For citation:
Agafonova N. Y. On the L1-convergence of Series in Multiplicative Systems. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 4, pp. 371-377. DOI: 10.18500/1816-9791-2016-16-4-371-377, EDN: XHPYFF
On the L1-convergence of Series in Multiplicative Systems
In the paper two analogs of Garrett – Stanojevic´ trigonometric results are established for multiplicative systems {χn} ∞n=0 of bounded type. First, the modified partial sums of a series P∞ k=0 akχk with coefficients of bounded variation converge in L 1 [0, 1) to its sum if and only if for all ε > 0 there exists δ > 0 such that R δ 0 ¯ ¯ ¯ ¯ P∞ k=n (ak − ak+1)Dk+1(x) ¯ ¯ ¯ ¯ dx < ε, n ∈ Z+, where Dk+1(x) = Pk i=0 χi(x). Secondly, if limn→∞ an ln(n + 1) = 0 and P∞ k=n |ak − ak+1| 6 Can, n ∈ Z+, then the series P∞ n=0 anχn(x) converges to its sum f(x) in L 1 [0, 1) if and only if f ∈ L 1 [0, 1).
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