Geodesic
On the uniqueness of Haar series convergent in the metrics of $L_p[0,\,1]$, $0$, and in measure
Sbornik. Mathematics, Tome 54 (1986) no. 1, pp. 99-111 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is established that if the partial sums $S_n(x)$ of a Haar series $\sum a_n\chi_n(x)$ converge to $f(x)\in L_p[0,1]$, $0, at the rate $\int_0^1|S_n-f|^p\,dx=o\bigl(\frac1{n^{1-p}}\bigr)$, then $f(x)$ is $A$-integrable and $a_n=(A)\int_0^1f(x)\chi_n(x)\,dx$, for $n=1,2,\dots$. Analogous theorems are proved also for the case where Haar series converge in the metric of $L_p[0,1]$, $0, over some subsequences of partial sums. The sharpness of these theorems is also proved. Bibliography: 10 titles. 
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     url = {http://geodesic.mathdoc.fr/item/SM_1986_54_1_a4/}
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A. A. Talalyan. On the uniqueness of Haar series convergent in the metrics of $L_p[0,\,1]$, $0

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