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On the uniqueness of Haar series convergent in the metrics of $L_p[0,\,1]$, $0<p<1$, and in measure

A. A. Talalyan


Abstract: It is established that if the partial sums $S_n(x)$ of a Haar series $\sum a_n\chi_n(x)$ converge to $f(x)\in L_p[0,1]$, $0<p<1$, 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 $L_p[0,1]$, $0<p<1$, over some subsequences of partial sums. The sharpness of these theorems is also proved.
Bibliography: 10 titles.

UDC: 517.5

MSC: 42C10

Received: 13.01.1984

 English version: Mathematics of the USSR-Sbornik, 1986, 54:1, 99–111

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