Partition Theorems Uniform Convergence Repeated Averages of Real-Valued Functions Convergence Index Oscillation Index
Issue Date:
2000
Publisher:
Institute of Mathematics and Informatics
Citation:
Serdica Mathematical Journal, Vol. 26, No 2, (2000), 79p-104p
Abstract:
If ξ is a countable ordinal and (fk) a sequence of real-valued
functions we define the repeated averages of order ξ of (fk). By using a
partition theorem of Nash-Williams for families of finite subsets of positive
integers it is proved that if ξ is a countable ordinal then every sequence
(fk) of real-valued functions has a subsequence (f'k) such that either every
sequence of repeated averages of order ξ of (f'k) converges uniformly to zero
or no sequence of repeated averages of order ξ of (f'k) converges uniformly to
zero. By the aid of this result we obtain some results stronger than Mazur’s
theorem.