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

  • Walter R. Bloom (a1) and Joseph F. Sussich (a1)

Abstract

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 limnrTnf = f uniformly for all fC. 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.

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Copyright

References

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Freud, G. (1964), ‘Über positive lineare Approximationsfolgen von stetigen reelen Funktionen auf kompakten Mengen’, On approximation theory, proceeding of the conference held at the Mathematical Research Institute at Oberwolfach, 4–10 August 1963 (editors Butzer, P. L. and Korevaar, J.), pp. 233238 (Birkhäuser Verlag, Basel).
Hewitt, Edwin and Ross, Kenneth A. (1963), Abstract harmonic analysis, Volume I (Die Grundlehren der mathematichen Wissenchaften, Band 115, Springer-Verlag, Göttingen, Heidelberg).
Hewitt, Edwin and Ross, Kenneth A. (1970), Abstract harmonic analysis, Volume II (Die Grundlehren der mathematischen Wissenschaften. Band 152, Springer-Verlag, Berlin, Heidelberg, New York).
Heyer, Hebert (1977), Probability measures on locally compact groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 94, Springer-Verlag, Berlin, Heideberg, New York).
Kendall, David and Lamperti, John (1970), ‘A remark on topologies for characteristic functions’, Proc. Cambridge Philos. Soc. 68, 703705.
Korovkin, P. P. (1960), Linear operators and approximation theory (translated from the Russian edition (1959), Hindustan Publishing Corporation (India), Delhi).
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MSC classification

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

  • Walter R. Bloom (a1) and Joseph F. Sussich (a1)

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