ON QUASI-UNIVERSAL WALSH SERIESINLp[0,1],p∈[1,2]

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Submitted: 2025-02-11; Published: 2025-02-11

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Abstract

Let the sequence{ak}∞k=1,ak↘0 with{ak}∞k=1/∈l2,and Walsh system{Wk(x)}∞k=0be given.Then for anyε>0 there exists a measurable setE⊂[0,1]with measure|E|>1−εand numbersδk=±1,0 such that foranyp∈[1,2]and each functionf(x)∈Lp(E)there exists a rearrangementk→σ(k)such that the series∞∑k=1δσ(k)aσ(k)Wσ(k)(x)converges tof(x)in thenorm ofLp(E).