Quasiuniversal in Lp [0;1] Orthogonal Series

Grigorian, M. G.
August 2007
Global Journal of Pure & Applied Mathematics;2007, Vol. 3 Issue 2, p139
Academic Journal
Let {φn(x)}n=1∞ be an orthonormal complete in L² [0, 1] system, such that ∥φn∥p0 ≤ const, n ≥ 1 for some p0 > 2. In the paper a series ࢣ ckφk (x) is constructed, which is which is quasiuniversal with respect to rearrangements in Lp [0, 1] for any p ≥ 1. This means that for any ϵ > 0 there exists a measurable set E ⊂ [0, 1] with Lebesgue measure ∣E∣ > 1 - ϵ, such that for any p ≥ 1 and for each function f(x) ϵ Lp(E) the series ∑ ckφk(x) can be rearranged, so that the new series converges to f(x) in Lp(E) metric.


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