, cos and sin. then limn→rTnf = f uniformly for all f∈C. We extend this result to spaces of continuous functions defined on a locally compact abelian group G, with the test family {1, cos, sin} replaced by a set of generators of the character group of G.

MSC classification

Secondary: 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 2 , December 1980 , pp. 180 - 186
Copyright
Copyright © Australian Mathematical Society 1980

References

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