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10

Part of 1971 Miklós Schweitzer

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Miklos Schweitzer 1971_10

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10/29/2008
Let {ϕn(x)} \{\phi_n(x) \} be a sequence of functions belonging to L2(0,1) L^2(0,1) and having norm less that 1 1 such that for any subsequence {ϕnk(x)} \{\phi_{n_k}(x) \} the measure of the set {x∈(0,1):  ∣1N∑k=1Nϕnk(x)∣≥y } \{x \in (0,1) : \;|\frac{1}{\sqrt{N}} \sum _{k=1}^N \phi_{n_k}(x)| \geq y\ \} tends to 0 0 as y y and N N tend to infinity. Prove that ϕn \phi_n tends to 0 0 weakly in the function space L2(0,1). L^2(0,1). F. Moricz
functionreal analysisreal analysis unsolved