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On a norm inequality with respect to Vilenkin-like systems. (English) Zbl 0973.42020

G. Gát [“On \(C,1)\) summability of integrable functions on compact totally disconnected spaces”, Stud. Math. (to appear)] has introduced a complete orthonormal system which generalizes the Vilenkin systems, the additive characters of an \(m\)-adic field, and UDMD systems. Here, the author shows that in the bounded case, the partial sums of the Fourier series \(S_nf\) (with respect to the Gát system) of functions which belong to an atomic Hardy space satisfy \[ \lim_{N\to\infty} \frac {1}{\log N} \sum_{n=1}^N \frac {\|S_n f\|_1}{n} = \|f\|_H . \]

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
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